CS3500 Computer Graphics
			      Spring 2009

			   Home Assignment 4
		Transformations and Projective Geometry
		   **** Practice for Mid-Exam 1 ****
			 Due: February 10, 2006

This assignment aims to give you practice with 3D transformations
and a bit of projective geometry.

Projective Geometry of the Plane:
1. Give the representation of the line through the points
   [-1 1 1]' and [1 0 1]' (where ' indicates transposition).

2. Give the intersection of the lines given by [2 3 1]' and
   [1 3 2]'.

3. Give the conic representation as a 3x3 matrix of a circle
   with centre at (2, 3) and a radius of 2 units.

4. Consider a circle of radius R with centre at the origin.  Give its
   conic representation C.  Take a point x = (a, b) on the circle. Prove
   analytically that Cx gives the equation of the tangent line at x.
   
5. Verify this fact for the above circle for points (0, 3), (2, 5), and
   (2 + sqrt(2), 3 - sqrt(2))


6. Derive the complete rotation matrix to rotate about an axis given by
   the direction vector [1 2 3]' by an angle 30 degrees. (Normalize the
   vector yourself). Derive it in two ways by:
   i)  Aligning the axis with X axis     and
   ii) Aligning the axis with Y axis

   Are the final matrices identical?

7. Simplified solar system: Earth and Saturn revolve around the Sun in
   circular orbits of radius R1 and R2 respectively at angular
   velocities theta1 and theta2 respectively. The planes of revolution
   of these two make an angle of k degrees with each other.  Earth and
   Saturn are perfect spheres with radii r1 and r2 respectively. Each
   rotates about its own axis with angular velocities phi1 and phi2. The
   points on the surfaces of these spheres are specified using two
   angles a and b in polar coordinates.

   a) Draw a neat figure indicating the situation and the coordinate
      axes required to reason about the coordinates of points on
      the surfaces of Earth and Saturn.
   b) Give the coordinates of a point given by the angles (a1, b1) on
      Earth and Saturn with respect to the Sun's coordinate axes.
   c) Give the coordinates of the point given by (a1, b1) on the
      surface of Saturn with respect to the point given by (a1, b1)
      on the surface of Earth.

   Make reasonable assumptions, state them clearly, and draw as many
   illustrations as you can!