Suppose we have a function
sends any point
in the plane to another point
We can define smooth curves that pass through
The function
is conformal if the angle between any two paths throughis unchanged by the transformation. This is shown below. The angle between
and
is the same as the angle between
and
A function is conformal if it is conformal on its domain. In fact, any analytic function is conformal at any pointfor which
Proof:
higher order terms.
higher order terms.
Subtract these two to give
higher order terms. The tangent vectorsandare both rotated by
but the angle between them is unchanged. In general of course entire regions are mapped. Some examples are shown below. The grid lines in the plane are transformed onto the curves. Since the angle between any tow grid lines is a right angle, so is the angle between any two curves.
