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vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3011em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"3\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},\"\\u22C5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8444em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"b\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3011em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"3\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"}))))))))))),mdx(\"p\",null,\"where \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"a\"),mdx(\"mn\",{parentName:\"msub\"},\"1\")),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"a\"),mdx(\"mn\",{parentName:\"msub\"},\"2\")),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"a\"),mdx(\"mn\",{parentName:\"msub\"},\"3\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2208\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"double-struck\"},\"R\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"a_1, a_2, a_3 \\\\in \\\\mathbb{R}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.7335em\",\"verticalAlign\":\"-0.1944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3011em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"1\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},\",\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.1667em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3011em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"2\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},\",\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.1667em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t 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mtight\"},\"3\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},\"\\u2208\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.6889em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbb\"},\"R\"))))),\". In other words, \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"a\"),mdx(\"mn\",{parentName:\"msub\"},\"1\")),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"a\"),mdx(\"mn\",{parentName:\"msub\"},\"2\")),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"a\"),mdx(\"mn\",{parentName:\"msub\"},\"3\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"a_1, a_2, a_3\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.1944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3011em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"1\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},\",\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.1667em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3011em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"2\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},\",\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.1667em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3011em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"3\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.15em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\" are scalars. These scalars are \",mdx(\"strong\",{parentName:\"p\"},\"unique\"),\".\"),mdx(\"p\",null,\"If the vectors in the basis are mutually perpendicular, then they are \",mdx(\"strong\",{parentName:\"p\"},\"orthogonal\"),\".\"),mdx(\"p\",null,\"An \",mdx(\"strong\",{parentName:\"p\"},\"orthonormal basis (ONB)\"),\" is a basis which is orthogonal and each vector has length 1.\"),mdx(\"p\",null,\"An example of an ONB is the \",mdx(\"strong\",{parentName:\"p\"},\"standard basis\")),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"e\"),mdx(\"mn\",{parentName:\"msub\"},\"1\")),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\"),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\"),mdx(\"mspace\",{parentName:\"mrow\",\"width\":\"1em\"}),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"e\"),mdx(\"mn\",{parentName:\"msub\"},\"2\")),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\"),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\"),mdx(\"mspace\",{parentName:\"mrow\",\"width\":\"1em\"}),mdx(\"msub\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msub\"},\"e\"),mdx(\"mn\",{parentName:\"msub\"},\"3\")),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"mn\",{parentName:\"mrow\"},\"0\"),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\"),mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\")\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"e_1 = (1, 0, 0) \\\\quad e_2 = (0, 1, 0) \\\\quad e_3 = (0, 0, 1)\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.5806em\",\"verticalAlign\":\"-0.15em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"e\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.3011em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.55em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord 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mtight\",\"style\":{\"marginRight\":\"0.02778em\"}},\"D\")))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.497em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\", that is it goes from point D to point C