Problem 1. Given a triple of noncollinear points P0 , P1 , and P2 in the plane, any other point Q in the plane
can be expressed uniquely as an affine combination of these three
Q   0 P0  1 P1   2 P2
(where  0  1   2  1)
The triple ( 0 ,  1 ,  2 ) of scalars is called the barycentric coordinates of Q relative to the Pi ' s .
For example, in the figure below, Q coordinates are (0.5,0, 0.5).
P2
Q
P1
P0
(a) Give a drawing that illustrates the regions of the plane that are associated with each of the possible sign
classes (+ or -) of the 3 barycentric coordinates. That is, label the region in which all 3 coordinates are
positive with (+, +, +), label the region in which only the first coordinate is negative with (-, -, -). etc.
(b) Which sign class(es) are missing from the diagram? Why?
Problem 2. Using the vector representation of a quaternion as a generalized complex number

q  ( s, u )  s  u x i  u y j  u z k .
where the basis values i, j, and k satisfy the following multiplication rules:
i 2  j 2  k 2  1
ij  k , jk  i, ki  j
ji  k , kj  i, ik   j.
Prove that
 

  
q1q2  ( s1s2  (u1  u2 ), s1u2  s2u1  u1 x u2 )
(hint: Express q1 and q2 using the above form, multiply everything out, and collect terms that share
common basis values. Be very careful in your algebra, or it will take forever to get it right.)
Problem 3. It is a fact that projective transformations map conic (e.g. circles, ellipses, parabolas, or
hyperbolas) into conics. Consider the following projective transformation, which maps projective 3-space
to 2-space: f ( x, y , z )  x / z, y / z ) . If f is applied to a circle in 3-space, and a hyperbola is the result, what
can be said about the location of the circle, relative to the projection plane? How about if the circle is
mapped into a parabola?
Problem 4. Prove that translation in the plane preserves affine combinations. That is, given the transformation
T, defined T ( x, y )  ( x  t x , y  t y ) , and given any two points P  ( p x , p y ), Q  ( q x , q y ) and any scalar  to
prove that:
T ((1   ) P  Q )  (1   )T ( P )  T (Q ).
Problem 5. Consider the plane H in 3-space, defined by the following equation
H : ax  by  cz  d  0,
for some nonzero constants a, b, c, and d. Suppose that we project this plane using our standard projective
transformation
P( x, y, z )  ( x /(  z ), y /(  z ), 1),
which projects the points onto the projection plane z=-1. Points at infinity on plane H are mapped to a horizon
line on the projection plane. As a function of a, b, c, and d, derive the line equation (on the x, y-projection
plane) for the horizon line.
(Hint: Consider the projections of any two points at infinity on H).
Horizon line
Problem 6. Suppose that we join two Bezier curves of order 2 end-to-end, using the control points sequence
P0 , P1 , P2 , and P2 , P3 , P4 , respectively. Exactly what conditions must be satisfied by these five points for the
combined curve to have C 1 parametric continuity at the point at which they are joined. Prove your answer
carefully by showing the continuity of the derivatives at this point.
P1
P4
P2
P
0
P
3
Problem 7. Show that for u = 0 the tangent vector for the cubic Bezier curve defined by four points
P0 , P1 , P2 , P3 is collinear with the line segment P0, P1 .
Problem 8. Consider the rotation that maps shape A to B shown in the figure.
(a) Give a set of Euler angles that represents this rotation (Hint: Rotate about x.)
(b) Give a different set of Euler angles that represents this rotation. (Hint: Rotate
about y and z.)
(c) If we interpolate along the rotation given in (a) and interpolate along the rotation
(b) will the intermediate result of the interpolations be the same? Explain.
(d) Give a quaternion that represents the rotation from A to B.
B
z
y
x
A
Problem 9. Give an affine transformation that maps the 400 x300 viewport shown below left to the 8 x 6
window shown on the right (Note the orde reversal along the y-axis). You may express your transformation
either in equation form or as a 3 x 3 homogeneous matrix.
0
400
4
-4
0
3
300
-3
Problem 10. short answer question. Except where requested, no explanation is needed, but may be supplied for
partial credit.
Q
n
P2
P1
(1-a)P+aQ
P
P0
x=b
Problem (c )
(a) Consider the line y  2 x  5 in the 2-dimensional projective plane. What are the homogeneous
coordinates of the (infinite) endpoints of this line?
(b) You are given a vertical line x=b and a pair of points P and Q in the plane. As a function of b and the
coordinates of P and Q, compute the affine combination of P and Q that lies on this vertical line.
(c) Suppose that you have a triangle in 3-space, defined by the three point P0, , P1 , and P2 . As a function of

these three points, derive a formula for a normal vector n that is normal to this triangle. The normal
vector should be directed so that from a viewer on the same side as the normal vector, the points appear
in counterclockwise order.
(d) You want to know whether a point P lines on a given surface. From which representation of the surface
is this question easier to answer: implicit or parametric?
Consider the limit of the sequence shown in the figure below. What is the fractal dimension of the final object?
(e) Give the three blending functions for a Beizier cure of order 2
(f) Suppose that a triangle is clipped to a rectangular window. After being clipped against the window,
what is the maximum number of sides that the resulting clipped polygon might have? Draw an example
to illustrate.
(g) State clearly which properties of the Bezier blending functions, bi ,d (u ), guarantee that the Bezier curve
lies within the convex hull of the control points.
(h) Suppose that the polygon shown in the figure below is scan converted using the KrsU method. For
each of the points A,B, C, D, and E, indicate weather or not it will be drawn by this algorithm.
A
E
D
B
C
(i) Consider the transformation that maps a square into the trapezoid (shown above).
transformation affine? Explain.
Is this
(j) Consider the line 0  2 x  3 y  5 in the 2-dimensional projective plane. What are the homogeneous
coordinates of the (infinite) endpoints of this line?
(k) B-splines possess a property called local support. What is local support, and why is this property
desirable?
(l) Consider the quaternion (1 2 ,0, 1 2 ,0) . Is this a unit quaternion? Is it a pure quaternion? What is
the conjugate of this quaternion?
(m) Finish the following: “The complex point c=a+bi lies in the Mandelbrot set if and only if the Julia set
generated by c is __________________.
(n) What is a rational parameterization? Give an example of a shape that has a rational parameterization
but no polynomial parameterization? (You are not required to give the parameterization, just the shape.)