CS 418 Final Exam May 11 2005 Name NetID General Directions 1 Make sure your name is on every page 2 Make sure that you answer all the questions 3 Remember to write clearly and legibly Unreadable answers will receive 0 credit 4 Unless explicitly stated otherwise you may leave complex functions such as sin cross product u v matrix product A B matrix inverse A 1 etc unevaluated 5 When asked to write code you do not need to produce 100 syntactically correct C 6 Remember to time yourself You ll have 180 minutes to complete 100 points worth of questions So you should only be spending roughly 25 minutes on a 15 point question Question Points Out of 1 20 2 20 3 20 4 20 5 20 Extra 1 Total 100 Page 1 of 13 CS 418 Final Exam Name 1 20 points Selected Topics a Indicate which of the following visual phenomena can not be simulated by the classical radiosity algorithm discussed in class 1 2 3 4 5 6 7 mirror reflection color bleeding caustics diffuse inter reflection soft shadows specular highlights participating media b In the standard radiosity equation Bi Ai Ei Ai i X Fji Bj Aj j i is the reflectance of patch i The reflectance of a patch is constant precisely because of the fundamental assumption made by the radiosity algorithm that all surfaces are perfectly diffuse Explain why this implies that the reflectance i is a constant Page 2 of 13 CS 418 Final Exam Name c The standard ray tracing algorithm discussed in class is incapable of generating rainbows produced by prisms Give the two primary reasons why this is the case d Suppose we animate a propellor by drawing it at 30 frames second Gradually the propellor accelerates spinning faster and faster At some point it appears to start spinning in the opposite direction Explain why this occurs e Describe the standard approach for minimizing the artifact discussed in the previous part Page 3 of 13 CS 418 Final Exam Name 2 20 points We have a CRT monitor whose color primaries are R1 G1 B1 and a projection system whose color primaries are R2 G2 B2 Suppose that each primary of the projector can be written as an affine combination of the CRT primaries For example R2 r R1 g G1 b B1 where r g b 1 and r g b 0 a What is meant by the gamut of the CRT monitor b The color space defined by the primaries R1 G1 B1 is not perceptually uniform Explain what this means Page 4 of 13 CS 418 Final Exam Name c Are there colors which can be displayed on the CRT monitor that cannot be displayed on the projector Are there colors which can be displayed on the projector that cannot be displayed on the CRT Explain your answer d In contrast to red green blue cyan magenta yellow are referred to as subtractive primaries Explain what this means Page 5 of 13 CS 418 Final Exam Name 3 20 points Suppose that we are rendering a scene with an OpenGL like renderer Let P be the current Projection matrix and M be the current ModelView matrix A point in screen space is given by coordinates i j d The pair i j are the integer indices of the pixel in the image The d coordinate is a scalar depth value ranging from d 0 corresponding to the near clipping plane up to d 1 corresponding to the far clipping plane a Consider a screen space point q i j 0 1 being transformed by the matrix A M 1 P 1 Where in world space does the resulting point Aq lie b For a given pixel say the pixel 10 47 we can define a vector valued function p d Aq where q 10 47 d 1 which is a function of the depth value d The set of all points p d for d 0 1 forms a particular geometric shape What is it Be specific Page 6 of 13 CS 418 Final Exam Name c Given a light source L describe how to build a shadow map for this light d Suppose that we re trying to shade pixel i j Describe how to find the corresponding point in the shadow map Your answer should provide a specific mathematical procedure for performing this computation e Assume that we ve found the point in the shadow map that corresponds to pixel i j Describe the test you would need to perform to decide whether the pixel i j under consideration is in shadow with respect to the light L Page 7 of 13 CS 418 Final Exam Name 4 20 points Ray tracing a Consider the surface of an infinite cylinder of radius 1 centered along the z axis as shown in the figure below Give the implicit equation for this surface z r 1 x y b Suppose you are given a ray p td Using your implicit equation above derive an equation for the value s of t at which the ray may intersect this cylinder Page 8 of 13 CS 418 Final Exam Name c The line determined by the ray may intersect the cylinder in 0 1 or 2 locations Describe how to test whether this ray intersects the cylinder or not d Now suppose that we want to intersect a ray against an infinite cylinder in some arbitrary spatial position We can find a 4 4 matrix M that transforms this cylinder into the canonical cylinder used above If we are only interested in correctly computing the ray surface intersection and nothing else which of the following must be transformed by M 1 2 3 4 5 6 7 ray origin p ray direction d camera position lookFrom camera direction lookAt camera orientation vUp light position other objects in the world Page 9 of 13 CS 418 Final Exam Name e Suppose that you are given the value of t at which the ray intersects the cylinder Furthermore suppose that there is a single point light source located at the point q Give an expression for the shadow ray that we would use to determine whether the point on the cylinder is in shadow with respect to the point light at q f A basic non recursive ray caster loops over every pixel i j and computes a color for that pixel thus producing an output image Briefly outline the process by which it would compute this color Page 10 of 13 CS 418 Final Exam Name 5 20 points Texturing a Briefly describe the construction of an image pyramid and its primary use b Recall that normal mapping is a variant of texture mapping where the texture is used to encode unit surface normals Standard texture minification involves averaging together all texels covered by a particular pixel sample Explain why this produces incorrect results on normal maps Page 11 of 13 CS 418 Final Exam Name c Explain why the sphere maps used for environment mapping in OpenGL are view dependent i e they must be recomputed …