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Parameterized Surface with Level Curves

Parameterized surfaces and level curves

Let's consider a continuous surface \(z=f(x, y)\) defined for points \((x, y)\) in a domain \(D\) in the \(x-y\) plane, we can use level curves to visualize the function of two variables \(f(x, y)\) without leaving the plane. The level curves can be regarded as an implicit description of a surface.
For a given value of \(z=k\) in the value domain of \(z=f(x, y)\), the corresponding level curve is simply the cross section of the graph of \(z=f(x,y)\) and the plane \(z=k\). The equation of the corresponding level curve is defined as follows: \[f(x, y) -k =0\] We can select a series of suitable values of k in the value domain of \(f(x,y)\) for the equation of the level curve, then obtain a series of level curves to describe the surface.

The surface to be analyzed

The function of parameterized surface that we chose is defined as \[z=f(x,y)\] \[=-2x^2+2y^2 +(x^2+y^2)^2\] \[-1.5 \le x \le 1.5, -0.6 \le y \le 0.6\] The surface defined by the function is shown in the following picture:
We can choose 10 equally spaced values \(z=f(x,y)=k\), as \(k=-0.8, -0.4, 0, 0.4, 0.8, 1.2, 1.6, 2.0, 2.4, 2.8\), and get a series of level curves shown in the following picture:
For the value \(k=0\), we can get a special level curve which is called the lemniscate of Bernoulli. The function of the lemniscate of Bernoulli in the Cartesian coordinates can be defined as \[-2x^2+2y^2 +(x^2+y^2)^2 = 0\]
For the 3D print model, we add a base at the bottom to make the print process stable.

Why to use this example

This example of parameterized surface with level curves was chosen for the following reasons:
The function of the surface has a concise form, so that we can focus on the concept of visualizing the surface by level curves. For each \(k \lt 0\), the level curves for \(f(x,y)=k\) are two closed circle-like curves. For \(k = 0\), the level curve for \(f(x,y)=0\) is the lemniscate of Bernoulli. For each \(k \gt 0\), the level curve for \(f(x,y)=k\) is actually a closed curve. However, due to the constraints in the domain of the function \(f(x,y)\), \(-1.5 \le x \le 1.5, -0.6 \le y \le 0.6\), each level curve might be broken into four parts.
The parameterized surface can be better understand by using polar coordinates. For \(k = 0\), we get the lemniscate of Bernoulli as the level curve. Its function in polar coordinates can be defined as \[-2\rho^2cos(2\theta)+\rho^4=0\] \[-\pi \le \theta \le \pi\] The function of the lemniscate of Bernoulli in the polar coordinated has a more concise form than that in the Cartesian coordinates.

In geometry, the lemniscate of Bernoulli is an \(x-y\) plane curve defined from two given points \(F_1\) and \(F_2\), known as foci, at distance \(2c\) from each other as the locus of points. For a given point \(P\) on the curve, the distance between \(P\) and \(F_1\) is \(PF_1\), and the distance between \(P\) and \(F_2\) is \(PF_2\). We have \(PF_1PF_2 = c^2\).

This lemniscate was first described in 1694 by Jacob Bernoulli as a modification of an ellipse. The curve of the lemniscate of Bernoulli has a shape similar to the numeral 8 and to the ∞ symbol. The mechanism behind the lemniscate of Bernoulli can be applied to machines in the textile industry, to mechanical toys, etc.

In the above example, the function we chose made a surface with different kinds of level curves, and can be defined in different coordinate systems. This helps us to better understand the concept of parameterized surfaces and level curves.
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