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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.

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