Announcements•Midterms are back.•Will go over solutions.•More about grading curves later...•P3 Due Thurs 4/9•More about course evaluations later.Midterm1. (16 points) List all vectors used in the calculation of the Phong Illumina-tion Model. Draw a sketch and label each vector and congruent angles.Indicate which vectors are used in the diffuse term and which are used inthe specular term.2. (16 points) Bump mapping and displacement mapping are both used togenerate more s urface details. What are the pros and cons of each? Aspart of your explanation, describe at least one artifact in bump mappingthat is fixed by displacement mapping. (Two to four sentences shouldsuffice.)2Midterm1. (16 points) List all vectors used in the calculation of the Phong Illumina-tion Model. Draw a sketch and label each vector and congruent angles.Indicate which vectors are used in the diffuse term and which are used inthe specular term.2. (16 points) Bump mapping and displacement mapping are both used togenerate more surface details. What are the pros and cons of each? Aspart of your explanation, describe at least one artifact in bump mappingthat is fixed by dis plac eme nt mapping. (Two to four sentences shouldsuffice.)2Midterm3. (17 points) Draw the triangle with coordinates (1, 2), ( −2, 2), and (0, −2).Then draw the points (1, 0) and (0, 0). Indicate whether these points areinside or outside the triangle. Now justify your answer by computingbarycentric coordinates for these points and using the barycentric coordi-nate test to determine if these points lie ins ide the triangle. Explain whatconditions you are using to test whether the point is inside or outside.3Midterm4. (17 points) Describe how we can use distributed (stochastic) ray tracingto achieve the following effects:(a) anti-aliasing(b) soft shadows(c) motion blur4Midterm5. (17 points) In the famous “vertigo effect,” Alfred Hitchcock zoomed acamera out while dollying (moving) the camera forward. If the effect isdone properly, objects at a certain fixed distance will remain the samesize, like the window frame:To explain this effect we can draw the following diagram:Note that when the camera zooms out, its focal length (labelled f1 andf2) increases to keep the image plane at constant size.(a) Which of the images (A or B) was taken from distance D1, whichfrom D2?(b) Write a m athe matical expression which relates D1, D2, theta1 andtheta2. (Hint: Use one of the basic trigonometric functions to expressthe width of the subject plane W in terms of D1 and theta1, then dothe same using D2 and theta2.)(c) Suppose a new object were placed between the camera and the win-dow and the images were retaken in the same way (i.e. with the samecombinations of focal length and distance). In image B, would thisnew object appear larger, smaller, or the same size as it would inimage A? Why?5LinkMidterm6. (17 points) A cubic spline is defined by a polynomial of the formf(t)=a0+ a1t + a2t2+ a3t3.Catmull-Rom spline is a special type of cubic spline widely used in com-puter graphic s. To specify a Catmull-Rom spline, at least four controlpoints are needed. Let’s call the four control points p1, p2, p3, and p4.Catmull-Rom spline defines the beginning of the curve to be f(0) = p2,and the end f (1) = p3. The derivative at the beginning of the curve isf!(0) =12(p3 − p1), and the derivative at the end of the curve is half off!(1) =12(p4 − p2). Given this information, fill in all the coefficients ofthe following matrix:? ? ? ?? ? ? ?? ? ? ?? ? ? ?a0a1a2a3=p1p2p3p4This is the matrix whose inverse is the basis of Catmull-Rom spline.6RadiosityRay Tracing and RadiosityForm FactorsEnhancementsTwo-pass RenderingRay Tracing and RadiosityForm FactorsEnhancementsTwo-pass Rendering1515--462 Computer Graphics I462 Computer Graphics I33OutlineOutline••A Brief Review/Introduction to RadiosityA Brief Review/Introduction to Radiosity••The Radiosity Equation, Form FactorsThe Radiosity Equation, Form Factors••Putting it all together, and ImprovingPutting it all together, and Improving••More Realism: A digression, and TwoMore Realism: A digression, and Two--Pass Pass RenderingRendering1515--462 Computer Graphics I462 Computer Graphics I44Review: Local vs. Global IlluminationReview: Local vs. Global IlluminationLocal illumination: Local illumination: PhongPhongmodelmodel(OpenGL, most real(OpenGL, most real--time graphics)time graphics)––Light to single surface point to viewerLight to single surface point to viewer––FASTFAST––Vastly simplifiedVastly simplified––No representation of many natural No representation of many natural phenomena (shadows, interphenomena (shadows, inter--reflections) reflections) without additional hackswithout additional hacksLocal Illumination- fast- simple- light → surface → viewer- ignores many important effects1515--462 Computer Graphics I462 Computer Graphics I55Review: Local vs. Global IlluminationReview: Local vs. Global Illumination••Global illumination: Global illumination: Ray tracingRay tracing––Realistic specular reflection/transmissionRealistic specular reflection/transmission––Simplified diffuse reflection*Simplified diffuse reflection*••Global illumination: Global illumination: RadiosityRadiosity––Realistic diffuse reflectionRealistic diffuse reflection––DiffuseDiffuse--only: No specular interaction*only: No specular interaction*indirect direct both6Computer Graphics 15-462Beyond Ray TracingRay tracing ignores the diffuse component of incident illumination–to achieve this component requires sending out rays from each surface point for the whole visible hemisphereEven if you could compute such a massive problem there is a conceptual problem—loops:–point A gets light from point B–point B also gets light from point A7Computer Graphics 15-462Doing it RightThe real solution is to solve simultaneously for incoming and outgoing light at all surface pointsthis is a massive integral equationRadiosity deals with the relatively easy case of purely diffuse scenesOr, you can sample many, many complete paths from light source to camera (photon mapping)Key Idea•Model diffuse interaction only!incoming lightoutgoing light7Computer Graphics 15-462Doing it RightThe real solution is to solve simultaneously for incoming and outgoing light at all surface pointsthis is a massive integral equationRadiosity
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