Lab 2

Pen & Paper

Due: Wednesday, Nov 27th 2019, 23:55

No late submissions / grace days allowed!!


Please note:

This assignment has to be turned in twice:

If you miss to hand in both copies, no points will be awarded for this lab!!


The following questions are a compilation of exercises from the books this class is based upon.

Task 1

Find four points equidistant from one another on a unit sphere. These points determine a tetrahedron. Find the general solution with explanation how you developed it. (Hint: You can arbitrarily let one of the points be at (0, 1, 0) and let the other three be in the plane y = −d, for some positive value of d). (2 points)

Task 2

For many Virtual Reality installations, the COP can be at any point and the projection plane can be at any orientation. Derive the projection matrix for this general case. (2 points)

Task 3

As geometric data passes through the viewing pipeline, a sequence of rotations, translations, scaling, and a projection transformation are applied to the vectors that determine the cosine terms in the Phong reflection model.
  a) Which, if any, of these operations preserve(s) the angles between the vectors? (1 point)
  b) Estimate the amount of extra calculations required for Phong shading as compared to Gouraud shading, taking into account the previously mentioned transformations. (1 point)

Task 4

Using barycentric coordinates, how do you calculate an arbitrary point P in a triangle formed by the Vertices V1, V2, V3? What are some of the practical applications for barycentric coordinates in computer graphics? Name at least two and explain them using the above mentioned triangle. (2 points)

Task 5

Find the projection of a point onto the plane ax + by + cz + d = 0 from a light source located at infinity in the direction (dx,dy,dz). (2 points)