UM EECS 487 Fall 2008
Homework Assignment 2
DUE DATE Mon, 10/06/2008, 1:40 pm in lab
September 22, 2008
Show your work for all the questions below. If you don’t show your work, you will get only a third
of the points if your answers are correct.
1. Generalized Shear (10 pts)
Consider a unit cube between (0,0,0) and (1,1,1).
(a) What is the matrix that shears the top surface by a displacement (∆x, ∆y, 0) keeping the
bottom surface fixed? (2 pts)
(b) What is the matrix that shears the edge (1,1,0) −→ (1,1,1) by a displacement (0,0,∆z)
keeping the edge (0,0,0) −→ (0,0,1) fixed? (2 pts)
(c) What is the matrix that stretches the cube so that the point (0,0,0) is fixed, the point (1,1,1)
is displaced by (∆x, ∆y, ∆z)and all other vertices are displaced proportionally based on their
initial distance from the origin along the body diagonal (1,1,1), forming a 3D rhomboid? (6
pts)
2. Advanced Deformation (15 pts)
You are given a function named canonicalSquare() that draws a wire-frame square (using
GL LINES, for example) between (0,0,0) and (1,1,0).
(a) Using only this geometric primitive and transformation matrices, generate the figure shows in
Fig. 1. You may use any arbitrary 4×4matrix necessary. (12 pts)
(b) Having completed step 1, what is the quickest way to align the axis through the figure so that
it is now along the (1,1,1) direction instead of the z−axis as shown in the figure? (3 pts)
Note that the kind of deformation shown in Fig. 1 will only work if with the wire-frame and not with
a solid surfaces (created using GL QUADS, for example). Polygons must be decomposed into trian-
gles before rendering and for quadrilaterals, an arbitrary diagonal is picked to create 2 constituent
triangles per surface. In Fig. 1 the planarity of all quadrilateral surfaces and within each triangle
surfaces is destroyed in step 4. To get Fig 1 with lines only we only need transform the end-vertices
whereas while trying compose it using triangles, information about the new vertex at the intersec-
tion of the lines connecting the two square faces is missing. Without it a basic engine like OpenGL
will continue to draw triangles connecting the new vertex locations and we get a slightly different
solid that looks like the intersection of rectangles along face-diagonals and between opposite edges
in the original cube.
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