General 2-space to 2-space mapping

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General 2-space to 2-space mapping. The system

1) u = u(x, y)

v = v(x, y) ,

where x, y, u, v are real numbers, effects a mapping from the xy plane into the uv plane. It represents the general 2-space to 2-space mapping (or point transformation). It maps a point P(x, y) in the xy-plane into point P'(u, v) in the uv plane.

Let us take a simple mapping and inquire how it maps figures, areas, points. Let us pick the mapping

2) u = x² - y²

v = 2xy

and ask the following question: Into what does the mapping map the rectangular area in the xy plane bounded by the lines x = 1/2, x = 1, y = 1/2, y = 1 shown in the top left of Fig. 2? We answer the question by asking what it maps the lines x = 1/2, x = 1, y = 1/2, y = 1 into. Into what does system 2) map the line x = 1? We obtain the answer by replacing x with 1 in system 2) giving

3) u = 1 - y²

v = 2y ,

a parametric system with y as parameter. Each value of y corresponds to some point on the line x = 1. Eliminating y from system 3) gives

which is the parabola shown in the top right of Fig. 2. In the same way we map the other lines x = 1/2, y = 1 and y = 1/2. The top right figure in Fig. 2 shows the area that the rectangle maps into.

We can ask another question: What system of lines does the line x = c₁ map into, regarding the constant c₁ as a parameter? To answer that question we replace x with c₁ in system 2) giving the parametric system

5) u = c₁² - y²

v = 2c₁y .

We then eliminate y from that system to get

which defines a family of parabolas having the origin of the uv plane as focus, the line v = 0 as axis, and all opening to the left. In the same way we can ask what system of lines the line y = c₂ maps. We proceed in the same way. We substitute c₂ for y in system 2) to obtain the parametric equations

7) u = x² - c₂²

v = 2c₂x

and then eliminate x to obtain

which is the equation of a family of parabolas having the origin as focus, the line v = 0 as axis, and all opening to the right.

Let us now ask a different question. Let us ask this question: What region or regions in the xy plane map into the rectangular area in the uv plane bounded by the lines u = 1/2, u = 1, v = 1/2, v = 1 shown in the bottom right of Fig. 2? We proceed in a similar way. We ask ourselves what line or lines map into the line u = 1. The answer is found by replacing u with 1 in system 2). It is the hyperbola

x² - y² = 1

in the xy plane. What line or lines map into the line v = 1? We replace v with 1 in system 2) to get

xy = 1/2 .

We obtain the lines in the xy plane imaging into the lines u = 1/2 and v = 1/2 in the same way and the areas in the xy plane that map into the specified rectangular area of the uv plane are shown in the bottom left of Fig. 2.

Again we can ask the question: What system of lines in the xy plane map into the line u = k₁ in the uv plane? We substitute k₁ for u in system 2) to get

x² - y² = k₁ ,

a system of rectangular hyperbolas. What system of lines in the xy plane map into the line v = k₂ in the uv plane? We substitute k₂ for v in system 2) to get

xy = k₂/2 ,

another system of rectangular hyperbolas.

Suppose we ask the following question: Into what does system 2) map the line

9) y = 2x + 1

in the xy plane? The answer is found by solving system 2) and equation 9) i.e. the following system

y = 2x + 1

u = x² - y²

v = 2xy ,

simultaneously, eliminating x and y.

After going through the algebra the answer is found to be

16u² + 24uv + 9v² + 12u - 16 v = 4

which is the equation of a parabola in the uv plane.