Wednesday, Nov. 5, 2003, 7:00 PM

Family Name:_____________________________

First Name:    _____________________________

Student ID:    _____________________________

Instructions:

Attempt all questions.

There are four questions.

The total mark is 30.

You have 50 minutes to complete the test.

Aids allowed: Calculators (you should not need one).

Textbooks and notes are NOT allowed.

1:           /8

2:           /8

3:           /10

4:           /4

_________

Total:                 /30 + 4 extra credit

1. Projection, drawing primitives: [8 marks] An ellipse defined by semi-major and semi-minor axes vectors A, B is to be drawn using orthographic projection with view direction V :

a.      [2 marks] Explain intuitively why the projected curve must also be an ellipse or a straight line.

b.     [3 marks] Under what conditions can the projected curve be a circle and what is the radius of that circle as a function of A, B, V.

c.     [3 marks] Write pseudo code for a loop that plots the ellipse defined as (x/a)2 + (y/b)2  =1 as a sequence of n line segments.

d.     EXTRA CREDIT [2 marks] What property of the curve should determine where to sample points on the ellipse so that the n line segments that connect them best approximate the ellipse. What part of your pseudo-code in part c needs to change to reflect this.

2.     Illumination: [8 marks] Motion capture systems use a number of cameras, and spherical reflective markers to track the position of the markers in time. The figure below shows how it works. A marker reflection needs to be picked up by 2 or more cameras. Given the position of the camera and reflected spot in the camera image, the viewing ray for the camera is determined. Now the 3D position of the marker is the intersection point of the two viewing rays.

 P

 Y

 Camera1

 X

 Camera2

 Z

a.      [2 marks] Why are the markers spherical in shape and reflective?

b.     [4 marks] Two cameras positioned at <0,0,0> and <1,0,7> pick up a marker and their view rays are calculated to be  directed along <1,1,2> and <2,3,-1> respectively. What is the position of the 3D surface point P captured by this setup?

c.      [2 marks] Typical motion capture systems have many more than 2 cameras to track the motion of markers. Why?

d.     EXTRA CREDIT [2 marks] If view vectors 1..Vn for a marker are calculated for n cameras in the scene how will you calculate the 3D marker position?

3.     Visibility: [10 marks] Consider a set of polygons, represented in 2D cross-section as line segments. Arrows represent front-facing normals.

a.      [4 marks] Draw the BSP tree obtained by inserting the polygons in alphabetical order. Left subtrees correspond to the front side, and right subtrees to the back. If any polygons are split, label the fragments on the diagram and the tree.

b.     [4 marks] Consider a drawing primitive, represented as the intersection (hatched region in figure below) of a cylinder (defined by center C, axis A, radius r and height h with no caps) and  a plane with origin O and normal N such that N.A=0. Give the equations and the test by which one may determine if a point P is on one side of the primitive or the other, similar to the planar equation test for polygons in a BSP.

c.     [2 marks] Describe any changes needed to the BSP algorithm to adapt it to be used as visibility ordering algorithm for the partial-cylinder primitive above.

4.     True or False with a reason: [4 marks]

a.      [1 mark] The difference between Gouraud and Phong shading is likely to be more noticeable with diffuse objects than with specular objects.

b.     [1 mark] The difference between Gouraud and Phong shading is likely to be more noticeable with large polygons than with small polygons (size of the polygon here is its projected size in the 2D image).

a.      [1 mark] Real cameras use a lens to compensate for the fact that apertures larger than a pin-hole make the image blurry.

b.     [1 mark] The result of multiple rotations to a point is order dependent but combinations of scaling and translations can be applied in any order with the same result.