9. [20 points] A degree three Bezier curve q(u) has the four control points p 0 = 0,0, p 1 = 2,0, p 2 = 0,2, and p 3 = 4,4.

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1 Name: 8 7. [10 points] A color has RGB specification of R = 1 and G = 1 2 and B = 3 4. (R,G,B color values are in the range 0 to 1.) What is the hue value (H) of this color? Express the hue by a value in the range 0 to [20 points] Describe mipmapping. What is mipmapping? What is it useful for? What problems does it help solve? When does mipmapping not work so well?

2 Name: 9 9. [20 points] A degree three Bezier curve q(u) has the four control points p 0 = 0,0, p 1 = 2,0, p 2 = 0,2, and p 3 = 4,4. (a) Sketch and label the control points and the control polygon for the Bézier curve. (b) Use the de Casteljau algorithm to compute the the point q( 1 2 ). (c) Sketch the Bézier curve approximately. For both (a) and (c): Draw your sketch to scale as best you can and, when drawing the Bézier curve, show clearly the initial and final points and the initial and final slopes.

3 Name: [20 points] (Catmull-Rom interpolation.) Let p 0 = 0,0, p 1 = 2,0, p 2 = 0,2, p 3 = 2,0, and p 4 = 0, 2. (a) Draw the Catmull-Rom curve that is defined by these points. Be sure to show clearly the starting point, ending point, and the slopes of the curve at each point p i on the curve. p 2 p 3 p 0 p 1 p 4 (b) What are the control points for the first Bézier segment of the Catmull-Rom curve?

4 Name: 2 1. [20 points] This problem concerns transformations in R 2. Suppose you are given a function DrawCircle() that draws a unit circle centered at the origin (radius equals one). Give a code fragment that will draw an ellipse as shown in the figure. The length of the ellipse is l and the width is w. One endpoint of the ellipse is at x 0,y 0 in R 2, namely, one of the endpoints of the axis along which the length l is measured. The ellipsoid is tilted at an angle θ (measured in degrees). Your code fragment that draws the ellipse may use any of the following pseudo-opengl commands: glmatrixmode(), glloadidentity(), pglrotatef(), pgltranslatef(), pglloadmatrix(), pglmultmatrix(), pglscalef(), and DrawCircle().

5 Name: 3 2. [20 points] a. Let u be the vector u = 0, 1, 0. Let θ equal 135 degrees, that is 3π/4 radians. Give a 4 4 matrix that represents the linear transformation R θ,u. b. Now let u = 0, 2 2, 2 2 and θ equal 90 degrees; i.e., π 2 radians. Give a 4 4 matrix that represents the linear transformation R θ,u.

6 Name: 4 3. [20 points] A light source is placed at the origin in R 3, and it casts shadows onto the plane defined by x = 10. Thus, the plane is like an infinite wall parallel to the yz-plane, placed at x = 10. For x = x 1,y 1,z 1 apointinr 3 where x 1 > 0, let A(x) = x 2,y 2,z 2 be the point on the wall where the shadow of x is. This means that x 2 = 10. Give a 4 4 matrix that represents the transformation A over homogenous coordinates, or, prove that there is no such matrix.

7 Name: 5 4. [20 points] (a) What does the term shading mean in computer graphics? (b) Briefly describe why shading is important for rendering 3D graphics images. (c) Briefly describe Gouraud shading and Phong shading and how they are different. (d) Compare these two kinds of shading. What are the relative advantages and disadvantages of Phong and Gouraud shading?

8 Name: 6 5. [20 points] Consider the triangle lying in R 2 with vertices x = 0, y = 1, 3, andz = 5, 0. (a) What point in R 2 has barycentric coordinates α = 1 2, β = 1 3,andγ = 1 6 relative to this triangle? (b) What are the barycentric coordinates of the point 3, 1?

9 Name: 7 6. [20 points] A patch f(α, β) inr 3 is defined using bilinear interpolation on the four points x = 0, 0, 0, y = 4, 0, 1, z = 4, 4, 0, andw = 0, 4, 0. The points in counterclockwise order around the patch are x, y, z, w. (a) What is the point on this patch with bilinear coordinates α = 1 2 and β = 1 2? (b) Give a (non-zero) vector which is normal to the patch at this point. (Your normal vector does not need to be a unit vector.)

