Solutions:

look here.

Answers to frequently-asked questions.

Transformations [4 marks] Consider the following 2-dimensional coordinate frames A and B and C.
1. [2 mark] What are the coordinates of point P in frames A, B and C (note that the vectors are not all unit length)?
2. [2 marks] What is the 3x3 homogeneous matrix M_BC that, when multiplied by a point in frame B, yields that point's coefficients in frame C?
Viewing and Projection [6 marks]
1. [3 marks] A typical property of perspective projection is that families of parallel lines all seem to converge to a point on the image plane. Is this true for all families of parallel lines? If so prove it, if not, given a viewpoint <0,0,0> and a view vector <3,4,0> describe all families of parallel lines that do not converge to a point and prove that all other families do converge to a point. (Note: If a line is described by the parametric equation A+Bt, its family of parallel lines is described by the vector B).
2. [2 marks] Write the eye-space and projective transforms for the viewpoint and view vector above with up vector <0,0,1> and projection plane a distance of d from the viewpoint as 4x4 matrices.
3. [1 mark] Does the family of lines with direction vector <1,0,0> converge on the image plane defined above and if so write the point of convergence as a function of d.
Modeling [4 marks]
1. [2 marks] Given a triangulation of a polygon of n vertices, a new triangulation can be generated using an operation called a diagonal flip. Some four vertices of the polygon that form a quadrilateral with one diagonal (in the given triangulation) are chosen and the alternate diagonal is picked to generate a new triangulation. Given that the input triangulation is valid describe an algorithm to that return true/ false on whether a proposed diagonal flip results in a valid triangulation for a simple polygon.
2. [1 mark] Describe a more efficient algorithm if the input polygon is convex.
3. [1 mark] Given any two triangulations of a convex polygon with n vertices show that you can turn one triangulation into another using at most 2*(n-3) diagonal flips.
Cartoon shading [8 marks]
In cartoon tracing, silhouette edges of a model are drawn as a black outline. For this question we define a silhouette edge to be one for which one adjacent face is a front face and the other a back face with respect to a given viewpoint. Write a program to display a cartoon shading of a mesh specified to stdin as follows:
v 0 0 0

v 1 0 0

v 0 1 0

v 0 0 1

f 3 2 1

f 1 2 4

f 3 1 4

f 2 3 4

e 5 0 0

n -1 0 0

u 0 1 0

here lines beginning with v represent a vertex followed by its location in 3D, lines beginning with f represent faces with the numbers are an index to the vertices of the polygon (starting from 1), in order,

e represents the viewpoint, n the view direction and u the up vector.

In general the file is a number of vertices (one per line) written as v <x> <y> <z>, followed by a number of faces (one per line) written as

f <index1> <index2> ... <index n>, followed by a viewpoint e <x> <y> <z>, view direction n <x> <y> <z> and up vector u <x> <y> <z>.

EXTRA CREDIT: [2 marks] interactively pan the viewpoint based on mouse motion, in the plane defined by the given vewpoint and view direction.

Submit your code electronically, using one of the following commands:

submit -N a2q4 csc418h cartoon.c
submit -N a2q4 csc2504h cartoon.c

Submit a printed copy as well.

Hierarchies [8 marks] Write a program to draw and control this robot:
Skeleton code is provided at
- q5/index.html
Submit your code electronically, using one of the following commands:
- submit -N a2q6 csc418h robot.cpp
- submit -N a2q6 csc2504h robot.cpp
Submit a printed copy as well.