in a space \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"a\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.4306em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"))))),\", and we want to \",mdx(\"strong\",{parentName:\"p\"},\"rotate\"),\" it from space \",mdx(\"span\",{parentName:\"p\",\"className\":\"math 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mathnormal\",\"style\":{\"marginRight\":\"0.08125em\"}},\"H\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.453em\",\"marginLeft\":\"-0.0813em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\"))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.247em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\" to some point \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msup\"},\"\\u03C1\"),mdx(\"mi\",{parentName:\"msup\"},\"a\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\rho^a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8588em\",\"verticalAlign\":\"-0.1944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.6644em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\")))))))))))),\", then \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msup\"},\"\\u03C1\"),mdx(\"mi\",{parentName:\"msup\"},\"a\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\rho^a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.8588em\",\"verticalAlign\":\"-0.1944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.6644em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\")))))))))))),\" will first be rotated by \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\"},\"R\"),mdx(\"mi\",{parentName:\"msubsup\"},\"a\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"R_a^b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.0961em\",\"verticalAlign\":\"-0.247em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.00773em\"}},\"R\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.453em\",\"marginLeft\":\"-0.0077em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\"))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.247em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\", giving \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msup\"},\"\\u03C1\"),mdx(\"mi\",{parentName:\"msup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\rho^b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.0435em\",\"verticalAlign\":\"-0.1944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))))))))))),\". \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msup\"},\"\\u03C1\"),mdx(\"mi\",{parentName:\"msup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\rho^b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.0435em\",\"verticalAlign\":\"-0.1944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))))))))))),\" will then be translated by some vector according to the transformation matrix. Since \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msup\"},\"\\u03C1\"),mdx(\"mi\",{parentName:\"msup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\rho^b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.0435em\",\"verticalAlign\":\"-0.1944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C1\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))))))))))),\" is now rotated in the \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"b\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.6944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"b\"))))),\" coordinate system, the translation vector must also be in this \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"b\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.6944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"b\"))))),\" coordinate system. The vector is from B to A because a point (0, 0, 0) in space \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"a\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.4306em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"))))),\" should be moved to the origin of \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"a\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.4306em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"))))),\" in \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"b\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.6944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"b\"))))),\" space after transformation. Thus, the translation vector is \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"r\"),mdx(\"mrow\",{parentName:\"msubsup\"},mdx(\"mi\",{parentName:\"mrow\"},\"A\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\"/\"),mdx(\"mi\",{parentName:\"mrow\"},\"B\")),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{r}_{A/B}^b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.3461em\",\"verticalAlign\":\"-0.497em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"r\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.378em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"A\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"/\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.05017em\"}},\"B\")))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.497em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\".