conversion
matrix.
The general n-dimensional case is then
discussed. We
conclude with a discussion of
how to generate this conversion matrix.
On-Line Geometric Modeling Notes
In computer graphics we manipulate objects. These may include light sources, cameras, and objects in our scenes. Each of these is likely to be defined in its own coordinate system and then placed within the scene we are modeling. We must be able to relate these coordinate systems, both to a global coordinate system and to each other. We place coordinate systems into an affine space through the use of frames.
We first discuss the
definition of a frame and give several
examples of frames including the well known
Cartesian Frame.
A matrix representation of
points and vectors under a given frame is
developed and utilized to discuss the
conversion of coordinates between frames
The method of conversion is developed first through
a specific two dimensional example which shows
that the problem can be solved by developing a conversion
matrix.
The general n-dimensional case is then
discussed. We
conclude with a discussion of
how to generate this conversion matrix.
For a postscript version of these notes look here.
Let 
be an
affine space
of dimension n. Let 
be a point
in this space and let
be any basis for 
. We call the
collection 
a frame
for 
. Frames form coordinate systems in our affine space.
That is, if we consider a point 
, 
can be written as 
, where 
is the origin of the frame and 
is a vector.
Since the 
forms a basis for 
, we have
that the vector 
can be written uniquely as
and so the point 
can be written uniquely as
The set of values
are the coordinates of 
relative to the frame 
.
Examples of Frames
Two-dimensional examples of frames are fairly straightforward to
produce. We will denote the frames as 
.
, 
and
. Most readers will equate the coordinates (x,y)
to points in this frame given by
In fact, we can define a Cartesian frame for any dimension. It consists of the origin (0, 0, ..., 0), and the vectors <1,0,..,0>, <0,1,..,0>, ..., <0,0,..,1>. This is the frame that we normally refer to when discussing points in multiple dimensions.
and 
. The point 
that
has coordinates (5,3) in this frame, can be written as
and 
. The point 
that
has coordinates (5,3) in this frame, can be written as
Points and vectors can be uniquely identified by the coordinates
relative to a specific frame.
Given a frame 
in an affine space
, we can write a point 
uniquely as
This can also be written as
Similarly, since the vectors of our affine space form a
vector space,
we can write a vector 
uniquely as
and this can be written as
Thus, points are represented as row vectors whose last component is 1 and vectors are represented as row vectors whose last component is 0 (zero).
A useful technique is, when given two different frames, to take a point that has a certain set of coordinates in one frame and find its coordinates in the second frame. Most readers will do this routinely, relating points back to the Cartesian frame. But what if the second frame is not the Cartesian frame?
It is useful to study a two dimensional example of this process before proceeding in general.
Consider the two frames 
and 
where
These frames are shown in the figure below
Suppose we have the point 
, with coordinates ( 3, 2 ) in the
frame 
. What are the coordinates of the point
in the frame 
?
This is straightforward to calculate using the
matrix notation for the
points.
and therefore the new coordinate is (3,0) under the frame
.
Now carefully note what we did in this calculation. First we wrote
the point in the
matrix notation for
the frame 
.
Then we wrote the vectors of the first frame in terms of the vectors
of the second frame - and we can do this since the vectors of the
second frame (any frame actually) form a basis for the space of
vectors. Then we wrote the origin 
in terms of the origin and
vectors of the second frame. This column vector, expressing the first
frame in terms of the second, was converted to the product of
a 
matrix and the column vector representing the second
frame. We then
multiplied the 
matrix times the initial coordinate to
obtain the result.
This can be done with any coordinate from the first frame, as the
matrix generated will be the same in every case. That
is, given a coordinate (u,v) from the first frame, we can convert it
to a coordinate for the second frame by simply calculating
To convert coordinates from one frame to another in n-dimensional space proceeds in the same way as the example above.
Suppose a point 
has coordinates 
relative
to some frame
.
What would be
the coordinates of 
relative to another frame
?
Well since 
is a basis, we can write
each of the vectors 
uniquely in terms of
the 
. In
addition, since 
is a vector, we can also write 
uniquely
in terms of the 
and 
. Thus we can calculate coefficients 
defined by
for i = 1, 2, ..., n and
So now we can write
and since the vectors 
are linearly independent we have that the coordinates
of the point in the second frame is
Thus the change of coordinates is accomplished via a matrix
multiplication. We note that the rows of the matrix consist of the
coordinates of the elements of the old frame 
relative to the new
frame 
.
We also note that if we consider the frames in three dimensional space
(i.e. n=3), the matrix is 
.
So how do we calculate the matrix
which is required for the conversion of the coordinates of a point in
one frame to those of another? We use
Cramer's Rule.
Since most of the calculations we are required to do are in 3
dimensions, we will present the techniques for calculating the matrix
in this special case - where the frames will now be denoted as
and
and the matrix will be
So how do we find the values 
?
Given any frame 
, and a vector 
,
I know that 
for some u, v and w. Utilizing
Cramer's Rule,
I can calculate these coordinates
directly by the following process: If we define
then we can calculate
Utilizing this process, we can let
, and
we can let 
and calculate
,
and
;
we can let 
and calculate
,
and
;
we can let 
and calculate
,
and finally
we can let 
and calculate
,
and
.
So using Cramer's rule, the calculations are straightforward and consist of taking a few cross products and dot products.
See the section on Cramer's Rule for more information when the frames are orthonormal.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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