For 2D to 1D
Suppose now that f(x, y)
is a function of two variables. An example would be
f(x, y) = x2 + y2.
It turns out that near any particular 2D point
(x0, y0) the function
is approximately an affine function. In other words, there exist constants
A and B such that for small values of
and
f(x0 + , y0 + )
~ f(x0, y0)
+ A
+ B
For example
(x0 + )2 + (y0 + )2
= x02 + y02
+ 2 x0 + 2 y0
+ 2 + 2
~ x02 + y02
+ 2 x0 + 2 y0
so that here A = 2 x0
and B = 2 y0.
This example suggests how to compute the coefficients A
and B. To see what A is,
fix y temporarily to
be constant,
and view F(x, y)
as a function of x alone.
The coefficient A is then the derivative
with respect to x of this function. It is called
the partial derivative of f(x, y)
with respect to x Similarly for B:
fix x
and take the derivative with respect to
y. There are symbols for the partial derivatives:
fx = and
fy =
For example, if f(x, y) = (x2+y)2
then
the partial derivatives are
4x(x2 + y)
and
2(x2+ y).
These linear approximations can be seen in
a plot of level lines
f(x, y) = const.
If we look only near a fixed point (x0, y0)
then the level lines look very much like the level lines of an affine function - they
are close to being straight,
and they are evenly spaced. Here is what happens for the function
f(x, y) = x4 + y4 as we zoom in.
Gradients
If f(x, y) is a function of two variables, in the neightbourhood
of any point (x0, y0) it
is approximated by an affine function
f(x0, y0) + A (x - x0) + B (y - y0)
where A and B are the partial derivatives of f.
As a consequence, in small
regions its level lines are very nearly the same
as the level lines of this affine function. Now the level lines of the
affine function AX+By+C are straight lines, and they
are perpendicular to the vector
[A, B]. Therefore:
For 2D to 2D
A 2D-to-2D transformation is a transformation from 2D points (x, y) to
other 2D points (u(x, y), v(x, y)). We shall see examples later on,
in the analysis of lens systems. But here I just wwant to derive
a linear approximation formula.
Here, too, there is a linear approximation formula, and it follows from
the previous one. Here's how it goes:
f(x0 + , y0 + )
=
( u(x0 + , y0 + ),
v(x0 + , y0 + ) )
~
( u(x0, y0)
+ ( )
+ ( ) ,
v(x0, y0)
+ ( )
+ ( ) )
= f(x0, y0)
+
In other words, in small regions of the plane
such a transformation is
approximated by an affine transformation,
one obtained from a linear transformation by a translation.