Answer Happy

COMPUTER SCIENCE TRIPOS Part IB - 2015 - Paper 4 3 Computer Graphics and Image Processing (PR) Consider the transformations used in the construction and rendering of a three- dimensional model on a screen. (a) List the three principal transformations in the processing pipeline and explain their rôles. [6 marks] (b) Why is it convenient to represent the transformations as matrices? [2 marks] (c) What are homogeneous coordinates? Explain how they are used in modelling these transformations as matrices. [2 marks] (d) Derive the matrix to represent a perspective transformation for a viewer at the origin of a point in three dimensions to a point on a screen in the plane z = d. [5 marks] (e) Perspective in classical art has vanishing points towards which parallel lines converge. Explain mathematically why this is the case and show how to calculate the location on the screen of the vanishing point for lines in a particular direction. [5 marks] [Hint: It may be helpful to represent lines parametrically in vector form as P(s) = A + SV where V is a direction and A is any point on the line.]