Cynthia Vaskis
SLM521 Spring 2004
Dropin #2 Assignment
File: matrotR1.htm
Lesson Title:
Math Calculations to Rotate an Object in 3D Space – Rotation matrix
about the X axis
Let’s define an object in 3D
space as an example. Pick at least three
separate points in 3D space on graph paper and label
the points with their (x, y, z) values.
For the examples here, use these three points which represent a triangle
in 3D space. See diagram of this triangle in 3D space.
Point 1 = (x1, y1, z1) = (1,
1, 1)
Point 2 = (x2, y2, z2) = (2,
2, 2)
Point 3 = (x3, y3, z3) = (1,
1, 2)
To rotate an object around
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.
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.
For rotations about the X
axis, use the matrix equation (5) in the reference made above. To rotate an object about its X axis the
amount of angle alpha, notice that the one (1) in the first row and first
column indicates that the rotation does not change the X 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 our example we will use
the angle alpha = +/-180 degrees, Thus, the cosine of (+/-180)) = -1 and the
sine (+/-180 degrees) = 0.
The rotational matrix for a
rotation about the X axis of +/-180 degrees is below. The object’s points are Point 1 = (x1, y1,
z1) = (1, 1, 1),
Point 2 = (x2, y2, z2) = (2,
2, 2), and Point 3 = (x3, y3, z3) = (1, 1, 2) and they are rotated one at a
time.
Look here for basic matrix multiplication operations.
The first matrix
multiplication follows (rotating Point 1 about the X axis):
____________________________________________________
The second matrix
multiplication is rotating Point 2 about the X axis:
__________________________________________________
The third matrix
multiplication is rotating Point 3 about the X axis:
Remember
these resulting new vectors Point 1 = (1, -1, -1), Point 2 = (2, -2, -2), and
Point 3 = (1, -1, -2) since they will be used as the starting Point vectors for
the Y axis rotations (Pitch maneuvers) next.