Geometric Primitives - 3D Math Primer for Graphics and Game Development - page 340

2. Give the slope and y-intercept of the 2D line defined implicitly by 4x+7y =
42.
3. Consider the set of five points:

v 1 = (7, 11,−5), v 2 = (2, 3, 8), v 3 = (−3, 3, 1),

v 4 = (−5,−7, 0), v 5 = (6, 3, 4).

(a) Determine the AABB of this box. What are p min and p max ?
(b) List all eight corner points.
(c) Determine the center point c and size vector s.
(d) Multiply the five points by the following matrix, which we hope you
recognize as a 45 o rotation about the z-axis:

2 3

0.707 0.707 0

4 5

M = −0.707 0.707 0 .

0 0 1

(e) What is the AABB of these transformed points?
(f) What is the AABB we get by transforming the original AABB? (The
bounding box of the transformed corner points.)

4. Consider a triangle defined by the clockwise enumeration of the vertices
(6, 10,−2), (3,−1, 17), (−9, 8, 0).

(a) What is the plane equation of the plane containing this triangle?
(b) Is the point (3, 4, 5) on the front or back side of this plane? How far
is this point from the plane?
(c) Compute the barycentric coordinates of the point (13.60,−0.46, 17.11).
(d) What is the center of gravity?
(e) What is the incenter?
(f) What is the circumcenter?

5. What is the best-fit plane equation for the following points, which are not
quite coplanar?

p 1 = (−29.74, 13.90, 12.70) p 4 = (14.62, 10.64,−7.09)

p 2 = (11.53, 12.77,−9.22) p 5 = (−3.31, 3.16, 18.68)

p 3 = (9.16, 2.34, 12.67)

6. Consider a convex polygon P that has seven vertices numbered v 1 . . . v 7 .
Show how to fan this polygon.