\"),mdx(\"p\",null,\"This is illustrated in the following diagram.\"),mdx(\"img\",{\"src\":\"/72da7ded44652a76a7d82c57b1230d92/maths-translation.svg\",\"alt\":\"Two three dimensional coordinate spaces, one labelled as space 'a' and the other as space 'b'. There is a vector from space 'a' to some point 'c', and is labelled 'rCAb'. There is another vector from 'b' to 'a' and is labelled 'rABb'. There is a final vector from 'b' to 'c' which is labelled 'rCBb = rABb + rCAb'\"}),mdx(\"p\",null,\"This assumes the original vector, \",mdx(\"inlineCode\",{parentName:\"p\"},\"rCAa\"),\" \",mdx(\"strong\",{parentName:\"p\"},\"has already been rotated\"),\" into space \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"b\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.6944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"b\"))))),\", i.e. we have \",mdx(\"inlineCode\",{parentName:\"p\"},\"rCAb\"),\".\"),mdx(\"p\",null,\"Finding the inverse of \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\"},\"H\"),mdx(\"mi\",{parentName:\"msubsup\"},\"a\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"H^b_a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.0961em\",\"verticalAlign\":\"-0.247em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.08125em\"}},\"H\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.453em\",\"marginLeft\":\"-0.0813em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\"))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.247em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\" is made simpler by the properties of the matrix.\"),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\"},\"H\"),mdx(\"mi\",{parentName:\"msubsup\"},\"a\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\")),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"msup\",\"stretchy\":\"false\"},\")\"),mdx(\"mrow\",{parentName:\"msup\"},mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"mn\",{parentName:\"mrow\"},\"1\"))),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\"},\"H\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"),mdx(\"mi\",{parentName:\"msubsup\"},\"a\")),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"mrow\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},\"[\"),mdx(\"mtable\",{parentName:\"mrow\",\"rowspacing\":\"0.16em\",\"columnalign\":\"center center\",\"columnspacing\":\"1em\"},mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"R\"),mdx(\"mi\",{parentName:\"msubsup\"},\"a\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\")),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"msup\",\"stretchy\":\"false\"},\")\"),mdx(\"mi\",{parentName:\"msup\"},\"T\"))))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mrow\",{parentName:\"mstyle\"},mdx(\"mo\",{parentName:\"mrow\",\"stretchy\":\"false\"},\"(\"),mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"R\"),mdx(\"mi\",{parentName:\"msubsup\"},\"a\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\")),mdx(\"msup\",{parentName:\"mrow\"},mdx(\"mo\",{parentName:\"msup\",\"stretchy\":\"false\"},\")\"),mdx(\"mi\",{parentName:\"msup\"},\"T\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"mo\",{parentName:\"mrow\"},\"\\u2212\"),mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"r\"),mdx(\"mrow\",{parentName:\"msubsup\"},mdx(\"mi\",{parentName:\"mrow\"},\"A\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\"/\"),mdx(\"mi\",{parentName:\"mrow\"},\"B\")),mdx(\"mi\",{parentName:\"msubsup\"},\"b\")))))),mdx(\"mtr\",{parentName:\"mtable\"},mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\",\"mathvariant\":\"bold\"},\"0\"))),mdx(\"mtd\",{parentName:\"mtr\"},mdx(\"mstyle\",{parentName:\"mtd\",\"scriptlevel\":\"0\",\"displaystyle\":\"false\"},mdx(\"mn\",{parentName:\"mstyle\"},\"1\"))))),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},\"]\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"(H^b_a)^{-1} = H^a_b = \\\\begin{bmatrix} (\\\\mathbf{R}_a^b)^T & (\\\\mathbf{R}_a^b)^T \\\\cdot -\\\\mathbf{r}_{A/B}^b \\\\\\\\ \\\\mathbf{0} & 1 \\\\end{bmatrix}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.1491em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mopen\"},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.08125em\"}},\"H\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8991em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.453em\",\"marginLeft\":\"-0.0813em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\"))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.247em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mclose\"},\")\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8641em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 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if the \",mdx(\"strong\",{parentName:\"p\"},\"last two\\nletters\"),\" of the vector match. That is, the position they are going from should match the\\norigin of the space they are measured in. This means that \",mdx(\"inlineCode\",{parentName:\"p\"},\"Hab * rCBb\"),\" is\\nvalid, but \",mdx(\"inlineCode\",{parentName:\"p\"},\"Hab * rCDb\"),\" is invalid. Note that the first vector is rC\",mdx(\"strong\",{parentName:\"p\"},\"B\"),\"b and the second is rC\",mdx(\"strong\",{parentName:\"p\"},\"D\"),\"b.