Tentamen Computer Graphics – 2M370
maandag 12 maart 2001
Wiskunde en informatica
Technische Universiteit Eindhoven
Opgave I – homogeneous transformations
i. Given a linear transformation L, give the corresponding 4x4 homogeneous matrix.
ii. Give a 4x4 homogeneous matrix for a translation over the vector (a,b,c).
iii. Give a 4x4 homogeneous matrix for a perspective projection on the plane z=d with the eye point
in the origin, d > 0, and the view direction (0,0,1).
iv. Explain why the use of homogeneous matrices increases the speed of rendering.
Opgave II - rendering
Explain in a clear and concise way how 3-dimensional triangles can be rendered using Phong shading
and the z-buffer algorithm.
Opgave III - implicit surfaces & Lipschitz
i. What is an implicit surface ?
ii. Define the Lipschitz condition for a function. Which useful property of implicit surfaces follows
from this condition (using constant 1) ? Give and explain an application of this property.
iii. Give an implicit function with constant 1 for a sphere (radius R and center in the origin), a half
space (ax+by+cz-d > 0 ), and an axis-aligned cube (side 2R, and center in the origin).
Hint 1: Given a surface S. Let f be a function that equals in each point in space the distance to S. The function f is now
a function with Lipschitz constant 1 describing this surface.
Hint 2: A cube can be constructed using CSG operations.
Opgave IV – ray tracing
i. Which three factors largely determine the efficiency of a ray tracer ?
ii. Give three different ways of improving the speed of a ray-tracer and explain them in terms of these
factors.
iii. Explain how a ray tracer handles shadows. Especially, consider the case when a ray-traced scene
contains mirrors.