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UCSD CSE 167 - Midterm

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CSE 167: Computer Graphics Midterm Winter, 2006 NAME: A. Vectors (4 points) Given the following two vectors: !a =340!"###$%&&&!!!!!!b =010!"###$%&&& Compute the following: 1. The magnitude of !a 5 2. The magnitude of !b 1 3. The dot product !a !!b 4 4. The cosine of the angle between !a and !b 4/5 AKA .8 B. Normal (1 point) You are looking down at a flat quadrilateral, defined by four points that all lie in a plane: 5. Write an expression for the normal to this quadrilateral. The normal should be pointing towards you (out of the paper). (b-a) X (d-a) / |(b-a) X (d-a)| and similar a b c dNAME: page 2 of 4 C. Point and vector expressions (6 points) For each of the following expressions involving pointsaand b and vectors !u and !v, circle “Point” if the result is a point, “Vector” if the result is a vector, or “Invalid” if the expression is not legal (AKA not affine invariant). Hint: consider what happens to the w component in homogeneous coordinates. 6. a + b Point Vector Invalid 7. a ! b Point Vector Invalid 8. !u !!v Point Vector Invalid 9. a + 2!u Point Vector Invalid 10. 13a +23b Point Vector Invalid 11. 13a +23!v Point Vector Invalid D. Transformation matrix (2 points) Consider a homogeneous affine transformation matrix M, constructed from columns !a,!!b,!!c!and d: axbxcxdxaybycydyazbzczdz0 0 0 1!"####$%&&&& 12. Write the result of transforming 0101!"####$%&&&&byM, in terms of !a,!!b,!!c!and d. b+d 13. Suppose this matrix represents the local-to-world transform of an object that is not changing position, but is spinning about its local z-axis. Which column or columns ( !a,!!b,!!c!or d) of the matrix will not change as the object spins? c and dNAME: page 3 of 4 E. Scene Modeling (4 points) 14. Write the local-to-world transform that places the boat (left) in world coordinates in the scene (right), expressed as a composition of 2D translation T(x , y) and rotationR(angle). Give numbers for the parameters of the transformations you use. T(3,4) R(-45) In this scene, a sailor and a palm tree are on the deck of the boat. The sailor’s bird is on a branch of the tree. Write the following using composition of the local transformation matrices Tboat, Rboat, Tdeck, Tsailor, Tpalm, and Tbird and their inverses as needed: 15. The object-to-world transform for the bird Tboat Rboat Tdeck Tpalm Tbird 16. The object-to-world transform for the sailor Tboat Rboat Tdeck Tsailor 17. The transform expressing the bird’s frame in the sailor’s coordinates Tsailor-1 Tpalm TbirdNAME: page 4 of 4 F. Concepts (8 points) a. Camera Space b. Image Space c. Inner Space d. Normalized View Space e. Object Space f. Outer Space g. Projective Space h. World Space 18. Graphics pipeline: Select the correct spaces from above and list them in the order that we expect each vertex to go through in the traditional transformation process: Fill in letters here: 1. e 2. h 3. a 4. d 5. b Match the descriptions to the appropriate concept. a. A simpler object to make culling tests quicker b. Aim at an object of interest c. Draw or don’t draw triangles based on the order of their vertices d. Makes things farther away appear smaller e. Only draw objects that are in view f. Preserves parallel lines 19. c Backface culling 20. a Bounding volume 21. e Frustum culling 22. b Look-at transform 23. f Orthographic transform 24. d Perspective transform 25. The following pseudocode operations are the steps involved in recursive traversal of a scene hierarchy. Arrange them in proper order for a drawNode() method. a. Apply local transform b. Draw geometry (if any) c. Recursively draw children (if any) d. Pop Matrix e. Push Matrix drawNode(node) { e a b or c c or b d


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UCSD CSE 167 - Midterm

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