\")),mdx(\"p\",null,\"If you would like the vector \",mdx(\"strong\",{parentName:\"p\"},\"only rotated\"),\", extract the \",mdx(\"strong\",{parentName:\"p\"},\"rotational component\"),\" of the transform and apply it to the vector. The \",mdx(\"strong\",{parentName:\"p\"},\"spaces\"),\" that the vector and rotation matrix are in should \",mdx(\"strong\",{parentName:\"p\"},\"match\"),\".\"),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-cpp\"},\"Eigen::Vector3d rCDb = Hba.rotation() * rCDa;\\n\")),mdx(\"p\",null,\"Homogeneous transformation matrices can be \",mdx(\"strong\",{parentName:\"p\"},\"multiplied\"),\" together to get a \",mdx(\"strong\",{parentName:\"p\"},\"new homogeneous transformation\"),\" matrix.\"),mdx(\"div\",{\"className\":\"math math-display\"},mdx(\"span\",{parentName:\"div\",\"className\":\"katex-display\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\",\"display\":\"block\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"H\"),mdx(\"mi\",{parentName:\"msubsup\"},\"a\"),mdx(\"mi\",{parentName:\"msubsup\"},\"c\")),mdx(\"mo\",{parentName:\"mrow\"},\"=\"),mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"H\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"),mdx(\"mi\",{parentName:\"msubsup\"},\"c\")),mdx(\"mo\",{parentName:\"mrow\"},\"\\u22C5\"),mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"H\"),mdx(\"mi\",{parentName:\"msubsup\"},\"a\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{H}_a^c = \\\\mathbf{H}_b^c \\\\cdot \\\\mathbf{H}^b_a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.9614em\",\"verticalAlign\":\"-0.247em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"H\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.7144em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.453em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\"))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"c\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.247em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2778em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mrel\"},\"=\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2778em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.9614em\",\"verticalAlign\":\"-0.247em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"H\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.7144em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.453em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\"))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"c\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.247em\"}},mdx(\"span\",{parentName:\"span\"})))))),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mbin\"},\"\\u22C5\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.2222em\"}})),mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.1461em\",\"verticalAlign\":\"-0.247em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"H\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8991em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.453em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"a\"))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.113em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.247em\"}},mdx(\"span\",{parentName:\"span\"}))))))))))),mdx(\"p\",null,\"This is written in the codebase as\"),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-cpp\"},\"Eigen::Isometry3d Hca = Hcb * Hba;\\n\")),mdx(\"h3\",{\"id\":\"unit-vectors\",\"style\":{\"position\":\"relative\"}},mdx(\"a\",{parentName:\"h3\",\"href\":\"#unit-vectors\",\"aria-label\":\"unit vectors permalink\",\"className\":\"anchor before\"},mdx(\"svg\",{parentName:\"a\",\"aria-hidden\":\"true\",\"focusable\":\"false\",\"height\":\"16\",\"version\":\"1.1\",\"viewBox\":\"0 0 16 16\",\"width\":\"16\"},mdx(\"path\",{parentName:\"svg\",\"fillRule\":\"evenodd\",\"d\":\"M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z\"}))),\"Unit Vectors\"),mdx(\"p\",null,\"A \",mdx(\"strong\",{parentName:\"p\"},\"unit vector\"),\" is a vector with a \",mdx(\"strong\",{parentName:\"p\"},\"length\"),\" of \",mdx(\"strong\",{parentName:\"p\"},\"1\"),\". They are used to represent \",mdx(\"strong\",{parentName:\"p\"},\"directions\"),\" in space.\"),mdx(\"p\",null,\"They are written by describing their origin, the direction they are pointing and the space they are in.\"),mdx(\"p\",null,mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"u\"),mdx(\"mrow\",{parentName:\"msubsup\"},mdx(\"mi\",{parentName:\"mrow\"},\"A\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\"/\"),mdx(\"mi\",{parentName:\"mrow\"},\"B\")),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{u}^b_{A/B}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.3461em\",\"verticalAlign\":\"-0.497em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"u\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.378em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"A\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"/\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.05017em\"}},\"B\")))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.497em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\" is a unit vector at B pointing towards A measured in space b.