INT32GP Graphics Programming: Tutorial #11: Theory

1. Is the homogeneous 2D coordinate (1.23,5.9,1) a point or a vector?
2. Express the vector (-1.5,2.0,-10.5) as a homogeneous coordinate.
3. What are the Cartesian coordinates of the homogeneous point (2.0,-9.0,16.5,2.0)?
4. OpenGL uses 4D homogeneous for all its vertices. How would it represent the 2D vertex (2.0,5.5)?
5. Multiply A[4][4] by B[4][1] to obtain AB[4][1].

A 2 0 0 0 B 100 0 1 0 0 150 0 0 3 0 -200 0 0 0 1 1

6. Write down the 2*2, 3*3, and 4*4 identity matrices, I.
7. Multiply I[4][4] by the following matrix, A[4][1], to get IA[4][1]

A 0.25 1.5 -100.5 1

8. Use homogeneous coordinates and matrix multiplication to calculate the point that results when the translation (0.0,-10.0,-100.0) is applied to the origin, (0.0,0.0,0.0).
• Step 1: Multiply the Identity Matrix by the Translation Matrix.
• Step 2: Multiply the result by the origin, expressed as a homogeneous coordinate.
9. Use homogeneous coordinates and matrix multiplication to scale the 3D point, (1.5,2.0,3.5)by the scaling factors s[x]=2, s[y]=1, and s[z]=3.0.
10. A square is specified by the points (0,0),(0,1),(-1,1),(-1,0,). We wish to halve its size by scaling and translate it by (2,5) (to obtain the points (2,5),(2,5.5),(1.5,5.5),(1.5,5)). Carry out the calculation using homogeneous coordinates and matrix multiplication.
    • Multiply the Identity Matrix by the Scaling Matrix. Then multiply the result by the Translation Matrix. Then multiply the result by each point to get the resulting points. Is the result as expected?
    • Switch the order. Multiply the Identity Matrix by the Translation Matrix, the result by the Scaling Matrix, and the result by the points. Does this make a difference?
11. Given that the following matrix is applied to rotate anti-clockwise around the Z-axis, and that cos 90o=0 sin 90o=1. Use matrix multiplication to rotate the point (0.5,0.5) around the z-axis by 90o.

	    Rotate Around Z-axis
	    cos(ra)   -sin(ra)    0   0
	    sin(ra)   cos(ra)     0   0
	    0         0           1   0
	    0         0           0   1
			

12. An object is specified by the vertices (-1.0,1.5),(-3.0,2.0),(-3.5,1.5),(-3.0,1.0). Use homogeneous coordinates and matrix multiplication to calculate its position when it is rotated by 90o anticlockwise.
13. Use homogeneous coordinates and matrix multiplication to show how the same object may be rotated "in place" by 90o anticlockwise.