CS 3744, Assignment 3 - Due February 16, 2010

Please show all your work!

  1. Multiply the 2×3 matrix {{1, 0, -2},{4, 2, -1}} by the 3×3 matrix {{1, -1, 0},{0, 6, -2},{2, 2, 4}}.

  2. Let T be the 2D transformation matrix T = {{2, 0, -1},{0, 4, 4},{0, 0, 1}} and let P = (x, y) be a point in 2D.

    a) Write the algebraic equations corresponding to the matrix equation P' = T P, where P' is the 2D point (x', y').

    b) From your answer to a), write the algebraic equations for computing P from P'.

    c) From your answer to b), write the matrix equation for computing P from P'.

  3. Determine the rank of the matrix {{3, -6, -9},{1, -1, 1},{2, -5, -10}} (no guessing... work it out!).

  4. Suppose we are given the following graphic configuration, and assume the viewport is a square 100 units on a side and centered at (200, 200). Since the drawing is not necessarily accurate, you should rely on the specified coordinates. The rectangular area shown in world coordinates is the window.

    a) Find the instance transformations T1 and T2 from master coordinates to instances 1 and 2.

    b) Find the transformation from World to Screen coordinates, W (assume that the ys axis points up and is not inverted).

    c) Find the transformation from Screen to World Coordinates, W-1.

    d) Draw a carefully labeled diagram of the object in the viewport.

    e) Find the image of the viewport in Master coordinates for Instance 2.

  5. Suppose a point P in 2D undergoes a CCW rotation of 30°, then a translation by (-2, 1), and finally a scaling by (2, 2) to compute a vector Q. Give the product of matrices that implements this transformation. Then give the product of matrices that can be used to compute P given Q. Do not multiply out the component matrices. Finally, determine the forward transformation of the point P = (2, -1) and the inverse transformation of the point Q = (-2, -1).

  6. Suppose we have two basis sets, {v1, v2, v3} and {u1, u2, u3}. Suppose that u1 = (3 v1 + 2 v2 + v3), u2 = (-v1 + 2 v2 - v3), and u3 = (6 v1 - v2 + 2 v3). If v = (3, 4, -2) wrt the v-basis, determine the corresponding vector in the u-basis.

This assignment is due in class on the date above.