\"),mdx(\"p\",null,\"The \",mdx(\"strong\",{parentName:\"p\"},\"direction\"),\" of a vector can be found by \",mdx(\"strong\",{parentName:\"p\"},\"normalising\"),\" it. This is done by dividing the vector by its \",mdx(\"strong\",{parentName:\"p\"},\"length\"),\".\"),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-cpp\"},\"Eigen::Vector3d uABb = rABb.normalized();\\n\")),mdx(\"p\",null,\"Unit vectors can be rotated by rotation matrices but not translated. They should not be multiplied by a homogeneous transformation matrix.\"),mdx(\"h3\",{\"id\":\"velocity-and-acceleration\",\"style\":{\"position\":\"relative\"}},mdx(\"a\",{parentName:\"h3\",\"href\":\"#velocity-and-acceleration\",\"aria-label\":\"velocity and acceleration permalink\",\"className\":\"anchor before\"},mdx(\"svg\",{parentName:\"a\",\"aria-hidden\":\"true\",\"focusable\":\"false\",\"height\":\"16\",\"version\":\"1.1\",\"viewBox\":\"0 0 16 16\",\"width\":\"16\"},mdx(\"path\",{parentName:\"svg\",\"fillRule\":\"evenodd\",\"d\":\"M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z\"}))),\"Velocity and Acceleration\"),mdx(\"p\",null,\"The velocity and acceleration of some object is represented relative to some space.\"),mdx(\"p\",null,\"If we have the \",mdx(\"strong\",{parentName:\"p\"},\"velocity\"),\" of object A moving \",mdx(\"strong\",{parentName:\"p\"},\"relative\"),\" to space b \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"v\"),mdx(\"mi\",{parentName:\"msubsup\"},\"A\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{v}^b_{A}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.1244em\",\"verticalAlign\":\"-0.2753em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\",\"style\":{\"marginRight\":\"0.01597em\"}},\"v\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4247em\",\"marginLeft\":\"-0.016em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"A\")))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.2753em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\", then we write this in the codebase as \",mdx(\"inlineCode\",{parentName:\"p\"},\"vAb\"),\".\"),mdx(\"p\",null,\"If we have the \",mdx(\"strong\",{parentName:\"p\"},\"acceleration\"),\" of object A moving \",mdx(\"strong\",{parentName:\"p\"},\"relative\"),\" to space b \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"a\"),mdx(\"mi\",{parentName:\"msubsup\"},\"A\"),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{a}^b_{A}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.1244em\",\"verticalAlign\":\"-0.2753em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\"},\"a\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.4247em\",\"marginLeft\":\"0em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"A\")))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.2753em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\", then we write this in the codebase as \",mdx(\"inlineCode\",{parentName:\"p\"},\"aAb\"),\".\"),mdx(\"p\",null,\"Velocities and accelerations can only be \",mdx(\"strong\",{parentName:\"p\"},\"rotated\"),\", \",mdx(\"strong\",{parentName:\"p\"},\"not translated\"),\". They should \",mdx(\"strong\",{parentName:\"p\"},\"not\"),\" be multiplied by a homogeneous transformation matrix, but they can be multiplied by a \",mdx(\"strong\",{parentName:\"p\"},\"rotation matrix\"),\".\"),mdx(\"p\",null,\"If a velocity is multiplied with \",mdx(\"strong\",{parentName:\"p\"},\"time\"),\", it becomes a \",mdx(\"strong\",{parentName:\"p\"},\"displacement vector\"),\".\"),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-cpp\"},\"Eigen::Vector3d rA_tAb = vAb * t;\\n\")),mdx(\"p\",null,\"If an acceleration is multiplied with \",mdx(\"strong\",{parentName:\"p\"},\"time\"),\", it becomes a \",mdx(\"strong\",{parentName:\"p\"},\"velocity vector\"),\".\"),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-cpp\"},\"Eigen::Vector3d vA_tAb = aAb * t;\\n\")),mdx(\"p\",null,\"If you only have a partial velocity or acceleration, such as velocity in the direction of an object you can represent this by labelling the target the velocity is towards.\"),mdx(\"p\",null,\"For example \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"msubsup\",{parentName:\"mrow\"},mdx(\"mi\",{parentName:\"msubsup\",\"mathvariant\":\"bold\"},\"v\"),mdx(\"mrow\",{parentName:\"msubsup\"},mdx(\"mi\",{parentName:\"mrow\"},\"A\"),mdx(\"mi\",{parentName:\"mrow\",\"mathvariant\":\"normal\"},\"/\"),mdx(\"mi\",{parentName:\"mrow\"},\"B\")),mdx(\"mi\",{parentName:\"msubsup\"},\"b\"))),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\mathbf{v}^b_{A/B}\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1.3461em\",\"verticalAlign\":\"-0.497em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathbf\",\"style\":{\"marginRight\":\"0.01597em\"}},\"v\"),mdx(\"span\",{parentName:\"span\",\"className\":\"msupsub\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-t