Computer Graphics Homework Set #2

View Transform

1.

Given an eye point $e=(1,\,1,\,1)$, a look-at point $a=(1,\,-2,\,1)$ and an up-vector $\vec{v}_{up}=\langle 1,\,0,\,0 \rangle$, fill in the missing points in world or view space.

World Space Points	View Space Points
$(0,\,0,\,0)$
$(1,\,1,\,1)$
$(1,\,-2,\,1)$
$(2,\,7,\,1)$
	$(0,\,0,\,0)$
	$(1,\,1,\,1)$
	$(1,\,-2,\,1)$
	$(2,\,7,\,1)$

2.

Given an eye point $e=(3,\,1,\,4)$, a look-at point $a=(1,\,5,\,9)$ and an up-vector $\vec{v}_{up}=\langle 2/\sqrt{65},\,6/\sqrt{65},\,5/\sqrt{65} \rangle$, find a matrix that performs the view transformation from world to view space. (Sorry for the strange numbers -- 50 /cfor the first one who identifies them.)

3.

Find the perpendicular projection of $\langle 0\,3,\,3\rangle$ onto $\langle 9,\,8\,8 \rangle$.

4.

Find the inverse of the rotation matrix

$$\left[\begin{array}{ccc} \frac{1}{3} & \frac{-2}{3} & \frac... ...ac{-2}{3} & \frac{-2}{3} & \frac{-1}{3} \\ \end{array}\right]$$

Perspective Transform

1.

Given a object point $O=(5,\,7,\,7)$, a center of projection $C=(1,\,6,\,1)$, and a projection plane P=x+4y+z+2=0 find the perspective projection of O onto P.

2.

  Given a field of view angle $\alpha = \pi/3$, near plane z_n=6, z_f=11, find the perspective transform that, after the homogeneous divide, maps the viewing frustrum into the cube $-1\leq x \leq 1$, $-1\leq y \leq 1$, and $0\leq z \leq 1$.

3.

Using the matrix from problem 2, determine the perspective space location of the view space points. Identify which are inside the view volume.

View Space Points	Perspective Space Points	Inside/Outside
$(0,\,0,\,0)$
$(0,\,0,\,8)$
$(3,\,-2,\,6)$
$(2,\,7,\,11)$
$(4,\,-5,\,17)$

No References!