# CSE167 Introduction to Computer Graphics Final exam Wednesday June 13 Matthias Zwicker Please include all steps of your

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```					             CSE167, Introduction to Computer Graphics
Final exam, Wednesday June 13
Matthias Zwicker

Please include all steps of your derivations in your answers to show your understanding of the
problem. Try not to write more than the recommended amount of text. If your answer is a mix of
correct and substantially wrong arguments we will consider deducting points for incorrect state-
ments. There are thirteen questions for a total score of 100 points.

Your name:

1. Given two points in the two-dimensional plane, p0 = (2, 1) and p1 = (4, 3), that deﬁne a
line. Write down the implicit line equation. Remember that this is a scalar-valued function
f (p), p ∈ R2 that returns the signed distance of a point to the line. This means f (p) is zero
for all points p on the line, negative for all points on one side of the line, and positive for all
points on the other side. 6 points

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2. Assume you are working with three coordinate systems: object, world, and camera space.
The basis vectors of object space have world coordinates (1, 0, 0), (0, 0, −1), and (0, 1, 0).
The origin of object space has world coordinates (0, 0, 10). The basis vectors of camera space
√      √     √         √         √             √        √     √
have world coordinates (1/ 3, 1/ 3, 1/ 3), (−1/ 2, 0, 1/ 2), and (1/ 6, −2/ 6, 1/ 6).
The origin of camera space has world coordinates (5, 9, −4). Write down the 4 × 4 matrices
that transform object to world coordinates, camera to world coordinates, world to camera
coordinates, and object to camera coordinates. 10 points

2
3. Given the perspective view frustum shown in the ﬁgure below. The top bounding plane of
the view frustum is determined by the plane going through the points (0, 0, 0), (1, 1, −1), and
(−1, 1, −1) in camera coordinates. Note that the other bounding planes will not be relevant
to this problem. In addition, there is an object coordinate system deﬁned by basis vectors
(0, 1, 0), (1, 0, 0), (0, 0, −1) and the origin (1, 3, −8) in camera coordinates. Note that the
order of the basis vectors matters!
Assume there is an object with a bounding sphere with radius 2 centered at (8, 1, 1) in object
coordinates. Determine if this bounding sphere intersects with the top bounding plane of the
view frustum. You should do this by transforming the center of the bounding sphere from
object to camera coordinates. Then you need to compute the distance from the bounding
sphere center in camera coordinates to the top bounding plane. 10 points

3
4. The ﬁgure below shows the CIE RGB matching curves. Describe how these curves were
determined. Your answer should include an explanation of the meaning of the three values
r(λ), g(λ), b(λ) for any given wavelength λ. It should also explain how to interpret negative
values (5-6 sentences). 10 points

4
5. The Blinn shading model is given by the expression
s
c=       cli (kd (Li · n) + ks (hi · n) ) + ka ca .
i

Explain the meaning of all the terms (i.e., c, i, cli , kd , etc.) in this equation. 10 points

5
6. Sketch the graphs of the cubic Bernstein polynomials. Make sure to indicate the domain and
the range of the polynomials in your sketch. In addition, list the three main properties of
Bernstein polynomials. 8 points

6
e
7. The ﬁgure below shows the control points of a single segment of a B´zier curve. What is the
degree of the curve? Sketch the evaluation of the curve at (approximately) t = 0.75 using
the de Casteljau algorigthm. 6 points

e
8. List the two main disadvantages of B´zier curves and surfaces, which are overcome by the
generalization to NURBS curves and surfaces. 6 points

7
9. Given a cone that has its apex at (0, 0, 3), and whose intersection with the xy-plane is a
unit circle. Write a parametric equation of the form p(u, v) to describe the surface of this
cone. Remember that p(u, v) consists of three functions x(u, v), y(u, v), and z(u, v). The
curve p(u, 0), u ∈ [0 . . . 1] should map to a unit circle in the xy-plane, and p(u, 1), u ∈ [0 . . . 1]
should be the apex.
In addition, derive equations for two tangent vectors and the normal at any point (u, v). You
do not need to normalize the normal vector to unit length. 12 points

8
10. A bilinear patch p(u, v) is given by four control points p0 = (2, 1, 1), p1 = (5, 2, 3), p2 =
(3, 3, 0), and p3 = (6, 4, 4). Evaluate the patch at p(0, 0), p(1, 0), p(0, 1), p(1, 1), and
p(2/10, 5/10). 6 points

9
11. Describe the shadow mapping algorithm using a sketch and a few explanatory sentences. List
two potential problems or artifacts that may appear with shadow mapping. 10 points

12. The following L-system describes a variant of the Koch curve that uses only right angles. The
L-system has a single variable F , two constants + and −, and the rule F → F +F −F −F +F ,
where F means “draw forward”, + means “turn left ninety degrees”, and − means “turn
right ninety degrees”. The starting sequence is F . Draw this curve for one and two levels of
recursion. 6 points

10

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