Exercises Chapter 19

1. Sketch the path of the curve (t2,1+t) for t in the range [0,1].


2. Which of the following are multi-affine maps?

(a) f(t) = 1 + t + 3t

(b) f(t) = 1 + 4t + t2

(c) f(t1 ,t2 ) = -t 1 t2

(d) f(t1 ,t2 ,t 3 ) = 1 + 5t1 t2   + t1  + t3

(e) f(t1 ,t2 ,t 3 ,t4 ) = 1 + t1   + t1 t2  + 5t2 t3 t 4  - 11t1 t2 t3 t4

(f) f(t1 ,t2 ,t 3 ) = 1 + t1  + t 1 t2  + 5t2 t3 t4  - 11t 1 t2 t3 t4

(g) f(t1, t2, t 3 ) = 4 + 3t1  + 3t 2  + 3t3  + 6t 1 t2  + 6t1 t3  + 6t2 t 3  - t1 t2 t3


3. In general, how many terms are there in a multi-affine map with 4 arguments?


4. Consider a general 3 argument multi-affine map:

f(t1 ,t2, t 3) = c0  + c1 t1  + c2 t 2+ c3 t3 + c4 t1 t 2  +  c5 t1 t3+ c6 t2 t3+ c7 t1 t2 t3

Show that for this to be symmetric we must have c1 = c2= c3, and c 4= c5= c6 .


The following two exercises can be done without explicitly computing the affine functions involved:

5. Suppose f(t1, t2) is a multi-affine symmetric function, and r,s are numbers with r < s. The following values are known: x0  = f(r,r), x1  = f(r,s) and x2   = f(s,s). Suppose t is any number between r and s. Find f(t,t).

6. Similarly, suppose f(t1 ,t2 , t3) is a symmetic multi-affine function, and r < s, and the following values are known: x0 = f(r,r,r), x1 = f(r,r,s), x2   = f(r,s,s), x3  = f(s,s,s). Find f(t,t,t) where t is any number between r and s.


7. Construct the de Casteljau diagram for a 4 degree blossom given initial values f(r,r,r,r), f(r,r,r,s), ..., f(s,s,s,s) - in other words show how to evaluate f(t,t,t,t) in this case.


8. Consider the 2 parameter blossom with given values x0   = f(r,r), x1  = f(r,s)  and x 2  = f(s,s), and take the situation where r = 0 and s = 1. Find an explicit formula for f(t,t) in terms of x0 , x1 and x2 . This will be a second degree polynomial in t. Repeat the same calculation for the 3 parameter case.


9. Suppose we are given the parametric curve defined by:-

P(t) = (X(t),Y(t)) = (1 + t + t2, t 3)

with t in the range 0 to 1. Find the Bezier control points for this curve.


10. In the case of the degree 2 (quadratic) and degree 3 (cubic) Bezier curves, express P(t) in terms of the control points pi , by finding the polynomials in terms of t. Do this only for the case r = 0 and s = 1. Infer and prove the formula for an nth degree curve.


11. Given the curve defined over the range t in [0,1]

P(t) = (X(t),Y(t))  = (1 + 3t + 9t2  + 5t3  , 1 - 2t + t2   - 2t3)

find the tangent vector P'(t) = (X'(t),Y'(t)).

Use this, together with the start P(0) and end P(1) of the curve to roughly sketch the curve.


12. Find the Bezier control points of the curve in 11., and again sketch the curve.


13. (a) Given the Bezier curve based on control points (0,0) (0,1) (1,1) find the control points of the equivalent Bezier curve one degree higher.
  
(b) Once again, raise the degree, so that there are five control points for a Bezier curve of the same shape as the original quadratic curve.

(c) What are the advantages/disadvantages of degree raising as a method for introducing more flexibility into curve design?

(d) What would happen in the limit if you continued to degree raise again and again?

Note in answer to the last question you might come to the conclusion that each time there is a degree raise, the control polygon gets closer to the shape of the actual Bezier curve. While true, this does not lead itself to a method for rendering Bezier curves, since the convergence is exceedingly slow.


14. Suppose an additional knot is inserted into the sequence of knots in the case of a cubic B-Spline curve. Illustrate with a diagram the relationship of the new control points for the relevant segment of the curve in relation to the original control points.


15. Given the control points (0,0), (1,1), (1,2), (2,3), (3,4) and a uniform knot vector but adjusted so that the cubic B-spline curve starts at (0,0) and ends at (3,4), insert two knots, 1.2, 1.8 between the knots 1 and 2, using the Oslo algorithm.


16. Suppose in a k degree B-spline, m<k knots were to be introduced in a particular interval. These could be inserted one at a time using Boehm's algorithm, or simulataneously using the Oslo Algorithm. Which would be more efficient in terms of number of operations?


17. Consider a bi-cubic Bezier surface F(t,u) and find the partial derivatives of F with respect to t and u. Discuss how this could be used to find the normal at a point on the Bezier surface, and why this is important for rendering.

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Last modified: Tue Oct 02 22:44:15 GMT Daylight Time 2001