CS3241 NATIONAL UNIVERSITY OF SINGAPORE CS3241 — COMPUTER GRAPHICS (Semester 1 AY2019/2020) Time Allowed: 2 Hours INSTRUCTIONS TO STUDENTS 1. This assessment paper contains SEVEN (7) questions and comprises ELEVEN (11) printed pages, including this page. 2. This is an OPEN BOOK assessment. 3. Answer ALL questions within the space provided in this booklet. 4. The full score of this paper is 75 marks. 5. Write legibly with a pen or pencil. Untidiness will be penalized. 6. Do not tear off any pages from this booklet. 7. Write your Student Number below using a pen. Do not write your name. (write legibly with a pen) Student Number: A This portion is for examiner’s use only Q# 1 2 3 4 5 6 7 Σ MAX 8 15 9 10 10 13 10 75 SC ANSWERS CS3241 —— Page 2 of 11 —— QUESTION 1 [8 MARKS] (1A) [4 marks] The diagram on the right shows 3 quadrilaterals (P1, P2 and P3) from a mesh that approximates a curved surface. The vertex v is shared only by P2 and P3. Describe two problems that they may cause when they are being rendered with OpenGL. Assume OpenGL lighting computation and Gouraud shading are enabled. (1) A crack may appear between P1 and (P2 and P3). (2) The interpolated color in P1 may look inconsistent from that of P2 and P3 along its edge adjacent with P2 and P3. (1B) [4 marks] How would you subdivide quadrilateral P1 to avoid the problems in Part A? Clearly draw your new polygon mesh in the space below. You must guarantee that each newly created polygon is planar. Furthermore, we assume that P1 is sharing its left edge, bottom edge and top edge with some adjacent polygons, and we do not want those polygons to be modified. P1,1 P2 P3 P1,2 P1,3 P1 P2 P3 v CS3241 —— Page 3 of 11 —— QUESTION 2 [15 MARKS] (2A) [4 marks] Each of the following two diagrams shows a top view of a view frustum enclosing a square (only an edge of the square is seen from top view). The near planes in the two setups are positioned differently. The far planes are not shown. Given that the square is to be rendered to a rectangular viewport, draw and show how the square would appear in the viewport for each of the setups. (2B) [6 marks] We want to capture an environment cubemap of a virtual scene from the world position (cx, cy, cz). Write an OpenGL program fragment to set up the view to render the scene for the +x side of the cubemap (the side that faces in the +x direction of the world space). Use gluLookAt() and glFrustum(), and remember to set the correct OpenGL matrix mode. Assume the near and far plane distances are nearDist and farDist respectively, and the up- vector for the gluLookAt() function is (0, 1, 0). Function usage: gluLookAt(eyeX, eyeY, eyeZ, lookAtX, lookAtY, lookAtZ, upX, upY, upZ); glFrustum(left, right, bottom, top, near, far); double cx, cy, cz; double nearDist, farDist; // both are positive values ... glViewport( 0, 0, 512, 512 ); (more writing space next page) square viewpoint near plane (i) (ii) square viewpoint near plane viewport viewport CS3241 —— Page 4 of 11 —— glMatrixMode( GL_PROJECTION ); glLoadIdentity(); glFrustum( -nearDist, nearDist, -nearDist, nearDist, nearDist, farDist ); glMatrixMode( GL_MODELVIEW ); glLoadIdentity(); gluLookAt( cx, cy, cz, cx + 1, cy, cz, 0, 1, 0 ); (2C) [5 marks] Suppose in the camera coordinate frame, there is a disc in the z = 0 plane, centered at (100, 200, 0), and has a radius of 10. Initially, the disc fits exactly in a window of size 300 300. When the window is resized, you want the disc to appear the same size as before, even though it may be clipped; or if the window is too large, the disc always appears in the top-left corner (see diagram below). Complete the reshape callback function to set up the required orthographic projection. The viewport should always be the same size as the whole window. Function usage: gluOrtho2D(left, right, bottom, top); void MyReshape( int w, int h ) { glViewport( 0, 0, w, h ); glMatrixMode( GL_PROJECTION ); glLoadIdentity(); gluOrtho2D( 100 - 10, 100 - 10 + 20 * (float)w/300, 200 + 10 - 20 * (float)h/300, 200 + 10 ); glMatrixMode( GL_MODELVIEW ); } Initial window Enlarged window Shrunk window CS3241 —— Page 5 of 11 —— QUESTION 3 [9 MARKS] (3A) [4 marks] There are five triangles parallel to the z = 0 plane in the camera coordinate frame. Suppose the 5 triangles have the following colors and z coordinates: Triangle A B C D E Color Red Green Blue Pink White z –2 –3 –5 –7 –8 The 5 triangles are projected using the following OpenGL orthographic projection: glOrtho( –100, 100, –100, 100, 0, 10 ); Assuming that all five triangles are inside the viewing volume, write in the following table the z-buffer value (zbuf) for the fragments of each triangle after it has been rasterized. The z-buffer value ranges from 0 to 1. Triangle A B C D E zbuf 0.2 0.3 0.5 0.7 0.8 (3B) [5 marks] Suppose all the 5 triangles in Part A cover the pixel location (xpix, ypix). Given that the triangles are drawn in the order D, B, C, A and E, write in the following table the z-buffer value and the pixel color at location (xpix, ypix) in the framebuffer after each step. Event zbuf at (xpix, ypix) Color at (xpix, ypix) 1. After clearing buffers 1.0 Black 2. After drawing Triangle D 0.7 Pink 3. After drawing Triangle B 0.3 Green 4. After drawing Triangle C 0.3 Green 5. After drawing Triangle A 0.2 Red 6. After drawing Triangle E 0.2 Red CS3241 —— Page 6 of 11 —— QUESTION 4 [10 MARKS] (4A) [3 marks] A 3D square is projected on the screen as a 45-rotated 2D square as shown in the diagram on the right. The four