1.  (Parametric Curves)

In 2D design a 2 segment Bezier curve which joins the points (0,0)  (1,0) and (1,0) (2,0).
The tangent vector direction is vertical (direction 0,1) at the start of the first segment
and horizontal (direction  1,0 ) at the end of the 2nd segment.   At the point (1,0) the
tangent vector direction is (1,  -1) for both segments.     What are the 4 control points
for each curve? Derive the parametric equations for each curve.    Comment on what
other Bezier curves satisfy these conditions.  Derive the conditions that would give
2nd order continuity at the point (1,0).
 

2. (Perspective Transformation)

In the perspective transformation (viewbox transformation), a division by the homogeneous
coordinate W turns transformed eye coordinates into 2D screen coordinates.   Z_screen is
derived and can be calculated in terms of Z_eye.    Devise an example of  two triangles that
will show that linear interpolation in Z does not give the correct depth at (at least) one point
on the polygons.     Show that your answer is correct.
 

3. (Orthogonal Matrices and Homogeneous Coordinate Systems)

P₀ is the point (0,0,0),  P₁ is the point (1, 1, 1) and P₂ is the point (1, 0, 1).

The matrix:
 

r1x	r2x	r3x	0
r1y	r2y	r3y	0
r1z	r2z	r3z	0
0	0	0	1

rotates the unit vector in the direction of P₀  P₁  into the z-axis.
Calculate the above rotation matrix using the properties of orthogonal matrices.