CS 418: Homework #3
Assigned: April 17, 2006
Due: in class April 25, 2006
Please be organized when writing your answers to these questions. Make sure that all solutions are clearly indicated
and labelled with the question they are answering. Remember to write clearly and legibly. Unreadable answers will
receive 0 credit.
Hint: For questions that ask you to derive equations, work through the solution using matrix/vector equations.
Only expand vector equations in terms of individual coefficients if you want to do a lot of extra tedious work.
1. (20 points) You are given three RGB colors with corresponding alpha values:
๎๎๎๎๎๎๎
๎๎๎๎๎๎๎๎
๎๎ ๎๎๎๎๎
๎๎ ๎๎๎
,
๎
๎๎๎๎๎๎๎๎๎๎๎๎๎
๎๎
,
and
๎
๎๎๎๎๎
๎๎๎๎
๎๎๎๎๎๎
with alpha values
๎๎๎
๎ ๎
๎๎๎๎๎๎
๎"!
๎ ๎
๎๎ #
, and
๎%$
๎&๎
๎๎'๎
.
(a) Compute the color and alpha value for (
๎
over
๎
over
๎
).
(b) Compute the color and alpha value for (
๎
over
๎
over
๎
).
2. (20 points) Consider the (uncapped) unit cone whose apex is the origin and whose radius and height are both 1.
y
z
{
{
radius๎
r = 1
height๎
h = 1
x
(a) What is the parametric form of this surface? Make sure to indicate what your parameter values mean and
what their ranges are.
(b) Briefly describe how to generate this object as a generalized cylinder.
(c) What is the implicit form of this cone? You can assume that the coordinates will always be restricted to the
range
๎
)(+*,๎๎-.๎๎/0(1๎
.
3. (20 points) Suppose we want to construct a model of an ellipsoid centered at the origin, whose axes are the
*,๎๎-2๎๎/
coordinate axes and whose radii are
3
๎546๎๎7
, respectively.
(a) Write an implicit equation for this ellipsoid.
(b) Convert this equation into the standard quadratic form
8:92;<8
๎ ๎
.
(c) Show how to derive this matrix
;
via a transformation of the quadratic form for the unit sphere.
4. (20 points) Suppose that you are given a point
=
๎>๎?*@๎A-2๎๎/B๎
and a plane
C:DFE<GIH
๎&๎
where
J๎CKJ
๎L๎
.
(a) Let
M
be the perpendicular projection of
=
onto the plane. Derive an expression of the location
M
in terms of
=
,
C
, and
H
.
