Quiz #1 - CSU540 Computer Graphics - Spring 2005

Professor Futrelle

Linear Algebra and Geometric Transformations

Put your answers and all your calculations in your Blue Book. 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 diagrams based on such values, a reasonable approximation suffices.

For convenience in writing out your answers, vectors can be written in row format, as [x,y] or [x,y,z], and for homogeneous coordinates too, in contrast to the book's 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 = A + 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 [3,4] and show how you computed it.


4. Write out the general form of the rotation matrix R for θ= 3π/4. You might want to plot the functions sinθ and cosθ from θ=0 to θ=2π to be sure you're using the correct values.


5. Write out the rotation matrix R2 for θ=-π/2. Apply R2 to the endpoints of the line segment from [3,3] to [5,5]. Does the result agree with what you'd expect intuitively? Explain.


6. Write out the 3x3 translation matrix Td for a translation by [tx,ty]. By multiplying out the components, show in general that the product of Td and Td produces a translation of twice the extent.


7. Write out the 3D transform for +π rotation around the z axis Apply it to the point [7,9,14]. Explain why you expected the result you obtained.

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