Graphics Reference
In-Depth Information
Chapter 10
Frames of Reference in Space
10.1 Introduction
In Chap. 8 we discovered how to compute the coordinates of a point in a frame of
reference in the plane. In this chapter we study the same problem but in 3D space.
Again, we employ many of the concepts previously described in order to develop
the transforms for translating and rotating frames in space.
The relativity of motion, previously discussed, implies that we cannot absolutely
claim that one frame of reference is stationary whilst another is in motion - it is sim-
ply a question of interpretation and convenience. Fortunately, a matrix transform can
be used to support moving points and moving frames, which means that the trans-
form used for rotating a point in a fixed frame of reference, can be used for rotating
the frame of reference in the opposite direction, whilst keeping the point fixed.
In a 3D space context, this implies that the rotation transform R α,x , which rotates
a point α about the fixed x -axis, can be used to rotate the frame of reference
α
about the x -axis, whilst the point remains fixed. Similarly, the rotation transform
R
α about the fixed x -axis, can also be used to rotate
the frame of reference α about the x -axis, whilst the point remains fixed.
We employ a simple notation to distinguish transforms that rotate points from
those that rotate frames. For example, R α,x rotates a point α about the x -axis,
whilst R 1
α,x
α,x , which rotates a point
rotates a frame α about the x -axis. Similarly, R α,x rotates a point
α
about the x -axis, whilst R 1
rotates a frame
α about the x -axis. Also, T t x ,t y ,t z
α,x
translates a point (t x ,t y ,t z ) , whilst T 1
translates a frame (t x ,t y ,t z ) . Similarly,
t x ,t y ,t z
translates a point ( t x , t y , t z ) , whilst T 1
T
translates a frame
t x ,
t y ,
t z
t x ,
t y ,
t z
(
t x ,
t y ,
t z ) .
10.2 Frames of Reference
We have already discussed frames of reference in Chap. 8, and even though the
frames were 2D, the same ideas can be generalised to 3D space without having to
introduce any new concepts - we simply add an extra z -coordinate.
 
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