10 Name: 8 7. [10 points] A color has RGB specification of R = 1 2 and G =1andB = 4 5. (R,G,B color values are in the range 0 to 1.) What is the hue value (H) of this color? Express the hue by a value in the rangle 0 to 360.

11 Name: 9 8. [20 points] A degree three Bezier curve q(u) has the four control points p 0 = 0, 1, p 1 = 1, 0, p 2 = 3, 0, andp 3 = 2, 0. (a) Sketch and label the control points and the control polygon for the Bezier curve. (b) Sketch the Bezier curve approximately. For both (a) and (b): Draw your sketch to scale as best you can and, when drawing the Bezier curve, show clearly the initial and final points and the initial and final slopes. (c) Use the de Casteljau algorithm to compute q( 1 4 ).

12 Name: [20 points] (a) A particle is to be animated in such way that the following hold: (1) At time t = 0, the particle is at position 1, 0 and has velocity 0, 1. (2) At time t = 1, the particle is at position 0, 0 and has velocity 0, 1. Express the position q(t) of the particle as a function of time by using a degree three Bezier curve. Give your answer by specifying the four control points of the Bezier curve. (b) Repeat part (a), with the same initial position and velocity, and with the same ending position, but now with the ending velocity equal to zero. In other words, now the particle at time t = 1 has position 0, 0 and also has velocity at time t = 1 equal to zero. Find a degree three Bezier curve for these new contraints.

13 Name: 2 1. Let the Catmull-Rom curve q(u) be defined by the following control points: p 0 = 0, 0 p 1 = 0, 2 p 2 = 2, 2 p 3 = 2, 0 p 4 = 4, 0 y p 1 p 2 p 0 p 3 p 4 x Thus, q(i) interpolates some of the points, with q(i) =p i for i =1, 2, 3. For the problems below, show and label all your work, especially for part d. a. Give a freehand sketch of the Catmull Rom curve q(u) on the graph below. b. What is the value of q (1)? c. What is the value of q (2)? d. What is the value of q( 3 2 )?

14 Name: 2 1. (25 points) This problem concerns affine transformations of points that lie in the xy -plane. (a) The picture below shows a F in standard position and orientation and another F which has been moved by an affine transformation in 2-space. Give the 3 3 matrix that performs this transformation. 0,1 y 1,1 2,1 y 0,0 1,0 x = 2,0 1,0 x 0, 1 1, 1 1, 1 (b) Consider the following 3 3 homogenous matrix M which operates on points in 2-space as represented with homogeneous coordinates M = For each of the points in R 2 listed below, to what points in R 2 are they transformed by M? i. 0, 0. ii. 1, 0. iii. 1, 1.

15 Name: 3 2. (25 points) Give the OpenGL commands to draw the solar system as pictured below. The picture is shown from the top view, looking down the y -axis. Use commands drawsun(), drawearth() and drawmoon() which draw the planetary bodies at the origin (before being transformed by the modelview matrix, similar to the way glutwiresphere works). Your code should start off by initializing the model view matrix, and then use glrotate, gltranslate, glpushmatrix, and glpopmatrix as necessary. (Your code only draws the three objects, it does not draw the dotted lines, etc.) The earth is 3 units away from the sun, and the moon is 1.5 units away from the earth. Sun θ φ Earth x Moon z

16 Name: 4 3. (20 points) Barycentric coordinates. Let a triangle in the xy -plane be formed by the three points x =(0, 0), y =(3, 0) and z =(1, 3). a. Explain (one or two sentences) the definitions of barycentric coordinates for points u with respect to this triangle. b. What are the barycentric coordinates for the point (1, 3)? c. What are the barycentric coordinates for the point (2, 0)? d. What are the barycentric coordinates for the point (2, 1)?

17 Name: 5 4. (30 points) Give an description of how specular reflection is modeled in the Phong reflection model. Include a description of the various vectors and angles and their properties. Include the treatment of multiple light sources. Include formulas and explain the terms in the formulas. Explain common shortcuts used in the calculations. You should not include OpenGL specific enhancements such as attenuation and spotlights. It is best if your description is both complete and succinct. You do not need to explain things at great length, but your answer should make it clear you understand how specular reflection is modeled.