Assignment Task:

Task:

1. (a) Find the matrix that converts the coordinates a

T = [α1, α2, α3, α4] of a point P in the frame [u1, u2, u3, P0] into coordinates b

T = [β1, β2, β3, β4] in the new frame [v1, v2, v3, Q0] , given by v1 = u1, v2 = u2, v3 = u2 + u3, Q0 = P0 + u1

(b) Specify the vector b if a

T = [1, 1, 1, 1].

2. Explain how we can determine whether the following polygon specified by the ordered set of vertices in counterclockwise orientation

P = {(2, 1), (6, 4), (8, 3), (8, 8)} is convex or concave. If the polygon is determined to be concave by this method, how can it then be split into a set of convex polygons?

3. A pair of transformations is said to commute if the order in which you apply them does not matter. In terms of transformation matrices, that means that AB = BA.
Explain which of the following transformation pairs commute in 3D.
(a) translate - translate
(b) scale - scale
(c) rotate - translate
(d) rotate - scale
(e) rotate - rotate

4. (a) Find the matrix for mirror reflection with respect to the plane passing through the origin and having a normal vector n = (1, 1, 0).

N.B. I am asking for the mathematical and geometrical developments that lead to the construction of the matrix. Giving the end result matrix only will reduce your mark to a tiny fraction!

(b) Find the image of the pyramid defined by the vertices A(0, 0, 0), B(1, 0, 0), C(0, 1, 0) and D(0, 0, 1) by this transformation.

 

5. Given the following code, find the current Modelview matrix.
glMatrixMode(GL MODELVIEW);
glLoadIdentity();
glTranslatei( 5, 2, 1 );
glRotatef( 45.0f, 0.0, 0.0, 1.0f );

6. Figure 1 displays the result of the function draw house().

 

(a) Draw the pictures of house A and house B which are the transforms of the original house by the following OpenGL commands.
glTranslate(1,0,0);
glRotate(90,0,0,1);
glPushMatrix();
glTranslate(0,2,0);
draw house();
glPopMatrix();
glTranslate(-1,0,0);
draw house();

(b) Give the series of affine transformations (assuming post-multiplying) needed to create the picture in Figure 2, assuming the house started from the position shown in Figure 1.

 

7. Give a sequence of the following commands for drawing the scene depicted in the  below. Assume that the current transformation matrix is initialized to the identity. All transformation matrices are multiplied to current transformation matrix from the right:

drawSquare(): draw a 1 × 1 square, bottom left corner at origin
translate( x, y ): translation by x, y
scale( x, y ): scaling in x and y direction
rotateZ(θ ): rotation around z axis (i.e. in x − y plane)
pushMatrix(): duplicates top entry of matrix stack
popMatrix(): deletes top entry of matrix stack.

 

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  • Uploaded By : Pearl
  • Posted on : March 26th, 2019

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