Assignment 2 Solution

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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 , , and into points , , and , 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.

  1. 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 , a point along the viewing direction at the centre of the screen , and a vector parallel to the vertical axis of the screen , compute the world to camera transformation matrix.

[2 marks] Under what conditions will a family of lines parallel to the vector

= ( , , ) 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:

( , , ) = ( – ( 2 + 2))2 + 2 − 2 = 0, where > .

[3 marks] Give a surface normal at point = ( , , ), using the surface implicit equation.

[3 marks] Give an implicit equation for the tangent plane at .

[3 marks] Show that the parametric curve ( ) = ( cos , sin , ) lies on the surface.

[3 marks] Find a tangent vector of ( ) as a function of .

[3 marks] Show this tangent vector at ( ) 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 1( ) for 0 ≤ ≤ 1 using … , and 2( − 1) for 1 ≤ ≤ 2 using … .

[2 mark] What are the tangents 1′ and 2′ at the shared point ?

[2 mark] What are the second derivatives 1′′ and 2′′ at ?

[4 mark] Given to , are the values of , and fixed for the combined curve to be

2 continuous? If so, what value are these points constrained to?

[2 mark] Give 4 reasons why cubic Bezier curves are popular in graphics.