Angel and Shreiner: Interactive Computer Graphics, Seventh
Edition
Chapter 5 Odd Solutions
5.1 Eclipses (both solar and lunar) are good examples of the projection of
an object (the moon or the earth) onto a nonplanar surface. Any time a
shadow is created on curved surface, there is a nonplanar projection. All
5.3 Suppose that we want the view of the Earth rotating about the sun.
Before we draw the earth, we must rotate the Earth which is a rotation
about the y axis. Next we translate the Earth away from the origin.
5.5 Yes. Any sequence of rotations is equivalent to a single rotation about
a suitably chosen axis. One way to compute this rotation matrix is to form
the matrix by sequence of simple rotations, such as
5.7 Consider the line determined by the points (x1, y1, z1) and (x2, y2, z2).
Any point along can be written parametrically as
5.9 The specification used in many graphics texts is to use the angles the
projector makes with x,z and y, z planes, i.e the angles defined by the
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5.11 The CORE system used this approach. Retained objects were kept in
distorted form. Any transformation to any object that was defined with
5.15 If we use θ=φ= 45, we obtain the projection matrix
1 0 −1 0
5.17 All the points on the projection of the point (x.y, z) in the direction
dx, dy, dz) are of the form (x+αdx, y +αdy, z +αdz). Thus the shadow of
the point (x, y, z) is found by determining the αfor which the line
intersects the plane, that is
axs+bys+czs=d
Substituting and solving, we find
adx+bdy+cdz
However, what we want is a projection matrix, Using this value of αwe
find
xs=z+αdx=x(bdy+cdx)−dx(d−by −cz)
adx+bdy+cdz
with similar equations for ysand zs. These results can be computed by
multiplying the homogeneous coordinate point (x, y, z, 1) by the projection
matrix
bdy+cdz−bdx−cdx−ddx
−adyadx+cdz−cdy−ddy
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5.23 Probably the simplest approach would be the one in the previous
problem of repeating vertices at the end of each row to create a degenerate
polygon.
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