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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 (xi, yi, ui, vi) of S the pair (xi, yi) satisfies 2), which means it lies on C. Thus S contains those points and only those points which lie on C. Every quadruplet (xi, yi, ui, vi) satisfies system 1) which means that the pair (xi, yi) is mapped into (ui, vi). 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.