Rodrigues’ rotation formula<\/a>. If \u03a9 is a matrix, we define the exponential of \u03a9, denoted \\(e^\u03a9\\). Let \\(\u00a0 \\theta^2 =\u00a0\u03c9_x^2 +\u00a0\u03c9_y^2 + \u03c9_z^2 \\)<\/p>\n\\begin{align}
\n\\theta ad(l) = ad(\u03c9) = \u03a9 =
\n\\begin{bmatrix}
\n0 & -\u03c9_z & \u03c9_y \\\\
\n\u03c9_z & 0 & -\u03c9_x \\\\
\n-\u03c9_y & \u03c9_x & 0
\n\\end{bmatrix}
\n\\end{align}<\/p>\n
\\begin{align}
\np_1 &= e^{\\theta ad(l)} p = e^\u03a9 p \\\\
\n&= ( I_3 + \\frac{\\sin \\theta}{\\theta} \u03a9 + \\frac{1 – \\cos \\theta}{\\theta^2} \u03a9^2 ) p\\\\
\n&= ( I_3 + \\sin \\theta\u00a0ad(l) + ( 1 – \\cos \\theta ) ad^2 (l) ) p\\\\
\n&= p + \\sin \\theta ad(l) p + ( 1 – \\cos \\theta )ad^2 (l) p \\\\
\n&= p + \\sin \\theta (l \\times p) +( 1 – \\cos \\theta )l \\times ( l \\times p ) \\\\
\n&= p + \\sin \\theta (l \\times p) -(1 – \\cos \\theta )l \\times p \\times l \\\\
\n&= p + \\sin \\theta (l \\times p) -(1 – \\cos \\theta )((l\\cdot l)p – (l \\cdot p)l ) \\\\
\n&= p + \\sin \\theta (l \\times p) -(1 – \\cos \\theta )(p – (l \\cdot p)l ) \\\\
\n&= p \\cos \\theta + \\sin \\theta (l \\times p) + (1 – \\cos \\theta )(l \\cdot p)l )
\n\\end{align}<\/p>\n
2.\u00a0 Unit quaternions are in bijection with points of the real 3-sphere \\(S^3\\) in \\(\\mathbb R^4\\). Let \\( v =\u00a0l\\sin(\\frac{\\theta}{2}) \\)\uff0c\\( w=\\cos(\\frac{\\theta}{2}) \\)<\/p>\n
\\begin{align}
\nq &= \\cos(\\frac{\\theta}{2})+l\\sin(\\frac{\\theta}{2}) = w + v\\\\
\np_1 &= qpq^{-1} \\\\
\n&= (w^{2} – v \\cdot v)p + 2w(v \\times p) + 2(v \\cdot p)v \\\\
\n&= p+2w(v \\times p) +2v \\times (v \\times p)
\n\\end{align}<\/p>\n
<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
\u2606.\u00a0 Rotation about the x, y, z axis \\begin{align} R_x(\\theta) &= \\begin{bmatrix} 1 & 0 & 0\\\\ 0 & \\cos\\theta & -\\sin\\theta\\\\ 0 & \\sin\\theta & \\cos\\theta \\end{bmatrix} = exp \\left (\\theta \\begin{bmatrix} 0 & 0 & 0\\\\ 0 & 0 & -1\\\\ 0 & 1 & 0 \\end{bmatrix} \\right ) \\\\ \\\\ \\\\ R_y(\\theta) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8,7],"tags":[20],"class_list":["post-268","post","type-post","status-publish","format-standard","hentry","category-graphics","category-math","tag-geo"],"_links":{"self":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/268","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=268"}],"version-history":[{"count":0,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/268\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/media?parent=268"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/categories?post=268"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/tags?post=268"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}