Define Frame Rate

Midterm ITCS 4120 Spring 2005

Score is out of 100 points.

Name: ______

1) [2pts] Define frame rate.

2) [2pts] Define critical fusion frequency.

3) [2pts] What are the general geometric properties preserved by an affine transformation?

4) [2pts] Define aliasing.

5) [3pts] What are the four types of polygons we discussed in class and what properties does each have? Also provide example illustrations. 6) [3pts] Illustrate the basic components of a CRT. Described these components in a few sentences.

7) [3pts] Given a rotation through an angle θ about an axis described by vector v and through the coordinate origin what is the quaternion representation of this rotation? If you had two such rotations expressed as two different quaternions how would you compute the concatenation of these rotations in quaternion form (give equation)?

8) [3pts] Illustrate the basic architecture of a computer graphics workstation at the level of key processors, buses, etc. 9) [3pts] Illustrate and label in a much detail as possible the components and distinguished locations on the human eye.

10) [3pts] You have two coordinate systems A and B and an affine transformation Ti

that maps the indicated square to the indicated parallelogram. Matrix MiA is a matrix transformation that describes how a point p with coordinates known in A is

transformed by Ti. How would you compute the transformation matrix MiB that transforms by Ti a point p whose coordinates are given in B. You answer should be a one-line, high-level mathematical expression. You do not need to compute any exact, real number values, etc.

B 11) [18pts] Draw the global Edge Table for the polygon below. Next, draw the sequence values the Active Edge Table (AET) would be assigned during the rasterization of this polygon. Also show the y coordinate of the scan-line that the rasterization algorithm is working on, when the AET has each these different assigned values.

A B 12) [18pts] Given the two coordinate systems A and B shown in the diagram:

a) What is the matrix MB←A that transforms a point’s coordinate given in A into its coordinate in B? (From inspecting the diagram you should be able to give the exact real number values of the elements in the matrix—its ok to leave results in a form like cos 45 if you want).

b) If you had to compute MB←A by concatenating together only elementary matrices (translations and rotations, scales , and shears about origin) what would the sequence of these elementary matrices be? You should leave your answer as a sequence of matrices to be concatenated and for each matrix indicate what type of elementary matrix it is.

A 45◦

d B 13) [19pts] Line Clipping: a) What are region codes in the Cohen-Sutherland algorithm? b) How are they computed? c) Sketch an example clipping window and illustrate the region codes in the appropriate regions in the sketch. d) How are the region codes used the clipping algorithm? e) Describe in a few sentences how the Cohen-Sutherland algorithm clips a line segment. 14)

15) [18pts] Polygon Clipping (Sutherland-Hodgman). For the polygon and clipping window below make a set of tables showing the sequence of vertices generated by the Sutherland-Hodgman algorithm as it clips the polygon against the clip window’s left edge, right edge, bottom edge and then top edge. Start by clipping edges (a,b), (b,c) and then (c,a). Also add and label whatever extra geometry (points, line, vectors, etc.) to the diagram that you need to reference when creating your tables the vertex lists in the tables.

b *********************************************************************** *********************************************************************** [12pts] Assume we are rasterizing a circle whose center is at coordinate (0,0) and we’ve just drawn pixel Pk at coordinate (xk,yk). How do determine which pixel to step to next? In particular:

a. What is the basic equation we need to use to make this decision? b. What are the different possible values this equation can have and for each value (or value range) what does it tell us about which pixel to visit next? c. How does this equation relate to the variables we need to define in a program for rasterizing circles?

Also you should augment the diagram with any key geometry (lines, points, vectors, etc.) and explain how they relate to the above.

Pk P?α yk

P?β yk-1

xk xk+1

(=xk+1-1) 16) [12pts] Given the following four control points of a 2D Bezier curve, what is the coordinate of the point on the curve when you set u = ¾. (Assume the parametric curve equation is parameterized on variable u). Show work for full credit.

P0 = (0,0), P2=(3,2),P3=(7,2),P4=(10,0)

Recall the Basis matrix for Bezier curves is: 骣-1 3 - 3 1 琪琪3- 6 3 0 琪-3 3 0 0 琪桫1 0 0 0