CS4102 Computer Graphics 2018
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CS4102
Computer Graphics
2018
1. Geometric primitives
(a) Consider two quadratic Bezier curves in three dimensions (3D) specified by the control points「0 0 0]T,「1 0 0]T, and「1 1 1]T, and「0 0.5 1]T,「1 0.5 0]T, and 「1 1 1]T . Explain why each curve is confined to a plane (2 marks). Hence find the expression for the intersection of the two planes (6 marks). [8 marks]
(b) Write down the rotation matrix that effects clockwise rotation of 60 degrees about the z-axis (3 marks). Hence write down the expressions for the Bezier curves in (a) after the rotation (3 marks). [6 marks]
(c) Is the cubic Bezier curve specified by the control points「0 0 0]T,「1 0 0]T,「1 1 1]T, and「1 1 2]T confined to a plane (3 marks)? How about one specified by the control points「0 0 0]T,「1 0 0]T,「1 1 1]T, and「2 1 1]T (3 marks)? In both cases
explain your answer.
[6 marks]
[Total marks 20]
2. Shading
(a) Describe succinctly what a shading model is. [4 marks]
(b) Consider a straight line segment extending from「0 0]T to「1 0]T, illuminated
by a directional light source with the direction 「1 2]T / │5, and with normals
corresponding to the two end points respectively「一1 3]T / │10 and「0 1]T .
If the light source is monochromatic, its intensity 3, and the reflectance model diffuse with the diffuse coefficient equal to 0.5 across the line segment, what is the brightness of the point「1/3 0]T when each of the following three shading models is adopted: flat (3 marks), Gouraud (3 marks), and Phong (4 marks).
Sketch the brightness variation across the entire segment for each of the shading models above (1, 2, and 3 marks respectively) [16 marks]
[Total marks 20]
3. Reflectance modelling
(a) Explain succinctly what a bidirectional reflectance function is.
(b) What does it mean that a reflectance model is empirical?
[4 marks]
[3 marks]
(c) Explain the conceptual role of each of the three terms comprising Phong’s reflectance model (5 marks). Hence write down the equations corresponding to it, explaining the role and meaning of each symbol you introduce (8 marks)
[13 marks]
[Total marks 20]
4. Projections
(a) A camera with the focal length f, located at「0 0 0]T in the world coordinate system, is aligned with the positive direction of the z axis with the x axis pointing upwards. The coordinate system of an avatar is located at「0 1 2]T in the world coordinate system, and its z and x axes are respectively in the directions of 「0 1 一 1]T and「1 0 0]T in the world coordinate system.
Write down the transformation matrix which transforms the location of a point from the avatar’s to the world coordinate system (5 marks). [5 marks]
(b) Using homogeneous coordinates and your answer to (a), compute the location in the world coordinate system of the point located at「2 2 2]T in the avatar’s coordinate system (2 points), and hence its location on the screen using the orthogonal (3 marks) and perspective (4 marks) projections. [9 marks]
(c) The point in (b) is rotated clockwise by 45 О around the z axis in the avatar coordinate system. Compute the new location of the point in the avatar’s coor- dinate system (3 marks) and the screen (3 marks) using orthogonal projection.
[6 marks] [Total marks 20]
2022-04-11