Given two points P0 and P1 we can construct the line between them by linear
interpolation:
This idea extends nicely to the two dimensional case.
Let P00, P10, P01, and P11 be four points.
The line between P00 and P10 is
The line between P01 and P11 is
We can connect points on the first line to points on the second line
through the use of a second variable, that is:
![\begin{displaymath}
(1-s)[(1-t) P_{00} + t P_{10}] + s[(1-t) P_{01} + t P_{11}],\quad 0\leq s,\,t \leq 1.
\end{displaymath}](img4.gif) |
(1) |
One nice way to write this is in matrix form:
These very simple surfaces are used in numerous graphics algorithms.
Let's suppose that two of above the points are identical, say
P00 = P10.
In this case, the formula (1) reduces to:
or
A barycentric (or affine) combination of a collection of points is
defined by :
where the weights sum to one:
We see that linear interpolation and bilinear interpolation are examples of
barycentric combinations of points.
Affine maps can be defined as maps that leave combinations invariant,
that is, if
and M is an affine map then
Affine maps can be represented as the sum of a linear transformation and a translation vector:
Translations, scales, rotations, shears, and parallel projections are some
examples of affine maps.
William Shoaff
1999-02-08