Written Assignment #3 Solution15-462 Graphics I, Fall 2003Doug JamesDue: Tuesday, November 18, 2003 (before lecture)80 POINTSNovember 27, 2003• The work must be all your own.• The assignment is due before lecture on Tuesday, November 18.• Be explicit, define your symbols, and explain your steps. This will make it a lot easier for us to assignpartial credit.• Use geometric intuition together with trigonometry and linear algebra.• Verify whether your answer is meaningful with a simple example.11 Angel, Chapter 7 (Discrete Techniques), Exercise 7.8Suppose that we have two translucent surfaces characterized by opacities α1and α2. What is the opacity ofthe translucent material that we create by using the two in series? Given an expression for the transparencyof the combined material.2 Angel, Chapter 7 (Discrete Techniques), Exercise 7.10In Section 7.9 we used 1−α and α for the destination and source blending factors, respectively. What wouldbe the visual difference if we used 1 for the destination factor, and kept α for the source factor?3 Angel, Chapter 7 (Discrete Techniques), Exercise 7.14When we supersample a scene using jitter, why shoulud we use a random jitter pattern?4 Angel, Chapter 8 (Implementation of a Renderer), Exercise 8.12Devise a method for testing whether one planar polygon is fully on one side of another planar polygon, i.e.,so that there exists a separating plane.The original question was ambiguous about whether the answer was 3D or 2D, e.g., separatingplane vs separating line. Answering either case is sufficient, and involves similar algebra, e.g.,testing using an oriented half plane (3D), or an oriented half line (2D).25 Angel, Chapter 13 (Advanced Rendering), Exercise 13.3Derive an implicit equation for a torus whose center is at the origin. (You can derive the equation by notingthat a plane that cuts through the torus reveals two circles on the same radius.)6 Angel, Chapter 13 (Advanced Rendering), Exercise 13.4Using your previous result (from 13.3), show that you can ray trace a torus using the quadratic equation tofind the required intersections.The key here is that the equation for the torus is factorable into two quadratic parts, and thereforethe roots of intersection can be analytically determined using the roots of the quadratic polynomi-als.37 Angel, Chapter 13 (Advanced Rendering), Exercise 13.12Suppose that you have an algebraic function in which the highest term is xiyjzk. What is the degree ofthe polynomial that we need to solve for the intersection of a ray with the surface defined by this function.Provide a derivation.Substituting the equation for a ray, r = r0+ dt, into the polynomial, we see that the highest termdetermines the degree of the polynomial that must be solved,xiyjzk= (x0+ dxt)i(y0+ dyt)j(z0+ dzt)k(1)= dxdydzti+j+k+ O(ti+j+k−1) (2)and therefore the degree of the polynomial that must be solved is (i + j + k).8 Angel, Chapter 13 (Advanced Rendering), Exercise 13.13Consider again an algebraic function in which the highest term is xiyjzk. If i = j = k, how many terms arein the polynomial that is created when we intersect the surface with a parametric ray?If the highest degree of the polynomial is (i + j + k), and i = j = k, then the highest degree is 3i,and therefore there can only be as many as 3i + 1 terms in the
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