vertices have been assigned RGB colors as indicated in the diagram. Suppose the colors at the vertices are to be bilinearly interpolated over the interior of the square. Given that x is a pixel location at the center of the square, what would be the RGB color computed at x? Assume the rasterization is performed in horizontal scan lines. RGB color at x: [0, 0, 0] (4B) [3 marks] Suppose the square in Part A has been rotated 90 counterclockwise as shown in the diagram on the right. Given that everything is the same as in Part A, what would be the RGB color computed at x? RGB color at x: [0.5, 0.5, 0.5] (4C) [4 marks] Based on your results in Parts A and B, (i) what can you say about bilinear interpolation of vertex attributes over a quadrilateral? (ii) What undesirable effect can this cause? (iii) When will this effect be more observable? (i) Bilinear interpolation of attributes over a quadrilateral is not rotational invariant. (ii) This can cause undesirable effects because an interior point on the quadrilateral can be colored differently depending on the orientation of the quadrilateral. (iii) This effect is made obvious when there is rotational animation of the quadrilateral. [0,0,0] [0,0,0] [1,1,1] x [0,0,0] [0,0,0] [0,0,0] [0,0,0] x [1,1,1] CS3241 —— Page 7 of 11 —— QUESTION 5 [10 MARKS] (5A) [2 marks] What is the use of vertex normals? A vertex normal is used to specify the orientation of the surface at the vertex, and it is usually used in the lighting computation at the vertex. (5B) [4 marks] Explain why vertex normals of triangles are not suitable for performing back-face culling. (1) The 3 vertex normals of the triangle may be different from each other, so we may not know which one to use. (2) It is possible that the angle between a vertex normal and the view vector is greater than 90 degrees, and yet the triangle’s front face is still visible. (5C) [4 marks] A triangle, whose three vertices have exactly the same vertex normal, is passed into OpenGL for rendering. The OpenGL lighting computation has been enabled and a point light source has been set up. The material at the three vertices are also the same. Are the computed colors at the three vertices always be the same? Explain why. Not the same. Because the three vertices have different positions, therefore their vector L to the light source are different, and their vector V to the viewer are also different. These will produce different values of N.L and R.V in the Phong Illumination Equation. CS3241 —— Page 8 of 11 —— QUESTION 6 [13 MARKS] (6A) [4 marks] In Whitted Ray Tracing, suppose a shadow ray P(t) = O + tD (where D is a unit vector) is being shot from the surface point O towards the only light source at position S. If there is an intersection between the ray and the scene at a value of t > epsilon, can we be sure that the surface point should be in shadow? Why? Assume all objects are opaque. No. Because t > epsilon includes intersections that are beyond/behind the light source. (6B) [2 marks] When finding the intersection between a ray and a triangle using the barycentric coordinates approach, how can you tell that the ray is parallel or almost parallel to the plane of the triangle? When the value of β or γ or t is huge. CS3241 —— Page 9 of 11 —— (6C) [7 marks] Suppose there is an opaque sphere in an enclosed environment. All the other surfaces of the environment are opaque and have materials that have diffuse and specular components (kd, kr, and krg are greater than 0). However, the sphere’s material has only diffuse component (kr = 0 and krg = 0). There are two point light sources in the scene. Assuming we want to render a 100x100 pixels image of the scene using Whitted Ray Tracing, with one level of recursion, what would be the total number of rays that have to be shot? Assume the camera is within the environment but outside the sphere, and the sphere occupies 3000 pixels in the rendered image. Show your workings clearly. If all objects are opaque, then there is no need to spawn refraction rays. If the scene is enclosed, then all primary rays and reflection rays will hit some object. Rays shot per pixel on sphere = 1 primary ray + 2 shadow rays = 3 Total number of rays shot for pixels on sphere = 3000 * 3 = 9,000 Rays shot per pixel not on sphere = 1 primary ray + 1 reflected ray + 4 shadow rays = 6 Total number of rays shot for pixels not on sphere = (100*100 - 3000) * 6 = 42,000 Total rays shot = 9,000 + 42,000 = 51,000 eye pixel shadow rays eye pixel reflection ray shadow rays CS3241 —— Page 10 of 11 —— QUESTION 7 [10 MARKS] (7) [10 Marks] The diagram below shows two cubic Bezier curve segments with control points a1 a2 a3 a4 and b1 b2 b3 b4. Create two more cubic Bezier curve segments to join the given curves in order to maintain C1 continuity. Namely, create two curve segments with control points c1 c2 c3 c4 and d1 d2 d3 d4 such that the first curve segment connects a4 to b1 and the second one connects b4 to a1. Draw your new points accurately in the diagram in the correct locations, together with the broken lines indicating the control polygons. State the coordinates of the new points below the diagram. Coordinates of the new control points: c1: (8, 7) c2: (12, 8) c3: (8, 9) c4: (10, 7) d1: (10, 2) d2: (7, 1) d3: (5, 2) d4: (6, 3) b 3 : (13,3) x y a 1 : (6,3) a 2 : (7,4) a 3 : (4,6) a 4 : (8,7) b 4 : (10,2) b 2 : (12,5) b 1 : (10,7) CS3241 —— Page 11 of 11 —— (blank page) ——— END OF PAPER ———
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