We can compute a rotation matrix \\( R \\in SO(3) \\) from an angle \\(\\theta\\) and axis \\(l\\) (unit vector)<\/p>\n
\\[
\nR = e^{\\theta C} = I + \\sin(\\theta)C + (1-\\cos \\theta)C^2
\n\\]<\/p>\n
where \\(C\\) is the antisymmetric matrix:<\/p>\n
\\[
\nC = \\begin{bmatrix}
\n0 & -l_z & l_y \\\\
\nl_z & 0 & -l_x \\\\
\n-l_y & l_x & 0
\n\\end{bmatrix}
\n\\]<\/p>\n
Our proof will use the Taylor’s formula to define the exponential of a matrix \\(M\\)<\/p>\n
\\[
\ne^M = \\sum_{k=0}^{\\infty}\\frac{M^k}{k!}
\n\\]<\/p>\n
where \\(M^0\\) is defined to be the identity matrix \\(I\\). Because \\(C^3 = -C\\) and the Taylor expansions:<\/p>\n
\\begin{align}
\n\\sin \\theta &= \\theta – \\frac{\\theta^3}{3!} + \\frac{\\theta^5}{5!} – \\cdots + \\frac{(-1)^n}{(2n+1)!}\\theta^{2n+1} + \\cdots \\\\
\n\\cos \\theta &= 1 – \\frac{\\theta^2}{2!} + \\frac{\\theta^4}{4!} – \\cdots + \\frac{(-1)^n}{(2n)!}\\theta^{2n} + \\cdots
\n\\end{align}<\/p>\n
then we obtain<\/p>\n
\\begin{align}
\nR &= e^{\\theta C}\\\\
\n&= \\sum_{k=0}^{\\infty}\\frac{(\\theta C)^k}{k!} \\\\
\n&= I + \\frac{1}{1!}\\theta C + \\frac{1}{2!}(\\theta C)^2 + \\frac{1}{3!}(\\theta C)^3 + \\frac{1}{4!}(\\theta C)^4 + \\frac{1}{5!}(\\theta C)^5 + \\frac{1}{6!}(\\theta C)^6 + \\cdots \\\\
\n&= I + (\\theta – \\frac{\\theta^3}{3!} + \\frac{\\theta^5}{5!} – \\cdots)C + (\\frac{\\theta^2}{2!} – \\frac{\\theta^4}{4!} + \\frac{\\theta^6}{6!} – \\cdots)C^2 \\\\
\n&= I + \\sin\\theta C + (1-\\cos \\theta)C^2
\n\\end{align}<\/p>\n","protected":false},"excerpt":{"rendered":"
We can compute a rotation matrix \\( R \\in SO(3) \\) from an angle \\(\\theta\\) and axis \\(l\\) (unit vector) \\[ R = e^{\\theta C} = I + \\sin(\\theta)C + (1-\\cos \\theta)C^2 \\] where \\(C\\) is the antisymmetric matrix: \\[ C = \\begin{bmatrix} 0 & -l_z & l_y \\\\ l_z & 0 & -l_x \\\\ […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8,7],"tags":[],"class_list":["post-485","post","type-post","status-publish","format-standard","hentry","category-graphics","category-math"],"_links":{"self":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/485","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/comments?post=485"}],"version-history":[{"count":0,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/485\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/media?parent=485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/categories?post=485"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/tags?post=485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}