course, exam #, semester/year CS 184, Midterm #1, Fall 1997)CS184, Fall 1997Midterm #1Professor Brian A. BarksyFall 1997 Foundations of Computer Graphics Professor Brian A. Barsky and Adrian SfartiOctober 9, 1997 CS184 T.A.'s: Lilian Chu, Dan Garcia and Aaron IsaksenMIDTERM EXAMNAME (LAST, FIRST):CS184-:Relax. You have 80 minutes. Remember to pace yourself. Feel free to use the back of each page foradditional answer space. Note that not all questions have equal value. Do not panic.GOOD SKILL!(write any comments about the exam here)How well do you believe you did?(Circle the appropriate picture which represents your perceived midterm rank);) :) ;} :| :{ :( X#Question Possible Your Scorecourse, exam #, semester/year CS 184, Midterm #1, Fall 1997)MIDTERM EXAM 11 152 253 254 35Total 100Question 1: Hardware and Scan Conversion (15 points)The diagram shown above shows a colormap systemwith a N bit deep W wide and H high framebuffer,digital-to-analog converters (DAC) for red, green andblue which are A, B and C bits wide respectively, anda screen with S triads wide and T triads high. Usingthe variables above (and assuming nothing about N,W, H, A, B, C, S or T - the diagram above is not toscale), derive an expression for:1.1) ...the # of pixels on the screen. [1 pt]1.2) ...the # of colors that could ever land on pixel pover time. [2 pts]1.3) ...the # of possible colors on the screen at once.[2 pts]1.4) Up to now, you have always been given a sample point (and two tie-breaking directions) and asked torender polygon primitives. Here we give you the polygon primitives and rendered pixels and ask you tofigure out the sample point and two tie-breaking directions. The sample point lies on the edge of thepolygon. Hint: The tie-breaking driections do not necessarily have to be horizontal or vertical. [10 pts]course, exam #, semester/year CS 184, Midterm #1, Fall 1997)Question 1: Hardware and Scan Conversion (15 points) 2Answer 1: Hardware and Scan Conversion1.1) 1.4) (Indicate sample point and two tie breakingdirections on the picture below.)1.2)1.3)Question 2: Transformations (25 points)The figure to the right shows a 2-D triangle PQR transformed by atransformation matrix T to a new position P''Q''R''. Pay carefulattention to the fact that T must transform P to P'', Q to Q'' and R toR''.2.1) Using row-vector notation, show T as a composition of matricesScale(sx, sy), Translate(tx, ty), Rotate(theta), Shearx (a) or Sheary(b)with appropriate arguments. [10 pts]E.g., T = Rotate(30o) * Shearx(5)*Translate(10,3)2.2) Write down the 3x3 transformation matrix T. Show all work. [10pts]2.3) Show the inverse matrix T-1 (which transforms the triangleP''Q''R'' to PQR) using a composition of matrices Scale(sx,sy),Translate (tx, ty), Rotate(theta), Shearx(a) or Sheary (b) withappropriate arguments. You may find your answer to question 2.1 usefulhere. [5 pts]Answer 2: Transformations2.1) T =2.2)2.3) T-1 =Question 3: Hierarchies and Change of Bases (25 points)course, exam #, semester/year CS 184, Midterm #1, Fall 1997)Answer 1: Hardware and Scan Conversion 3This diagram show how the upper triangle can be instanced five times to form the middle group G1 (G1left,G1top and G1right) and part of the bottom group G2 (G2top and G2right). G1 can be instanced once to helpform the bottom group G2 (component G2left). Coordinates refer to the verices of the subtending triangle.3.1) Find the 6 transformations labeled M1 thorugh M6 in the diagram above. You do NOT need to createthe matices for the transformations, you can just write them as products of row vector transformationsScale(sx, sy), Translate(tx, ty), Rotate(theta), Shearx(a) or Sheary(b) with appropriate arguments. [6 pts]3.2) Draw the hierarchy tree, labeling the branches with the appropriate concatenated row-vectortransformations M1 through M6. Show the state of the stack entering each node. [15 pts]Captain Matrix takes your ordinary X, Y, and Z axes, rotates them about theorigin in some unknown way until he obtains three new axes, a, b, and c,such that a used to be X, b used to be Y, and c used to be Z. He thenpolitely gives you the change of basis matrix such that the expression to theright holds:3.3) What are the numerical values of e, f, g, and h? [4 pts]Answer 3: Hierarchies and Changes of Bases3.1)M1=M(delta->G1left)=M2=Mdelta->G1top)=M3=M(delta->G1right)=M4=M(G1->G2left)=course, exam #, semester/year CS 184, Midterm #1, Fall 1997)Question 3: Hierarchies and Change of Bases (25 points) 4M5=M(delta->G2top)=M6=M(delta->G2right)=3.2)3.3)e=f=g=h=Question 4: Projection (35 points)Your hovercraft floats above your lake as in assignment 4. Here you decide to place the center of projection(designated by the eye in the diagram below) on the hovercraft (not shown) itself. There is NO clipping.Your projection and geometry parameters:The origin is designated by the label O near the small axes in each diagram.• Center of projection = (2,0,1)• The lake is defined by A (0,-1,0) B(3,-1,0) C(3,-1,3) and D(0,-1,3)• Projection plane is the z=5 plane• Point C after projection and homogenous division maps to world coordinate (4,-2,5) as shownbelow on the projection plane z=5 in world coordinates (not viewing coordinates).• Each grid line in all of the figures below is one unit (including the answer plot on the bottom)• 4.1) Project A, B and D onto the projection plane diagram z=5 shown on the bottom left on the previouspage. We've already projected C for you. Then copy your answer to the right diagram (you'll use onediagram for the answer to 4.6 and one for the answer to 4.7 below). [3 pts]4.2) Write down the row-vector notation tranformation matrix (you may express it as a single 4x4 matrix oras a composition of 4x4 matrices) that, after homogeneous division, has the effect of projecting ABCD ontothe projection plane z=5. HINT: You may wish to verify it with point C whose coordinates are given beforeand after projection.4.3) Assuming ABCD was the bottom face of a cube, what classification of projection is this? List the entirehierarchy (E.G., planar-parallel-orthographic-axonometric-trimetric) [3 pts]4.4) How many vanishing points are there with this projection? [2 pts]4.5) Where are the vanishing points? [4 pts]course, exam #, semester/year CS 184, Midterm #1, Fall 1997)Answer 3: Hierarchies and Changes of Bases 54.6) Let's say we project points ABCD onto the projection plane (as you have done in 4.1) and then call asimple 2D polygon wireframe
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