Written Assignment #1 - CSU540 Computer Graphics - Fall 2005

Professor Futrelle

Due in class, Monday, September 26th (Quiz on the 29th)

Do the following problems by hand, on paper. No calculator is needed or should be used. For example, a value such as √2 should be kept in that form and certainly not written down as 1.41421356237309504880, etc. In drawing items based on such values, a reasonable approximation suffices. The first three questions can be done based on material in Chapter 2. The remainder are based on the material in Chapter 5, and especially, the material in Chapter 6.

For convenience in writing out the problems below, vectors are written in row format, in contrast to the book's column format, as [x,y] or [x,y,z] or for homogeneous coordinates [x,y,1] or [x,y,z,1]. You should write your answers (and exam answers) using vectors in column format.

When you are asked to do a transformation step-by-step, write, draw and, comment on the results of each step.

1. Draw the vectors A = [8,0], B = [0,6] and the sum C of A and B. Using the coordinates of C, compute its length.
2. As in Problem 1, but draw and compute the difference, C - B.
3. Write out the normalized form of the vector [7,2] and show how you computed it.
4. Write out the general form of the rotation matrix R, as well as its value R_b for θ = -3/4π.
5. Assume you have a triangle whose vertices are A=[6,0], B=[4,3] and C=[4,-3]. Draw it and then draw its appearance when rotated by -3/4π around the origin.
6. Show that applying R_b in Problem 4 to the vertices of the triangle in Problem 5 gives the same result as what you drew in 5.
7. Write the value of the rotation matrix R_a, for θ = +3/4π and show that R_a x R_b is the identity by computing each element. Show that R_b x R_a gives the same result.
8. Write out the 3x3 translation matrix T_a for a translation by [tx,ty]. By multiplying out the components, show in general that the product of T_a and T_b (for a translation [-tx,-ty]) is the identity matrix.
9. For a translation T_c, by [5,5], show that the product Prod_A = R_a x T_c is not equal to to the product Prod_B = T_c x R_a. Draw how the point p = [3,2] should be transformed by the two transforms matrices, Prod_A and Prod_B, doing them step-by-step, using R_a and T_c separately. Then transform p by actually multiplying it properly by the two different composite matrices, Prod_A and Prod_B. Your results should agree with what you just drew.
10. Consider the two points [5,6] and[5,4]. Use a single matrix that is the product (in order applied) of a translation by [-5,-5], a rotation by +π/2, and a translation by [5,5] and apply it two both points. What did you expect to happen and did you get the result you expected?
11. Write out the 3D transforms for +π/2 rotations around the z axis (transform T_z) and around the x axis (transform T_x). Draw and compute the result of applying T_z and then T_x to the point P₅=[5,5,5]. Apply the matrices in sequence one-by-one and then compute their product and apply it to P₅ also.
12. Write out the translation matrix for [5,-4,3] and apply it to the 3d point P₃=[4,6,-3]. Is one of the transformed coordinates 0, as you'd expect?

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