Homework 4
CS 354, BME 345
1. (15 points) Consider a line from point (0.1, 0.7) to (5.3, 2.3). Compute which pixels should be filled in
according to Bresenham’s algorithm by filling out the table below. Show your work.
k x d y
0 1/2
1 3/2
2 5/2
3 7/2
4 9/2
5 11/2
2. (15 points) You wish to determine the two intersection points between a ray R and a sphere S. Ray
R starts at (1, 2, 1) and heads in the direction (0.3, 0.8, 0.5). Sphere S is centered at the point (6, 7, 4)
and has a radius of length √
26. At what two points does the ray intersect the sphere? Which point is
closer to the origin of the ray? Hint: First parameterize the ray as R(t) and then solve for the value
of t for each point of intersection.
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3. (10 points) Consider 4 squares each with side length = 2. Their center points and normals are given
below:
name center normal
A (0, 1, 1) (1, 0, 0)
B (1, 0, 1) (0, 1, 0)
C (0, 3, 1) (1, 0, 0)
D (1, 1, 1) (−1, 0, 0)
Let fAB be the form factor between squares A and B. Let the other form factors be named similarly.
Give fAB, fAC , and fAD in sorted order from least to greatest. Explain why they are sorted in that
order. You do not need to compute the form factors.
4. (10 points) A quadratic Bezier curve has the form
P(t) = (1 − t)
2P0 + 2t(1 − t)P1 + t
2P2 for 0 ≤ t ≤ 1
Given control points P0 = (0, 0), P1 = (0, 1), P2 = (1, 0), sketch the Bezier curve. Label and give exact
coordinates for points at t = 1/4, t = 1/2, and t = 3/4.
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