vlist-t2\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.8491em\"}},mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-2.378em\",\"marginLeft\":\"-0.016em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"A\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mtight\"},\"/\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\",\"style\":{\"marginRight\":\"0.05017em\"}},\"B\")))),mdx(\"span\",{parentName:\"span\",\"style\":{\"top\":\"-3.063em\",\"marginRight\":\"0.05em\"}},mdx(\"span\",{parentName:\"span\",\"className\":\"pstrut\",\"style\":{\"height\":\"2.7em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"sizing reset-size6 size3 mtight\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal mtight\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-s\"},\"\\u200B\")),mdx(\"span\",{parentName:\"span\",\"className\":\"vlist-r\"},mdx(\"span\",{parentName:\"span\",\"className\":\"vlist\",\"style\":{\"height\":\"0.497em\"}},mdx(\"span\",{parentName:\"span\"})))))))))),\" would describe the velocity of B towards the point A.\"),mdx(\"h3\",{\"id\":\"examples\",\"style\":{\"position\":\"relative\"}},mdx(\"a\",{parentName:\"h3\",\"href\":\"#examples\",\"aria-label\":\"examples permalink\",\"className\":\"anchor before\"},mdx(\"svg\",{parentName:\"a\",\"aria-hidden\":\"true\",\"focusable\":\"false\",\"height\":\"16\",\"version\":\"1.1\",\"viewBox\":\"0 0 16 16\",\"width\":\"16\"},mdx(\"path\",{parentName:\"svg\",\"fillRule\":\"evenodd\",\"d\":\"M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z\"}))),\"Examples\"),mdx(\"p\",null,\"Using \",mdx(\"strong\",{parentName:\"p\"},\"Eigen\"),\", we define our \",mdx(\"strong\",{parentName:\"p\"},\"vectors\"),\" as \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Vector3d\"),\", \",mdx(\"strong\",{parentName:\"p\"},\"rotation matrices\"),\" as \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Matrix3d\"),\", and \",mdx(\"strong\",{parentName:\"p\"},\"homogeneous transformation matrices\"),\" as \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Isometry3d\"),\". The \",mdx(\"inlineCode\",{parentName:\"p\"},\"d\"),\" denotes \",mdx(\"inlineCode\",{parentName:\"p\"},\"double\"),\" and instead can be \",mdx(\"inlineCode\",{parentName:\"p\"},\"f\"),\" for \",mdx(\"inlineCode\",{parentName:\"p\"},\"float\"),\". \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Isometry3d\"),\" allows us to do computations \",mdx(\"strong\",{parentName:\"p\"},\"easier\"),\" than with \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Matrix4d\"),\".\"),mdx(\"p\",null,\"It allows \",mdx(\"strong\",{parentName:\"p\"},\"multiplication\"),\" with a \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Vector3d\"),\" even though the dimensions do not match. Using \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Isometry3d\"),\" makes it quicker to find the \",mdx(\"strong\",{parentName:\"p\"},\"inverse\"),\", as it employs the techniques mentioned above. \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Isometry3d\"),\" has functions to \",mdx(\"strong\",{parentName:\"p\"},\"individually return\"),\" the \",mdx(\"strong\",{parentName:\"p\"},\"rotational component\"),\" of the matrix and the \",mdx(\"strong\",{parentName:\"p\"},\"translation component\"),\" of the matrix, by using \",mdx(\"inlineCode\",{parentName:\"p\"},\"Htw.rotation()\"),\" and \",mdx(\"inlineCode\",{parentName:\"p\"},\"Htw.translation()\"),\" respectively, where \",mdx(\"inlineCode\",{parentName:\"p\"},\"Htw\"),\" is of type \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Isometry3d\"),\". It also makes it easy to individually \",mdx(\"strong\",{parentName:\"p\"},\"assign\"),\" the \",mdx(\"strong\",{parentName:\"p\"},\"rotational\"),\" and the \",mdx(\"strong\",{parentName:\"p\"},\"translational\"),\" components of the matrix by using \",mdx(\"inlineCode\",{parentName:\"p\"},\"Htw.linear() = Rtw\"),\" and \",mdx(\"inlineCode\",{parentName:\"p\"},\"Htw.translation() = rWTt\"),\" respectively.\"),mdx(\"p\",null,\"When we get a message that is a homogeneous matrix it will be of type \",mdx(\"inlineCode\",{parentName:\"p\"},\"mat4\"),\". This can then be cast to \",mdx(\"inlineCode\",{parentName:\"p\"},\"Eigen::Isometry3d\"),\". Let the matrix be called \",mdx(\"inlineCode\",{parentName:\"p\"},\"Htw\"),\" (\",mdx(\"strong\",{parentName:\"p\"},\"world\"),\" to \",mdx(\"strong\",{parentName:\"p\"},\"torso\"),\" transform) and the message be called \",mdx(\"inlineCode\",{parentName:\"p\"},\"sensors\"),\" (see: \",mdx(\"a\",{parentName:\"p\",\"href\":\"https://github.com/NUbots/NUbots/blob/105b78ad6655801f5f74d029cbd169ba8e8cb5e8/shared/message/input/Sensors.proto#L68\"},\"Sensors.proto\"),\"). Then this is written as\"),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-cpp\"},\"Eigen::Isometry3d Htw(sensors.Htw);\\n\")),mdx(\"p\",null,\"If you have a transformation matrix \",mdx(\"inlineCode\",{parentName:\"p\"},\"Htw\"),\" (\",mdx(\"strong\",{parentName:\"p\"},\"world\"),\" to \",mdx(\"strong\",{parentName:\"p\"},\"torso\"),\" transform) and you want the matrix \",mdx(\"inlineCode\",{parentName:\"p\"},\"Hwt\"),\" (\",mdx(\"strong\",{parentName:\"p\"},\"torso\"),\" to \",mdx(\"strong\",{parentName:\"p\"},\"world\"),\" transform), this is can be achieved by calling \",mdx(\"inlineCode\",{parentName:\"p\"},\"Htw.inverse()\"),\". The inverse function \",mdx(\"strong\",{parentName:\"p\"},\"swaps\"),\" the letter order. The returned inverse matrix is not \",mdx(\"strong\",{parentName:\"p\"},\"mutable\"),\".