Computer Graphics

Week 3 Solution

Question 1

$T=\left[\begin{array}{ccc}1/2&-\sqrt 3/2&\sqrt 3\\\sqrt 3 /2&1/2&1\\0&0&1\end{array}\right]$

Question 2

If you first rotate and then translate you get

$A(\sqrt 2/2+1,\sqrt 2/2)\quad B(1,\sqrt 2)\quad C(1-\sqrt 2/2,\sqrt 2/2)$

If you first translate and then rotate you get

$A(\sqrt 2,\sqrt 2) \quad B(\sqrt 2/2,3\sqrt 2/2)\quad C(0,\sqrt 2)$

Question 3

$T=\left[\begin{array}{ccc}1&h&0\\0&1&0\\0&0&1\end{array}\right]$

Question 4

The parametric equation of a line segment joining a and b is

$L(t) = (1-t)a + tb\quad 0\le t\le 1$

This is true whether or not we use homogenous coordinates.

If T is a transform the the transform of the line is

$\begin{array}{rcl}TL(t)&=&T((1-t)a + tb)\\&=&(1-t)Ta + tTb\end{array}$

(Matrix multiplication obeys distributive law)

Which is the line segment connecting the transformed endpoints.

A scaling by (2,1) (i.e. double x values and leave y unchanged) turns circles into ellipses so it is not true for circles.

Question 5

$T=\left[\begin{array}{ccc}a&0&c\\0&e&f\\0&0&1\end{array}\right]$

where

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