Institutionen för systemteknik

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Code: Day: Lokal: M7002E 19 March E1026 Institutionen för systemteknik Examination in: M7002E, Computer Graphics and Virtual Environments Number of sections: 7 Max.

1 Code: Day: Lokal: M7002E 19 March E1026 Institutionen för systemteknik Examination in: M7002E, Computer Graphics and Virtual Environments Number of sections: 7 Max. score: 100 (normally 60 is required to pass) Teacher: Matthew Thurley Result announced: Allowed aids: As soon as possible after the exam. Drawing and writing tools. An English/(Native Language) dictionary. Calculators are not permitted or required in the exam Answers must be given in English (Obs! Derive betyder härleda ) Section 1 Representation and Transformations (a) Given the following homogenous coordinate representation [ ] T, explain if this representation is a point or a vector in 3D space and give its 3D representation. (2 points). (b) Using the points p and q, and a vector n, explain which of the following operations are valid in an affine vector space. Hint: if you are unsure, perform these operations using homogeneous coordinates to see what they generate. (3 points) a. p + n b. p + q c. p - q (c) Derive the parametric equation of a line between 2 points P and Q. Draw a diagram that supports your answer. (2 points). (d) Derive the matrix representation, T, of a translation by the vector d = (d x, d y, d z ). Hint solve the matrix equation Tp = p + d. (2 points) (e) What is the matrix representation of a rotation about the z-axis of an angle θ. (2 points) (f) Show that the following pair of matrices commute. An arbitrary rotation about the z axis and a scaling of all axes using the same constant. ie. R z (θ) S = S R z (θ) (3 points) (g) Write the following transformations in the correct order so that they represent a rotation about the z-axis of an angle θ, R z (θ), followed by a scaling S, followed by a translation T. (1 point) 1

2 Section 2 Shading Given a triangle with vertices (points) P 1, P 2, and P 3 ; 20 Points (a) What is an expression for n, the vector normal to the plane determined by the triangle with the counter-clockwise order of vertices P 1 P 2, P 3? (2 points) (b) If C is the location of the camera and M is the center of the triangle, what is the vector, v from the centre of the triangle to the camera position along which the camera is looking.? (1 point) (c) Given a normal vector to the triangle n, a light at point L, a vector l from the midpoint of the triangle to L, the Phong reflection vector r, draw a picture with the triangle, P 1, P 2, P 3, v, n, l, and r. (1 point) (d) The Phong reflection model has three types of reflection from a surface: ambient, diffuse, and specular. Provide a short description for each type of reflection. Describe what sort of materials the diffuse and ambient reflection types are designed to model? (3 points) (e) Provide a mathematical expression for each of the listed components of the Phong lighting model for the triangle. You may assume any vectors you require are unit vectors. Assume a single light source L with separate ambient, diffuse, and specular intensity components. Reminder: Do not forget to include the material properties of the triangle in the model. a. Ambient reflection intensity (1 point) b. Diffuse reflection intensity. Explain when the diffuse reflection intensity is brighter and when it is weaker. (2 points) c. Specular reflection intensity. Explain when the specular reflection intensity is brighter and when it is weaker. (2 points) d. Provide a mathematical expression for a distance correction for the light source L and explain how it would be applied to the three types of reflection in the model. (3 points) (f) What is the difference between FLAT shading, GOURAUD shading, and PHONG shading? (5 points) Section 3 Scenegraphs and Object-Oriented Models 5 Points What sort of structure is a scenegraph and what are the advantages of such a structure? 2

3 Section 4 Clipping and Hidden Surface Removal 10 Points (a) Give a step-by-step description of how Sutherland-Hodgman polygon clipping would work on the polygon in Figure 3. (6 points) (b) Explain whether the Sutherland-Hodgman clipping algorithm is adapted for the graphics pipeline? (1 point) (c) What is back-face removal? (1 point) (d) Given an arbitrary triangle with a view vector v from midpoint of the triangle to the camera; how would the view vector v and triangle normal vector n be used to remove a hidden surface? Give an expression for a test. (2 points) V 4 V 3 V 1 V 2 Figure 3: Polygon to be Clipped Section 5 Implementation 20 Points (a) Why does it matter what order vertices are used when constructing polygons? (1 point) (b) What is double-buffering and what problem is it designed to overcome? (1 point) (c) Write a pseudo code description of the z-buffer hidden surface removal algorithm. (4 points) (d) Describe the Painter s algorithm. What factors complicate it? How can they be handled? (4 points) (e) Given a texture image, describe how you could map that texture onto the surface of an irregularly shaped polygonal model of an apple. Your answer should include diagrams but does not require any mapping equations. (3 points) (f) If a model of an apple was required to be highly polished and partially reflective of the environment, how could that be combined with the model of the apple in (e)? What additional steps are necessary? (3 points) (g) Consider that you are implementing a graphics application on an embedded device that does not have OpenGL Picking and Selection implemented. Given that you know the position of the camera, and the position that the user has selected on the image plane, how would you construct a ray to project into your 3D world, describe an efficient way to determine which objects this ray intersects, and how would you determine which object is closest to the camera? (4 points) 3

4 Section 6 Rasterization A digital difference analyzer (DDA) is a drawing algorithm that moves one step in either the x or y directions and computes the corresponding point in the other direction based on an incremental change from the last drawn point. Bresenham s algorithm is a DDA that uses only integer addition after initialization. (a) Derive Bresenham s algorithm for a line segment from point P 0 to point P 1 assuming that the gradient of the line is between 0 and 1 inclusive. Explain the decision variable used in Bresenhams algorithm and the necessary conditions for how this decision variable is updated. The following diagram may be useful aid. You do not have to produce the initial value of the decision variable. (8 points) (b) What is anti-aliasing and how could it be applied to improve the appearance of a diagonal line? (2 point) (c) Consider that we are scan converting a polygon in order to determine which pixels on a each scanline (row) are inside the polygon. Describe an algorithm or test we can use to determine whether we are inside or outside of the polygon during scan conversion. Describe one situation in which this algorithm will work not correctly and how this situation can be avoided? (5 points) Bonus marks: If you can describe a second situation where this algorithm does not work you can get up to 2 bonus marks. 4

5 Section 7 Viewing Transformations Given a perspective view with the camera at (-5, -5, 5) and pointing at (-15, 5, -5), the near clipping plane 20 units from the camera, the far clipping plane 100 units from the camera, a field-of-view in the y axis of 90º, and the aspect ratio of 4:3 (x width is 4/3 times the y width). Compute the matrices necessary to transform the viewing volume into the standard OpenGL viewing volume. This answer requires you to provide a series of transformation matrices. There are many ways to determine the necessary rotation matrix. If your technique calculates two separate rotation matrices make sure you multiply them together to provide a single final rotation matrix. All of the other matrices you do not need to multiply together, simply provide then in the correct sequence. Note that you do not need a calculator to determine the values of cosine and sine in any rotation matrices you generate. Recall that for any given right angle triangle, you know the cosine and sine of the angles based on the length of the sides of the triangle. Bonus marks: If you derive the perspective normalization matrix you can get up to 4 bonus points. c a cos θ = b/c sin θ = a/c tan θ = a/b θ b 5

CS 130 Final. Fall 2015

CS 130 Final. Fall 2015 CS 130 Final Fall 2015 Name Student ID Signature You may not ask any questions during the test. If you believe that there is something wrong with a question, write down what you think the question is trying