\"),mdx(\"p\",null,\"If you have a vector \",mdx(\"inlineCode\",{parentName:\"p\"},\"rTWf\"),\" (\",mdx(\"strong\",{parentName:\"p\"},\"world\"),\" to \",mdx(\"strong\",{parentName:\"p\"},\"torso\"),\" in \",mdx(\"strong\",{parentName:\"p\"},\"foot\"),\" space) and you want the opposite letter order (\",mdx(\"strong\",{parentName:\"p\"},\"torso\"),\" to \",mdx(\"strong\",{parentName:\"p\"},\"world\"),\" in \",mdx(\"strong\",{parentName:\"p\"},\"foot\"),\" space), this is done by negating the vector, \",mdx(\"inlineCode\",{parentName:\"p\"},\"rWTf = -rTWf\"),\".\"),mdx(Alert,{mdxType:\"Alert\"},mdx(\"p\",null,\"Negating does not change the space the vector is measured in.\")),mdx(\"p\",null,\"If you have two transformation matrices and you would like to multiply them to get a new transformation matrix, the \",mdx(\"strong\",{parentName:\"p\"},\"inner two letters\"),\" should be \",mdx(\"strong\",{parentName:\"p\"},\"identical\"),\", i.e. the same space.\"),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-cpp\"},\"Eigen::Isometry3d Htf = Htw * Hwf;\\n\")),mdx(\"p\",null,\"This gives the \",mdx(\"strong\",{parentName:\"p\"},\"foot\"),\" to \",mdx(\"strong\",{parentName:\"p\"},\"torso\"),\" transform by \",mdx(\"strong\",{parentName:\"p\"},\"multiplying\"),\" the \",mdx(\"strong\",{parentName:\"p\"},\"world\"),\" to \",mdx(\"strong\",{parentName:\"p\"},\"torso\"),\" transform with the \",mdx(\"strong\",{parentName:\"p\"},\"foot\"),\" to \",mdx(\"strong\",{parentName:\"p\"},\"world\"),\" transform.\"),mdx(\"p\",null,\"These two \",mdx(\"strong\",{parentName:\"p\"},\"inner\"),\" letters are \",mdx(\"strong\",{parentName:\"p\"},\"cancelled\"),\", with the outer two remaining. This also works for rotation matrices.\"),mdx(\"p\",null,\"If you have two \",mdx(\"strong\",{parentName:\"p\"},\"vectors\"),\" in the \",mdx(\"strong\",{parentName:\"p\"},\"same space\"),\" and their \",mdx(\"strong\",{parentName:\"p\"},\"inner\"),\" (or \",mdx(\"strong\",{parentName:\"p\"},\"outer\"),\", since vector addition is \",mdx(\"strong\",{parentName:\"p\"},\"commutative\"),\") \",mdx(\"strong\",{parentName:\"p\"},\"letters\"),\" are both the same then they can be \",mdx(\"strong\",{parentName:\"p\"},\"cancelled\"),\" in a similar way to the transformation matrices.\"),mdx(\"pre\",null,mdx(\"code\",{parentName:\"pre\",\"className\":\"language-cpp\"},\"Eigen::Vector3d rXZa = rXYa + rYZa;\\n\")),mdx(\"h3\",{\"id\":\"reference-frames\",\"style\":{\"position\":\"relative\"}},mdx(\"a\",{parentName:\"h3\",\"href\":\"#reference-frames\",\"aria-label\":\"reference frames permalink\",\"className\":\"anchor before\"},mdx(\"svg\",{parentName:\"a\",\"aria-hidden\":\"true\",\"focusable\":\"false\",\"height\":\"16\",\"version\":\"1.1\",\"viewBox\":\"0 0 16 16\",\"width\":\"16\"},mdx(\"path\",{parentName:\"svg\",\"fillRule\":\"evenodd\",\"d\":\"M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z\"}))),\"Reference Frames\"),mdx(\"p\",null,\"The NUbots codebase uses a number of different reference frames (spaces) for different purposes. The following is a list of the commonly used spaces in the codebase.\"),mdx(\"table\",null,mdx(\"thead\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"thead\"},mdx(\"th\",{parentName:\"tr\",\"align\":null},\"Space\"),mdx(\"th\",{parentName:\"tr\",\"align\":null},\"Letter\"),mdx(\"th\",{parentName:\"tr\",\"align\":null},\"Description\"))),mdx(\"tbody\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":null},mdx(\"inlineCode\",{parentName:\"td\"},\"camera\")),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"C\"),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"Located at camera (left/right) with x pointed \\\"outwards\\\", y to the \\\"left\\\" and z \\\"up\\\"\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":null},mdx(\"inlineCode\",{parentName:\"td\"},\"world\")),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"W\"),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"Located on the ground below the robot on startup. (Fixed inertial frame)\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":null},mdx(\"inlineCode\",{parentName:\"td\"},\"torso\")),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"T\"),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"Fixed midway between the robots hip yaw joints, with x \\\"forward\\\", y to the \\\"left\\\" and z \\\"up\\\"\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":null},mdx(\"inlineCode\",{parentName:\"td\"},\"robot\")),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"R\"),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"Located directly beneath torso projected onto the ground, with x \\\"forward\\\", y to the \\\"left\\\" and z \\\"up\\\"\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":null},mdx(\"inlineCode\",{parentName:\"td\"},\"field\")),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"F\"),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"Located at the centre circle of the field, with x pointing towards the \\\"own\\\" goal and z \\\"up\\\"\")))),mdx(\"img\",{\"src\":\"/opt/build/repo/src/book/02-system/01-foundations/images/spaces.png\",\"alt\":\"NUbots spaces\",\"title\":\"NUbots spaces\"}),mdx(\"h3\",{\"id\":\"summary\",\"style\":{\"position\":\"relative\"}},mdx(\"a\",{parentName:\"h3\",\"href\":\"#summary\",\"aria-label\":\"summary permalink\",\"className\":\"anchor before\"},mdx(\"svg\",{parentName:\"a\",\"aria-hidden\":\"true\",\"focusable\":\"false\",\"height\":\"16\",\"version\":\"1.1\",\"viewBox\":\"0 0 16 16\",\"width\":\"16\"},mdx(\"path\",{parentName:\"svg\",\"fillRule\":\"evenodd\",\"d\":\"M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z\"}))),\"Summary\"),mdx(\"table\",null,mdx(\"thead\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"thead\"},mdx(\"th\",{parentName:\"tr\",\"align\":null},\"Item\"),mdx(\"th\",{parentName:\"tr\",\"align\":null},\"Description\"))),mdx(\"tbody\",{parentName:\"table\"},mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":null},mdx(\"inlineCode\",{parentName:\"td\"},\"Hab\")),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"Denotes a 3D affine transformation matrix going from space \",mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"b\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.6944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"b\"))))),\" to space \",mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"a\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.4306em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"))))),\".\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":null},mdx(\"inlineCode\",{parentName:\"td\"},\"Rab\")),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"Denotes a 3D rotation matrix from space \",mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"b\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"b\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.6944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"b\"))))),\" to space \",mdx(\"span\",{parentName:\"td\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"a\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"a\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.4306em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"a\"))))),\".\")),mdx(\"tr\",{parentName:\"tbody\"},mdx(\"td\",{parentName:\"tr\",\"align\":null},mdx(\"inlineCode\",{parentName:\"td\"},\"rABb\")),mdx(\"td\",{parentName:\"tr\",\"align\":null},\"Denotes a vector from point 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math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},\"(\"),mdx(\"mi\",{parentName:\"mrow\"},\"r\"),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03C6\"),mdx(\"mo\",{parentName:\"mrow\",\"separator\":\"true\"},\",\"),mdx(\"mi\",{parentName:\"mrow\"},\"\\u03B8\"),mdx(\"mo\",{parentName:\"mrow\",\"fence\":\"true\"},\")\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\left(r, \\\\varphi, \\\\theta\\\\right)\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"1em\",\"verticalAlign\":\"-0.25em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"minner\"},mdx(\"span\",{parentName:\"span\",\"className\":\"mopen delimcenter\",\"style\":{\"top\":\"0em\"}},\"(\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},\"r\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},\",\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.1667em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C6\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mpunct\"},\",\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mspace\",\"style\":{\"marginRight\":\"0.1667em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\",\"style\":{\"marginRight\":\"0.02778em\"}},\"\\u03B8\"),mdx(\"span\",{parentName:\"span\",\"className\":\"mclose delimcenter\",\"style\":{\"top\":\"0em\"}},\")\")))))),\", where \",mdx(\"inlineCode\",{parentName:\"p\"},\"r\"),\" is radial distance, \",mdx(\"span\",{parentName:\"p\",\"className\":\"math math-inline\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex\"},mdx(\"span\",{parentName:\"span\",\"className\":\"katex-mathml\"},mdx(\"math\",{parentName:\"span\",\"xmlns\":\"http://www.w3.org/1998/Math/MathML\"},mdx(\"semantics\",{parentName:\"math\"},mdx(\"mrow\",{parentName:\"semantics\"},mdx(\"mi\",{parentName:\"mrow\"},\"\\u03C6\")),mdx(\"annotation\",{parentName:\"semantics\",\"encoding\":\"application/x-tex\"},\"\\\\varphi\")))),mdx(\"span\",{parentName:\"span\",\"className\":\"katex-html\",\"aria-hidden\":\"true\"},mdx(\"span\",{parentName:\"span\",\"className\":\"base\"},mdx(\"span\",{parentName:\"span\",\"className\":\"strut\",\"style\":{\"height\":\"0.625em\",\"verticalAlign\":\"-0.1944em\"}}),mdx(\"span\",{parentName:\"span\",\"className\":\"mord mathnormal\"},\"\\u03C6\"))))),\" is the polar angle, and \",mdx(\"span\",{parentName:\"p\",\"className\":\"math 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