Spring 2000

Department of Computer Science

Homework Assignment 2

Write and draw your answers to these questions by hand. Clearly show each step, including the answers in which you must multiply matrices---show the resulting matrix of each matrix multiplication. This assignment is due at the beginning of the class period noted in the class schedule.
  1. Show that rotating all 2-D points about the origin by 90 degrees is equivalent to rotating it twice by 45 degrees. Use homogeneous matrices.
  2. In 2-D, what angle of rotation is equivalent to a reflection through the origin, meaning a reflection through both the x axis and the y axis? Show it by writing the 2-D, homogeneous matrices for the reflection and the rotation.
  3. What sequence of transformations is required to rotate this 2-D object about the door knob by 20 degrees? List the steps using notation like T(1,2) for a translation by (1,2). Then write the full expression of matrix multiplications and a general point as a homogeneous column vector that performs this transformation. Now compose all matrices into one matrix and write that matrix. Then apply this single matrix to the point that is the peak of the roof and write your answer. Check that it makes sense.

  4. Now let's move on to 3-D. What is the sequence of transformations you would use to transform points in (x',y',z') into points in (x,y,z)? Write the matrices you would use, then compose them into one matrix and apply that matrix to transform the point in red, (x',y',z') = (1,2,3), into a point in (x,y,z).

  5. Show how you would construct the rotation matrix to rotate this vector down to the x axis. Describe each step. Show the final matrix, and test it by applying it to point (4,3,2).
  6. In the figure for Question 3, assume the z axis is pointing out of the screen at you. Let's say we want to rotate this object about the line formed by left side of the roof by 20 degrees, into the screen. So, the initial and final picture should look something like

    Write the sequence of transformations required to do this. Use the T, R, etc., notation. Describe in detail how you came up with the rotation matrix for aligning the roof side with one of the axes. Then write down what each matrix is. You don't have to compose all of the matrices this time, but do apply the sequence of transformations to the door knob coordinates and report what you get as an answer.