CIS881
Homeworks

• Discussion of how to do homeworks among students in the class is acceptable, even encouraged - but when it comes time to write it up, do your own work.
• Homeworks are due at the beginning of class - we will discuss the answers the day that you hand in the homework. Late homeworks will not be accepted!

Homework #1
Due: Wednesday, April 28

1. Derive a modified Hermite curve based on the following information (the diagram shows Hermite, but use Bezier formulation).
   • P(0) = P0
   • P'(0) = P0'
   • P''(0) = P0''
   • P(1) = P1
   Give the matrix form P(u) = UMB What is P'(1)?

   Basic idea:

   • p(u) = au³ + bu² + cu + d
   • p(0) = d = P0
   • p'(0) = c = P0'
   • p''(0) = 2b = P0''
   • p(1) = a + b + c + d = a + P0''/2 + P0' + P0 = P1
   • a = P1 - P0''/2 - P0' - P0
   • B = [ P0 P1 P0' P0'']^T
   • M =
     

     -1	1	-1	-1/2
     0	0	0	1/2
     0	0	1	0
     1	0	0	0

     
     
   • P'(u) = [3u² 2u 1 0]MB
   • P'(1) = [3 2 1 0]MB
   • P'(1) = 3(P1-P0) -2P0' - (1/2)P0''
   
   
2. For a blended parabolic curve passing through points P0, P1, P2, etc., what is the tangent of an internal control point equal to? Give is as an equation in terms of the other control points. See the diagram.
   • for any set of four control points: C(t) = (1/2)TMP
   • the curve runs between two internal control points from C(0) to C(1)
   • P = [ P_i-1 P_i P_i+1 P_i+2 ]
   • T = [ t³ t² t 1 ]
   • M =
     

     -1	3	-3	1
     2	-5	4	-1
     -1	0	1	0
     0	2	0	0

     
     
   • C'(t) = (1/2)T'MP
   • T' = [3t² 2t 1 0]
   • C'(0) = (-P_i-1 + P_i+1)/2
   • You can also use the equation for a single parabolic arc since the tangent at an internal control point is constructed so that the tangent there is maintained. However, the magnitude of the tangent will differ by a factor of 2 (virtual extra point question: why?)
3. How would you fit a cubic Bezier curve through 4 given points: Q0, Q1, Q2, Q3? (where P(0)=Q0 and P(1)=Q3). Give a procedure to form the Bezier control points of P0, P1, P2, P3 which would pass through the four points Q0, Q1, Q2, Q3.
   
   • First come up with estimates for the u values for Q1 and Q2
     • u₁ = |Q1-Q0|/(|Q1-Q0|+|Q2-Q1|+|Q3-Q2|)
     • u₂ = (|Q1-Q0|+|Q2-Q1|)/(|Q1-Q0|+|Q2-Q1|+|Q3-Q2|)
   • Two possible ways to go from here
     1. User four point method and convert to Bezier
        • Use the four point method from the book to derive p(u)=UM_FQ (Section 3.7, p. 61) where Q = [ Q0 Q1 Q2 Q3 ]^T
        • Convert to Bezier
          • p(u) = UM_FQ = UM_BP where P = [ P0 P1 P2 P3]^T
          • M_FQ = M_BP
          • P = M_B^-1M_FQ
     2. Alternative Solution: set up two cubic Bezier equations and two unknows and crank through the solution for P1 and P2
        • p(u₁) = UM_BP
        • p(u₂) = UM_BP
4. In Catmull-Rom, the tangent at an interior point is totally determined by the position of adjacent points. Consider these sets of three sequential points:
   1. (0,0), (5,2), (10,0)
   2. (0,0), (1,2), (10,0)
   3. (0,0), (10,2), (10,0)
   In each of these cases the tangent computed at the middle point is the same. Discuss the problems this might create and come up with other suggestions on ways of computing internal tangents and discuss why they might be better.
   
   • For direction of tangent (T):
     • Calculate bisector of angle made by vectors
       • A = P_i-1 - P_i
       • B = P_i+1 - P_i
       • C = A/|A| + B/|B|
     • Calculate normal to plane of A and B: N = A x B
     • T = N x C
   • For length, use something like a fraction of min(|A|,|B|)
5. I want to try to create a cubic Bezier segment which approximates a semicircular arc. I start out with P0=(0,0), P3 = (2*d,0), P1=(0,y), P2 = (2*d,y). What should 'y' be in order for P(0.5) = (d,d)? What is the value of P(0.25) and how much does it deviate from the desired semicircle?
   
   • p(u) = UMP
   • P = [ P0 P1 P2 P3 ]
   • U = [ u³ u² u 1 ]
   • M =
     

     -1	3	-3	1
     3	-6	3	0
     -3	3	0	0
     1	0	0	0

   • we want the y coordinate of the p(0.5) point to be equal to d by adjusting the y coordinates of P1 and P2 given that the y coordinates of P0 and P3 are zero.
   • p_y(0.5) = d = [.125 .25 .5 0]M[0 y y 0]^T
   • d = -3y*.25 + 3y*.5 = -.75y + 1.5y = .75y
   • y = (4/3)d
   • p(0.25) = < you can plug in the values and crank out the result >
   • compare distance of p(0.25) to the point (d,0) - remember that p(0.25) is not guaranteed to be at a 45% angle from the center point!
6. For an open, uniform B-spline curve with k=4, what is P(0)? Give it as an equation in terms of the control points and use geometric construction to show it's position diagrammatically.
   
   • p(u) = UM_SP
   • P = [ P0 P1 P2 P3 ]
   • U = [ u³ u² u 1 ]
   • M = (1/6) *
     

     -1	3	-3	1
     3	6	3	0
     -3	0	3	0
     1	4	1	0

   • p(0) = [0 0 0 1]M_SP = (1/6)(P0+4P1+P2)
   • as James said in class: p(0) = P1 - (1/6)(P0-P1) - (1/6)(P2-P1)

Homework #2
Due: Wednesday, May 26

1. Given an open, uniform, cubic B-spline with control points p₀, p₁, ...., p_n, for an interior control point, p_i, derive the equation for the point on the curve which is maximally influenced by p_i. (Hint: refer to Figure 5.7 on page 130). Show how you derived the equation.
2. Given: a cubic Hermite curve defined by three points p₀, p₁, and q where p₀' = a*(q-p₀) and p₁' = b*(p₁ - q). Determine what a and b have to be in order for p₀ and p₁ to be points of inflection (i.e., for p^u and p^uu to be colinear at 0 and 1. (Hint: derive expressions for p^u and p^uu at the endpoints and see what values of a and b make the respective vectors colinear - other than a=b=0.)
   
3. Given a surface definition, p(u,w) = UMBM^TW^T and a position on that surface defined by specific values for u and w, how would you determine which way a ball bearing would roll if placed at that position. Assume y is up. Give the direction in terms of both parametric coordinates (u,w) as well as model space (x,y,z space in terms of p(u,w), p^u(u,w), etc.) .