1) Write the parametric form of a ray with source p and direction d.

2) Assuming d in the above ray is a unit vector, write an algorithm that, for an arbitrary non-negative integer n, generates n evenly spaced points along the ray at 1 unit intervals.

3) Write the parametric form of the unit circle.

4) Given an arbitrary positive integer n, write an algorithm that generates n evenly spaced points that are sampled along the unit circle.

5) Suppose you have access to the functions scale(sx, sy), translate(tx, ty), and rotate(α) which generate the corresponding 2D homogeneous transformation matrices. Using these functions, write the expression of a transformation T such that if p lies on the unit circle, Tp will lie on an ellipse centered at c, with a major radius r1, minor radius r2 and rotated at an angle of α. You do not have to compute the full matrix. You may leave it expressed using the above functions. Hint: remember that order is important and that operations associate from the "inside out".

6) Write the implicit form of the unit sphere.

7) Given an arbitrary point p, write a test to determine if p is inside the unit sphere.

8) Suppose you are given an affine transformation T that maps the unit sphere to some arbitrarily located and oriented ellipsoid. Give an expression which, given an arbitrary point p, determines if p is inside said ellipsoid. Hint: use the result from the previous exercise.

9) You are given the vertices of a convex polygon in the 2D plane in counter-clockwise order as (p1, ..., pn). The coordinates of vertex pi are (xi, yi).

9a) Give an expression for the coordinates of the outward-facing normal ni of the edge connecting pi and pi+1.

9b) Let q = (xq, yq) be an arbitrary point on the plane containing the given polygon. Let l be the line containing pi and pi+1, and let ni be the outward-facing normal vector as given in part a). What is a test that determines whether or not q lies on the outward side of l (the side toward which ni points)?

9c)  Provide an algorithm that determines whether a 2D point q is inside, outside, or on the boundary of the given polygon. Hint: Each edge is contained in an infinite line. Each infinite line divides the 2D plane into two half-planes: the "left" half-plane and the "right" half-plane (left and right are defined with respect to a counter-clockwise direction of traversal of the vertices). The key insight you should use is that the interior of a convex polygon is the intersection of the left half-planes of each edge of the polygon.

10) In stereo rendering, two cameras are needed, with slightly different vantage points and view directions; each is used to render an image for the corresponding eye. This can be specified with the following parameters:

• c: The center of interest, a point in world space that lies along the optical axis of both cameras.

• em: The midpoint between the eye positions of each camera.

• t: An "up" vector that allows us to specify a tilt rotation about the axis passing through c and em. The z axis is a special case.

• s: The distance between the two eyes.

All answers should be given in terms of the above stereo parameters, as well as any intermediate quantities you specify.

10a) Given an expression for the unit vector d that is perpendicular to both t and c - em such that (c - em, t, d) is a right-handed coordinate frame.

10b) Give expressions for eL and eR, the eye positions of each camera.

10c) Give expressions for the basis vectors that make up the two cameras' coordinate frames: (uL, vL, wL) and (uR, vR, wR)

10d) Give a test that determines whether or not a polygon face with normal n is a back face that can be culled when rendering from both cameras.

11) This illustration shows a top-down view of a 3D scene where each blue edge corresponds to a planar square perpendicular to the image plane (i.e. coming out of the page) and the eye e of a possible viewpoint lies in the image plane. The short vectors are normal vectors.

11a) Assuming the particular scene and camera placement shown above, is it possible to exclude any polygons from rendering? Explain your answer.

11b) Draw the BSP tree for the above scene that would result from adding the polygons to the tree in the order (S1, S2, S3, S4, S5, S6).

11c) Describe how your tree will be traversed when rendering the scene from the eye location specified.