Talk:Extra:Intersecting planes - Revision history

KevinYager at 21:26, 23 December 2015

2015-12-23T21:26:24Z

KevinYager at 22:52, 21 December 2015

2015-12-21T22:52:20Z

KevinYager: Created page with "==Rotate $\mathbf{v}_{2b}$ about $\mathbf{n}_{2}$ == In general, rotation of a vector $\scriptstyle \mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x..."$

2015-12-21T22:41:08Z

Created page with "==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{2}</math>== In general, rotation of a vector <math>\scriptstyle \mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x..."

New page

==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{2}</math>==

In general, rotation of a vector <math>\scriptstyle
\mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x & y & z \end{bmatrix}</math>about an arbitrary unit-vector <math>\scriptstyle
\mathbf{n} = \begin{bmatrix} u & v & w \end{bmatrix}</math> gives ([https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle 1], [http://inside.mines.edu/fs_home/gmurray/ArbitraryAxisRotation/ 2]):
::<math>
\mathbf{v}_{\mathrm{end}} = \begin{bmatrix} u(ux +vy+wz)(1-\cos \theta) + x \cos \theta + (-wy+vz)\sin \theta \\
v(ux+vy+wz)(1 - \cos \theta) + y \cos \theta + (wx-uz) \sin \theta\\
w(ux+vy+wz)(1-\cos \theta) + z \cos \theta + (-vx+uy) \sin \theta \end{bmatrix}
</math>
In this particular case, we thus expect:
::<math>
\begin{alignat}{2}
\mathbf{v}_{2} & = \begin{bmatrix} (-wy+vz)\sin \phi \\
v(vy+wz)(1 - \cos \phi) + y \cos \phi \\
w(vy+wz)(1-\cos \phi) + z \cos \phi\end{bmatrix}
\\
& = \begin{bmatrix} (- \cos\alpha q \cos\alpha + -\sin\alpha q \sin\alpha)\sin \phi \\
-\sin\alpha(-\sin\alpha q \cos\alpha+\cos\alpha q \sin\alpha)(1 - \cos \phi) + q \cos\alpha \cos \phi \\
\cos\alpha(-\sin\alpha q \cos\alpha+ \cos\alpha q \sin\alpha)(1-\cos \phi) + q \sin\alpha \cos \phi\end{bmatrix}
\\
& = q \begin{bmatrix} -(\cos^2 \alpha +\sin ^2 \alpha)\sin \phi \\
\sin^2 \alpha(\cos\alpha-\cos\alpha )(1 - \cos \phi) + \cos\alpha \cos \phi \\
\cos^2 \alpha(-\sin\alpha + \sin\alpha)(1-\cos \phi) + \sin\alpha \cos \phi\end{bmatrix}
\\
& = q \begin{bmatrix} -\sin \phi \\
\cos\alpha \cos \phi \\
\sin\alpha \cos \phi\end{bmatrix}
\end{alignat}
</math>

@@ Line 54: / Line 54: @@
 \cos\alpha \cos \phi \\
 \sin\alpha \cos \phi\end{bmatrix}
 \end{alignat}
 </math>

← Older revision		Revision as of 22:52, 21 December 2015
Line 1:		Line 1:
		+	==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{1}</math>==
		+	In general, rotation of a vector <math>\scriptstyle
		+	\mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x & y & z \end{bmatrix}</math>about an arbitrary unit-vector <math>\scriptstyle
		+	\mathbf{n} = \begin{bmatrix} u & v & w \end{bmatrix}</math> gives ([https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle 1], [http://inside.mines.edu/fs_home/gmurray/ArbitraryAxisRotation/ 2]):
		+	::<math>
		+	\mathbf{v}_{\mathrm{end}} = \begin{bmatrix} u(ux +vy+wz)(1-\cos \theta) + x \cos \theta + (-wy+vz)\sin \theta \\
		+	v(ux+vy+wz)(1 - \cos \theta) + y \cos \theta + (wx-uz) \sin \theta\\
		+	w(ux+vy+wz)(1-\cos \theta) + z \cos \theta + (-vx+uy) \sin \theta \end{bmatrix}
		+	</math>
		+	In this particular case, we thus expect:
		+	::<math>
		+	\begin{alignat}{2}
		+	\mathbf{v}_{2} & = \begin{bmatrix} -y\sin \phi \\
		+	y \cos \phi \\
		+	z(1-\cos \phi) + z \cos \phi\end{bmatrix}
		+	\\
		+	& = \begin{bmatrix} -q \cos \alpha\sin \phi \\
		+	q \cos \alpha \cos \phi \\
		+	q \sin \alpha(1-\cos \phi) + q \sin \alpha \cos \phi\end{bmatrix}
		+	\\
		+	& = q \begin{bmatrix} - \cos \alpha\sin \phi \\
		+	\cos \alpha \cos \phi \\
		+	\sin \alpha\end{bmatrix}
		+	\end{alignat}
		+	</math>
		+
		+
	==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{2}</math>==		==Rotate <math>\mathbf{v}_{2b}</math> about <math>\mathbf{n}_{2}</math>==

Talk:Extra:Intersecting planes - Revision history

KevinYager at 21:26, 23 December 2015

KevinYager at 22:52, 21 December 2015

KevinYager: Created page with "==Rotate \mathbf{v}_{2b} about \mathbf{n}_{2}== In general, rotation of a vector \scriptstyle \mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x..."

KevinYager: Created page with "==Rotate $\mathbf{v}_{2b}$ about $\mathbf{n}_{2}$ == In general, rotation of a vector $\scriptstyle \mathbf{v}_{\mathrm{start}} = \begin{bmatrix} x..."$