Rotation of a figure about a point p in R^2 is accomplished by first translating the figure by –p, rotating about the origin, and then translating back by p. See Figure 7. Construct the 3 × 3 matrix that rotates points -30° about the point (-2, 6), using homogeneous coordinates.

Question

Rotation of a figure about a point p in [latex] \mathbb{R} ^{2} [/latex] is accomplished by first translating the figure by –p, rotating about the origin, and then translating back by p. See Figure 7. Construct the 3 × 3 matrix that rotates points -30° about the point (-2 , 6), using homogeneous coordinates.

Accepted Answer

Assemble the matrices right-to-left for the three operations. Using p = (-2 , 6), cos(-30°) = [latex] \sqrt{3}/2 [/latex], and sin(-30°) = -.5 , we have

[latex] \overset{\overset{Translate}{_{back by p}} }{\left [ \begin{matrix} 1 & 0 & -2 \ 0 & 1 & 6 \ 0 & 0 & 1 \end{matrix} ight ] } \overset{\underset{the origin}{_{Rotate around}}}{\left [ \begin{matrix} \sqrt{3}/2 & 1/2 & 0 \ -1/2 & \sqrt{3}/2 & 0 \ 0 & 0 & 1 \end{matrix} ight ] } \overset{\overset{Translate}{_{by -p}}}{\left [ \begin{matrix} 1 & 0 & 2 \ 0 & 1 & -6 \ 0 & 0 & 1 \end{matrix} ight ] } [/latex]

[latex] = \left [ \begin{matrix} \sqrt{3}/2 & 1/2 & \sqrt{3}-5 \ -1/2 & \sqrt{3}/2 & -3\sqrt{3}+5 \ 0 & 0 & 1 \end{matrix} ight ] [/latex]

Question 2.7.PP.1: Rotation of a figure about a point p in R^2 is accomplished ...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions