Website owner:  James Miller

Mapping of figures

Let

1)        u = u(x, y)

            v = v(x, y)

represent a mapping from the xy plane into the uv plane. Let

2)        f(x, y) = 0 

be the equation of some curve C in the xy plane. See Fig. 1.

Q. What is the equation of the image C' of C in the uv plane?

A. The equation of the image of f(x, y) = 0 in the uv plane is obtained by solving the system

3)        u = u(x, y)

            v = v(x, y)

f(x, y) = 0

simultaneously and eliminating x and y, to produce an equation g(u, v) = 0. The equation g(u, v) = 0 is the equation of the image C' of C.

Proof. Let S be the solution set of system 3). It consists of all quadruplets (x, y, u, v) that satisfy the system. Every point on C is mapped into the uv plane and is thus included in the solution set. Now in every quadruplet (x_i, y_i, u_i, v_i) of S the pair (x_i, y_i) satisfies 2), which means it lies on C. Thus S contains those points and only those points which lie on C. Every quadruplet (x_i, y_i, u_i, v_i) satisfies system 1) which means that the pair (x_i, y_i) is mapped into (u_i, v_i). Every pair (u, v) that satisfies g(u, v) = 0 is in S and is the image of a point on C.

Note. The above theorem can be generalized to mappings in 3-space or n-space.