CS4102 Computer Graphics 2019
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CS4102
Computer Graphics
2019
1. B
zier curves
(a) Write down the expression for the n-th order B
zier curve (3 marks) and thus for the n-th order B
zier surface (2 marks), explaining what each of the symbols used represents. [5 marks]
(b) Explain what it means that a B
zier curve is enclosed by the convex hull of its
control points. [3 marks]
(c) Explain why the interior of a convex polygon can be described as the set of all points which can be expressed as a linear combination of vectors corresponding to its vertices such that all coefficients are positive and sum up to unity (i.e. 1).
[5 marks]
(d) Prove for the specific case of a cubic B
zier curve that it is indeed enclosed by the convex hull of its points. [7 marks]
[Total marks 20]
2. Non-Lambertian reflectance
(a) Explain what the bidirectional reflectance function (BDRF) is. [4 marks]
(b) With reference to the form of the BDRF, explain what unisotropic and specular surfaces are (3 marks each), clearly highlighting the key differences. [6 marks]
(c) A child is standing at the origin of the world coordinate system and looking down the wet road extending along the z axis in the positive direction. The direction towards the Sun is「2 − 1 1|T . If the child’s eyes at the height of 1m in what direction does the child need to look at the road to observe the point of maximum glare? [10 marks]
[Total marks 20]
3. Geometry and transformations
(a) In the context of computer graphics, what does the term ‘homogeneous coordi-
nates’ refer to (3 marks)? Why are homogeneous coordinates used (3 marks)? [6 marks]
(b) In the context of homogeneous coordinates, what does ‘dehomogenization’ refer
to (2 marks)? Can this process be expressed through matrix multiplication? Explain your answer (2 marks). [4 marks]
(c) What transformation is effected by the following transformation matrix in
(d) An avatar’s coordinate system origin is at the locus「0 2 1|T in the world coor- dinate system. The x axis of the avatar is aligned with the x axis of the world coordinate system, and its y axis is pointing in the direction「0 1 2|T in world coordinates. Derive the transformation matrix in homogeneous coordinates which transforms the locus of a point in the avatar’s coordinate system to the locus in world coordinates (5 marks). Then derive the transformation matrix
which performs the opposite (2 marks).
4. Phong’s model
[7 marks] [Total marks 20]
(a) Phong’s reflectance model comprises three additive terms. Write down expres- sions for each of them explaining the meaning of all symbols used, and explain what the purpose of each term is. [6 marks]
(b) What is meant by the term ‘empirical’ when discussing Phong’s reflectance
model? [2 marks]
(c) One of the three terms comprising Phong’s model is not viewing direction dependent. State which one and briefly explain (2 marks). Sketch the function capturing the dependence of the amount of light reflected on the viewing direction, making sure that all axes and salient features are labelled clearly and unambiguously (2 marks). Sketch the physically realistic behaviour of real-world surfaces that this term is aiming to mimic (2 marks). Comment on the difference (2 marks).
Finally, derive (rather than merely write down) the expression for the only quantity in this term that is not explicitly given by the geometry of the scene (4 marks). [12 marks]
[Total marks 20]
2022-04-11