Cynthia Vaskis
SLM521 Spring 2004
Dropin #2 Assignment
File: matrotY1.htm
Cynthia Vaskis
Lesson Title:
Math Calculations to Rotate an Object in 3D Space – Rotation matrix
about the Z axis
Use graph
paper to draw the resulting Points from the Y axis
rotations. The new Point 1 = (1, -1,
1) from the Y axis rotations where Point 2 and Point 3 must also have been
calculated.
The original Point vectors
were:
Point 1 = (x1, y1, z1) = (1,
1, 1)
Point 2 = (x2, y2, z2) = (2,
2, 2)
Point 3 = (x3, y3, z3) = (1,
1, 2)
See diagram of the original triangle in 3D space.
For reference see how an
object is rotated about its X, Y and Z axes, use the three rotation matrix
equations in (5), (6), and (7) (click and move down to see the rotating
boxes about each axis and matrix equations (5), (6), and (7)). The first box rotates about the X axis, the
second box rotates about the Y axis, and the third box rotates about the Z
axis. The rotation matrices, listed in
matrix equations (5), (6), and (7), are listed again below. We are using equation (7) here for Z axis
rotations.
To rotate an object about
one axis the set of equations is written in the form called a rotational
matrix. It is really a list of
simultaneous equations with their coefficients taken out and placed in a box
structure.
The matrix for rotating an
object about the Z axis is found in equation (7) from the reference above.
To rotate an object about
its Z axis the amount of angle gamma, notice that the one (1) in the third row
and third column indicates that the rotation does not change the Z values in
the object’s definition. The positive
rotation direction is clockwise as you look down the coordinate axis vector
that you are going to rotate around toward the origin of the coordinate system.
To rotate an object about
just one of its axes at a time is done by multiplying the associated rotational
matrix with each of the object’s definition vectors to obtain new definition
vectors that are in the rotated position.
The rotational matrices use
the trigonometry functions of sine and cosine of the rotational angle. To understand how we calculate values for the
sine and cosine we must look at the unit circle with the sine and cosine values
for several angles around the circle. Of
course, you could use any angle amount between -180 degrees and + 180 degrees
but the value would have to be looked up in a trigonometry table. For simplicity, the examples use angles that
produce well know whole number (integer) sine and cosine values such as -1, 0
and +1.
To multiply a matrix times a
vector you first write the matrix down and then write the vector in a vertical
position to the right of the matrix.
For this example, the angle
gamma = -90 degrees. The cosine of -90
degrees = 0 and the sine -90 degrees = -1.
The matrix for a rotation
about the Z axis of -90 degrees is below.
Look here for basic matrix multiplication
operations.
Exercises:
a) First, look at the values in the Yaw matrix
to see if they are correct based on the general Z axis rotational matrix
mentioned above. Then multiply the Z
rotational matrix with the resulting Point 1, 2, and 3 vectors from the Pitch
axis rotations. The resulting vectors
will be the final position of the triangle.
Use graph paper to draw where the final
triangle’s Point vectors (corners) after all of the rotations are performed.
b) Compare where the object’s Point vectors first
started (original Points listed above) and where they ended up?
Hint
– Rotational exercise answers
Summary
You just performed all three
rotations on the triangle’s points by rotating them about each axis
separately. You can accomplish the same
thing by multiplying the first matrix M(Roll) by the second matrix M(Pitch)
and then multiplying the resulting
matrix by the third matrix M(Yaw) to get just one Full Rotational matrix that
could rotate the object’s points to the final position in one matrix multiply
per vector. This is written as M(Yaw) *
M(Pitch) * M(Roll) to produce M(Full Rotation) matrix that could be used to
multiply the point vectors just once to get the final fully rotated point
vectors.
If a matrix has all ones
down the diagonal from the top left down to the bottom right corner and all
zeros elsewhere it is called an Identity matrix. When it is used to multiply a vector it does
nothing to the vector. It is the matrix
equivalent of multiplying by the number one.