W5

Written Homework 5

Due at 9:30am, November 14th

Question 1: Bezier Curves / DeCasteljau Algorithm

• 1A) Given the 3rd order (2nd degree) Bezier curve segment with control points (0,0), (0,8), (8,0), divide the curve into two new Bezier segments at (u=.25). Your answer should give the 3 control points for each of the 2 curve segments.
• 1B) Given the 5th order (4th degree) Bezier curve segment with control points (0,0), (0,4), (4,8), (8,4), (8,0), divide the curve into two new Bezier segments at (u=.5). Your answer should give the 5 control points for each of the 2 curve segments.

Question 2: Bezier curves, continuity

• 2A) We have a 3rd order (2nd degree) Bezier curve segment with control points (0,0), (0,4), (4,0). We want to connect a degree 3 (cubic) Bezier to this curve so that the two curves form a closed loop (the beginning of one curve is the end of the other), that has C(1) continuity. Where would the 4 control points of the curve go?
• 2B) We have a 5th order (4th degree) Bezier curve segment with control points (0,0), (0,2), (2,4), (4,2), (4,0). We want to connect a degree 3 (cubic) Bezier to this curve so that the two curves form a closed loop (the beginning of one curve is the end of the other), that has C(1) continuity. Where would the 4 control points of the curve go?

Question 3: B-Splines

We will use the notation as in the text (see 15.6) where blending function i is defined to have its non-zero "range" begin at i.

Consider a uniform, quadratic B-Spline (k=3) like on page 337 of Shirley. We will start numbering the points from 0 to n-1, so that the knot values are [0,1,...n+2]. (note: this is different from the book since in the book most things begin at 1).

We will make a curve with 7 control points. The points are at (0,0), (1,1), (2,3), (3,3), (4,2), (5,2), (6,0). Therefore, the curve is defined over the range t=2 to 6 (since we started at zero.

• 3A) What will the values of each of the 7 blending functions be for t=2.5? What will the value of the curve be?
• 3B) For all of the integer values of t in the range [2,7] inclusive, what will the value of the curve be?

Question 4: Matrix Form of B-Splines

If you were to encode the Quadratic (k=3) B-Spline as a matrix, what would the matrix be?

Question 5: Local Lighting

In the scene below, there is a single directional light shining downward (the negative Y direction). We are looking at a single point at the origin, and that point is on a polygon rotating around the z axis (which is pointing towards you in the picture). Notice that the angle is measured clockwise (so in the picture A is 45 degrees).

Suppose that the light has intensity 1 for its diffuse and specular components, and .5 for its ambient component. The light is white.

If the material of the polygon has diffuse color red (1,0,0), specular color green (0,1,0) and ambient color blue (0,0,1), what color would the object be when:

You can use either the half-angle or the relfectance vector for the specular component. Be sure to state which one you do (since you get different answers)

• 5A) A = 0 (e.g. horizontal)
• 5B) A = 45 degrees (as in the picture)
• 5C) A = 90 degrees