Computer Graphics, Assignment 2

Part A [45 marks in total]
Below are 4 exercises covering different topics from the weeks of class upto reading week. They require
thought, so you are advised to consult the relevant sections of the textbook, the online lecture notes and
slides, and your notes from class well in advance of the due date. Your proofs and derivations should be
clearly written, mathematically correct, and concise.
All questions require showing the steps toward the solution, and marks will be subtracted if this is
not the case. Even if you cannot answer a question completely, it is very important that you show your
(partial) answers and your reasoning. Otherwise your TA will not be able to award you partial marks.
Both Part A and Part B must be done individually and electronically submitted. Part A must be in PDF
format, by scanning your handwritten solution or by using LaTeX/word to typeset it. Please make sure
that you mention your name, student number, and CDF ID on the first page of your submission for part
A. Otherwise, expect a delay in receiving your grades.
1. Transformations [10 marks]
[4 marks] A 2D Affine transformation is completely specified by its effect on three non-colinear
points, i.e., by how it maps a triangle into another triangle. Find the 2D affine transformation that
maps points 𝒂𝟏, 𝒂𝟐, and 𝒂𝟑 into points 𝒃𝟏, 𝒃𝟐, and 𝒃𝟑, respectively.
Under what conditions (for the points), is this mapping fully determined?
[2 marks] How many point mappings need to be specified to completely determine a general 2D
Homography? A 2D similarity transform?
[4 marks] Are the centroid (average of the three points) and circumcenter (intersection point of
the perpendicular bisectors) of a triangle affine invariant? Prove or provide a counterexample.
2. Viewing and Projection [10 marks]
[3 marks] Why is the image formed in a pinhole camera inverted? (no more than a few sentences)
[3 marks] Given a 3D camera position 𝒄, a point along the viewing direction at the centre of the
screen 𝒑, and a vector parallel to the vertical axis of the screen 𝒖, compute the world to camera
transformation matrix.
[2 marks] Under what conditions will a family of lines parallel to the vector
𝒗 = (𝑣𝑥, 𝑣𝑦, 𝑣𝑧) remain parallel after this perspective projection?
[2 marks] When this condition is not met, do all lines in the family converge at a single 2D point?
If so, which point? If not, provide a counterexample.
3. Surfaces [15 marks]
The tangent plane of a surface at a point is defined so that it contains all tangent vectors. In this exercise,
you will verify that a specific tangent vector is contained in the tangent plane. Let the surface be a torus in
3D (Figure 1) defined by the implicit equation:
𝑓(𝑥, 𝑦, 𝑧) = (𝑅 – 𝑠𝑞𝑟𝑡(𝑥
2 + 𝑦
2
))
2
+ 𝑧
2 − 𝑟
2 = 0, where 𝑅 𝑟.
[3 marks] Give a surface normal at point 𝒑 = (𝑥, 𝑦, 𝑧), using the surface implicit equation.
[3 marks] Give an implicit equation for the tangent plane at 𝒑.
[3 marks] Show that the parametric curve 𝒒(𝜆) = (𝑅 cos𝜆, 𝑅 sin𝜆, 𝑟) lies on the surface.
[3 marks] Find a tangent vector of 𝒒(𝜆) as a function of 𝜆.
[3 marks] Show this tangent vector at 𝒒(𝜆) to lie on the implicit equation of the tangent plane.
Left: Torus, showing a tangent plane, normal and 3D curve at a point. Right: Close-up.
4. Curves [10 marks]
Consider a curve made up of two cubic Bezier segments 𝐵1(𝑡) for 0 ≤ 𝑡 ≤ 1 using 𝑷𝟏 … 𝑷𝟒, and
𝐵2(𝑡 − 1) for 1 ≤ 𝑡 ≤ 2 using 𝑷𝟒 … 𝑷𝟕.
[2 mark] What are the tangents 𝐵1′ and 𝐵2′ at the shared point 𝑷𝟒?
[2 mark] What are the second derivatives 𝐵1′′ and 𝐵2′′ at 𝑷𝟒?
[4 mark] Given 𝑷𝟏 to 𝑷𝟒, are the values of 𝑷𝟓, 𝑷𝟔 and 𝑷𝟕 fixed for the combined curve to be
𝐶
2
continuous? If so, what value are these points constrained to?
[2 mark] Give 4 reasons why cubic Bezier curves are popular in graphics.