CSCI 250 Computer Graphics

Fall 2006 Exam #1

 

Instructions:  Read through the entire test before answering any question in order to maximize your score by answering the easiest questions first.  This test is closed book, closed notes, closed neighbor, open mind.  There are 110 points to the test, not counting bonus questions.  Remember if I can’t read it, I have to assume it is incorrect.  Good luck, and have fun!

 

1.    (60 points) Math stuff:

a.    What is the dot product of [3 7 2] and [1 2 4]?

 

 

 

 

 

 

 

 

 

 

b.    What is the cross product of [3 7 2] and [1 2 4]?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c.    Find a vector perpendicular to the vector [1 2 3].  How do you know it is perpendicular?

 

 

 

 

 

 

 

 

d.    What are the parametric equations for the coordinates of the points P(t) on a ray starting at [2 1 5] and passing through [3 3 3]?

 

 

 

 

 

 

 

 

 

 

 

e.    For the ray in part d, above, what are the coordinates when t=0?  What are the coordinates when t=1?

t=0:   [                              ]

 

 

            t=1:   [                              ]

 

 

 

 

f.     What is the sum of the matrices  and ?

 

 

 

 

 

 

 

 

 

 

 

 

 

g.    What is the product of the matrices   and ?

 

 

 

 

 

 

 

 

 

h.    For a two dimensional transformation, using homogeneous coordinates, what is the matrix that translates all points [x y] by 3 units in the x direction and -2 units in the y direction?

 

 

 

 

 

 

 

 

 

i.      For a two dimensional transformation, using homogeneous coordinates, what is the matrix that rotates all points by 90 degrees?

 

 

 

 

 

 

 

 

 

 

j.      For a two dimensional transformation, using homogeneous coordinates, what is the matrix that scales all points by 2 units in the x direction and 6 units in the y direction?

 

 

 

 

 

 

 

 

 

 

k.    For a two dimensional transformation, using homogeneous coordinates, what is the matrix that reflects all points about the y axis?

 

 

 

 

 

 

 

 

 

l.      Find a vector perpendicular to the plane containing the points (0,0,0), (1,4,2) and (3,-1,5).  Is this vector unique?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m.   Find the equation of a plane whose normal is [2 -1 4] and that passes through the point (1, 1, 1).  Is this equation unique?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n.    Find an implicit equation of a sphere centered at (2, 1, 5) with a radius of 3.  (Remember an implicit equation is of the form f(x,y,z)=0.)

 

 

 

 

 

 

 

 

 

 

 

 

 

o.    Matrix multiplication is not generally commutative.  Give an example where it is commutative and another where it is not.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.    (10 points) Describe a sequence of transformations that would allow you to rotate an object at (3, 7, 2) around the direction vector [2 1 4].  You do NOT have to produce the actual transformation matrices, just describe the sequence of transformations (eg scale the x axis by 4 units, then rotate around the y axis to get the object to lie along the y axis, etc).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.    (15 points)  Let’s work in 2D to make the math a bit easier for this problem.  Consider an ellipse, f(x,y) =  = 0.  Calculate the closest point of intersection of this ellipse with a ray that starts at (2, 5) and passes through (3, 6).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.    (10 points) Illumination model:

a.    Which part computes the ambient light?  What does that computation approximate?

 

 

 

 

 

 

 

 

 

 

 

 

 

b.    What real life phenomenon does the term fatt approximate?

 

 

 

 

 

 

 

 

 

 

 

 

 

c.    Describe what purpose the dot product serves in the specular term.

 

 

 

 

 

 

 

 

 

 

d.    Does the diffuse term depend on the position of the viewer?

 

 

 

5.    (15 points)  The basis matrix for a Bezier curve is .

a.    What are the four blending functions that are multiplied by P1, P2, P3, and P4?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b.    For the four control points (0,0,0), (0,0,1), (1,1,1), and (1,0,1), what are the coordinates of the curve at t = ½ ?

 

 

 

Bonus: (5 points) Mickey and the Four Winds band had a jam session last night at the local pub. The band debuted five new songs during the evening and to everyone’s surprise, each member of the band took the lead singer’s role for one of the songs. By the end of the night, the crowd was on its feet in a standing ovation for the band, whose star was clearly on the rise. The band members were thrilled with the crowd’s approval of their new songs – they were afraid that having all of them taking turns in the lead singer role would turn off the crowd. But they were earning a reputation for performing their music in new ways and their fans accepted their performance with eager demands for more. Determine the full name of each band member, what instrument each played, and the title of the song each member performed as lead singer.

 

1. Wanda, whose last name wasn’t East, was the lead singer for the song, “Girls in the Band”. Tim didn’t play the keyboard.

 

2. The lead singer for the song, “Rockin’ on Down”, played the lead guitar.

 

3. Stella played the bass guitar. The lead singer for “When it Rains” had a last name of West.

 

4. The three guitar players were the lead singer for “Lucky Day”, the person whose last name was South, and Mickey Trent.

 

5. Daniel, whose last name wasn’t North, didn’t play the drums. Wanda’s last name wasn’t North. The band member who played the drums also had a last name of East.

 

6.     The lead singer for “Breakup Blues” didn’t play any of the guitars.

 

First Name

Last Name

Instrument

Song Title

Daniel

 

 

 

Mickey

 

 

 

Stella

 

 

 

Tim

 

 

 

Wanda

 

 

 

 

Last names: East, North, South, Trent, West

Instruments: bass guitar, drums, guitar, keyboard, lead guitar

Song titles: Breakup Blues, Girls in the Band, Lucky Day, Rockin’ on Down, When it Rains

 

Bonus: (5 points) Sudoku!  This one is only Moderate, instead of Tough or Diabolical like the other classes got!

 

 

 

2

 

 

 

8

 

 

 

8

7

 

 

 

 

1

5

 

 

 

4

 

9

 

 

 

 

2

 

7

6

1

 

3

 

 

 

 

 

 

 

 

 

 

 

7

 

3

9

2

 

8

 

 

 

 

6

 

8

 

 

 

5

4

 

 

 

 

3

6

 

 

 

1

 

 

 

9