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52 CS488/688 Introduction to Computer Graphics 1. Primary rays 2. Shadow rays 3. Reflected rays 17.11 Distribution Ray Tracing [Last Used: Winter 2006 Final] In distribution ray tracing, by casting many rays in a random fashion (i.e., “distributing them”) we can simulate many effects. The “random fashion” part is what allows us to achieve our effects. Describe what types of rays get “distributed” and what they “get distributed over” to achieve each of the following effects. We have provided a model answer for “Anti-aliasing” to better indicate what you should give for your answers. Anti-aliasing — distribute primary rays over pixel area. 1. Soft shadows — distribute rays over 2. Motion blur — distribute rays over 3. Depth of field — distribute rays over 17.12 Ray tracing lighting [Last Used: Winter 2006 Final] Consider a ray (P, ⃗v) that hits a surface S at a point Q, where the normal to S at Q is ˆn, and where S has only diffuse reflection material components (d r , d g , d b ). Further, assume there is a white point light at position L with RGB intensity values (I, I, I), and a white ambient light with RGB intensity values (A, A, A). In ray tracing, we want to know the intensity of light that travels back along this ray and eventually reaches the eye. For the situation described above, explain how we compute this intensity, noting any additional rays we cast and any local lighting computations we perform. 18 Splines 18.1 Bézier Curves: Evaluation and Properties [Last Used: Fall 2009 Final] The following pictures show possible arrangements of control points for a cubic Bézier curve. P ❜ 3 ❜ P ❜ 2 ❜ P 0 ❜ P 1 ❜ P 2 P 0 ❜ P 3 ❜ P 1
CS488/688 Introduction to Computer Graphics 53 P ❜ 1 ❜ P 3 ❜ P 0 ❜ P 2 P ❜ 1 P 0 ❡❜ P 3 ❜ P 2 (a) In the case of each picture, make as accurate a sketch of the curve as possible. A cubic Bézier curve has the mathematical form 3∑ C(t) = P i b i (t) (b) Give explicit formulas for b 0 (t), . . . , b 3 (t). (c) Show that b i (t) ≥ 0 for 0 ≤ t ≤ 1. (d) Show that ∑ 3 i=0 b i (t) = 1 for all t. (e) What implications do (c) and (d) have in locating the points of C(t)? (f) Suppose R is a rotation matrix. Show that RC(t) is given by the Bézier curve that has control points RP i . i=0 (g) Sometimes a Bézier curve is presented in the format ⎡ C(t) = [t 3 , t 2 , t 1 , 1]B ⎢ ⎣ ⎤ P 0 P 1 ⎥ P 2 ⎦ P 3 Show how to find the matrix B from the formulas in part (b) above. (h) Subdivision of the segment at the parametric value a is given by a process that can be visualised by means of a pyramid of calculations: P 0123 ❅ ❅ P012 P123 ❅ ❅ ❅ ❅ P01 P12 P23 ❅ ❅ ❅ ❅ ❅ ❅ P0 P1 P2 P3 Complete this pyramid diagram by putting appropriate expressions on the edges. Make an enlarged copy of picture 1 and draw in and lable the locations where the points P 0 , P 01 , P 012 , P 0123 would appear when a = 1/3.
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