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Homework 4 Problems

1) Given the three-dimensional point located at coordinates [5, 3, 4] in a right-handed cartesian system, what would its position be after rotation about the X axis by 30 degrees?

x' = x
y' = cos Q * y - sin Q * z
z' = sin Q * y + cos Q * z

x' = 5
y' = cos 30 * 3 - sin 30 * 4 = .598
z' = sin 30 * 3 + cos 30 * 4 = 4.96

[ 5, .598, 4.96]

2) Show the transformation matrix needed to scale an object (centered about the origin) so that its Y dimension increases by 75%, its Z dimension doubles, and its X dimension remains the same.

I	1	0	0	I
I	0	1.75	0	I
I	0	0	2	I

3) Adding the vector [4, 6, 2] and the vector [6, 4, 8] results in what vector?

[4, 6, 2] + [6, 4, 8] = [(4+6) (6+4) (2+8)] = [10, 10, 10]

4) Given a three-dimensional object defined in a right-handed cartesian coordinate system, what would be required to describe this same object, in the same spatial position, if the right-handed system were replaced by a left-handed system? Assume the origins of both systems are the same.

The difference between the left and right- handed systems is the direction of the z axis. Therefore, you would change the sign (+,-) of one of the three axis. For example, if the x and y coordinates remained the same, you would mutitply the z coordinate by a -1.

5) What is the dot or inner product of the two vectors [1, 0, 4] and [5, 4, 1] ?

Dot product of [1, 0, 4] · [5, 4, 1] = [(1*5)+(0*4)+(4*1)] = 9

6) How many coordinate values are needed to locate a point in a four dimensional space?

Four coordinate values are needed to locate point in 4D space.

7) Given the point [5, 6, 3], what is its orthogonal projection onto the x,y image plane?

Drop the Z coordinate. [5, 6]

8) Given the same point [5, 6, 3], what is its perspective projection onto the image plane if the image plane is located 1 unit in front of the eye (in the Z direction) and the eye is located at the origin?

f = 1 [5, 6,3]

x' = X' / w	y' = Y' / w	z' = Z' / w
x' = x / ( z/f )	y' = y / ( z/f)	z' = f
x’ = 5/(3/1)	y’ = 6/(3/1)	z’ = 1
x’ = 5/3	y’ = 2

[ 5/3, 2 ]

9) What is the transformation matrix needed to move a point +4 units in the X direction, +3 units in the Y direction, and -2 units in the Z direction?

I	1	0	0	4	I
I	0	1	0	3	I
I	0	0	1	-2	I
I	0	0	0	1	I

10) What is the transformation matrix needed to rotate an object 60 degrees about the Z axis in a homogeneous three-dimensional cartesian space?

x' = cos Q * x - sin Q * y
y' = sin Q * x + cos Q * y
z' = z

x' = cos 60 * x - sin 60 * y
y' = sin 60 * x + cos 60 * y
z' = z

I	cos 60	-sin 60	0	0	I
I	sin 60	cos 60	0	0	I
I	0	0	1	0	I
I	0	0	0	1	I