The Historical Timeline of Minimal Surfaces and the Mathematics behind its Defining Property
 |
| A photo of a catenoid. Image from Catenoid Minimal Surface |
What is a Minimal Surface?
As one might think, a minimal surface is a surface that minimizes its surface area for given boundary conditions. Moreover, the more mathematical definition of a minimal surface is a surface with zero mean curvature (Weisstein). In this post, I will not only expound the mathematical definition of a minimal surface, but I will also provide a brief history of curvature and minimal surfaces.The Study of Curvature and Minimal Surfaces Throughout the Years
The history of curvature goes back a long way, but for this post I will begin telling its story in the 18th century. Leonhard Euler (1707-1783), being the first to study curvature of surfaces, provided the world of differential geometry the formula for normal curvature in 1760 (Schreck, 2016, p.120). This was seen as a huge breakthrough. Adding to the study of curvature in 1771 was Gaspard Monge (1746-1818) who studied space curves. He hinted at the concept of torsion, but it was one of his students, Michel Ange Lancret (1774-1807), who described “the curvature and torsion of a space curve as infinitesimal angles of rotation of the normal and osculating planes” and gave an explicit formula for torsion in 1806 (Schreck, 2016, p.108). Lancret gave credit to Joseph Fourier (1768-1830) because Fourier had established the relationship between curvature and torsion.
Carl Friedrich Gauss (1777-1855) also studied the curvature of surfaces as he recognized its importance in differential geometry. In his paper that was published in 1828, “Disquisitiones generales circa superficies curvas,” Gauss delineated his ideas on intrinsic properties of surfaces and in doing so, he presented the curvature K as a function of the coefficients of the first and second fundamental form (Schreck, 2016, p.122). His theorem, Theorema Egregium, proved that the principal curvatures individually depend on extrinsic properties of the surface, but Gaussian curvature, as the product of the principal curvatures, is an intrinsic measure of curvature (Schreck, 2016, p.123). Here, intrinsic properties are those that can be measured within the surface itself, such as arc length and surface area, while extrinsic properties provides information on how the surface is embedded in R3 (Schreck, 2019a).
An extrinsic measure of curvature is mean curvature, which was developed by Sophie Germain (1776-1831) in 1830 during her research. In her final paper on her theory of vibrating surfaces, “Mémoire sur la courbure des surfaces,” Germain defined mean curvature as half the sum of the principal curvatures (Bradley, 2006, p.11). Mean curvature was originally developed by Germain because it was vastly useful in her studies of elasticity theory, and now we see it play an essential role in defining minimal surfaces.
It is said that the history of the theory of minimal surfaces begins with Joseph-Louis Lagrange (1736-1813), who studied minimal surfaces in Euclidean 3-space; however, the first nontrivial minimal surface was discovered by Euler in 1744, which we now know as the catenoid (Dillen & Verstraelen, 2000, p.207). In 1762, Lagrange analytically derived the minimal surface equation utilizing methods of calculus of variations (Schreck, 2016, p.147). The equation of minimal surfaces is as follows:
for a surface defined by z=f (x,y) (Dillen & Verstraelen, 2000, p.207). Although it was Lagrange who discovered the equation of minimal surfaces, Jean-Baptiste Meusnier (1754-1793) provided the geometrical interpretation of the equation in 1776 (Schreck, 2016, p.147). He found that the equation implied that the surface has a “vanishing mean curvature function” (Dillen & Verstraelen, 2000, p.207). Furthermore, Meusnier is credited for verifying that the catenoid and helicoid satisfy Lagrange’s equation of minimal surfaces. It took decades before there was another major advancement and more minimal surfaces were discovered by Heinrich Scherk (1798-1885) in 1834, including Scherk’s First Surface and Scherk’s Second Surface (Dillen & Verstraelen, 2000, p.207).
In 1840, Joseph Plateau (1801-1883), a physicist, began his investigation on minimal surfaces and soap films when a servant spilled oil into a container holding a mixture of water and alcohol. Plateau observed the oil droplets forming into perfect spheres in the mixture; thus, he conducted several experiments by recreating the spilling of the oil into the water and alcohol mixture and studied the shape of the oil droplets while rotating the mixture (O'Connor & Robertson). Eventually, he started using soapy water with glycerine and he submerged wire contours into it. He came to the conclusion that the surfaces formed by the soap films were minimal surfaces; however, he did not possess the mathematical skills to study it theoretically (O'Connor & Robertson). Therefore, showing the existence of a minimal surface with a given boundary became known as Plateau’s problem, even though it was Lagrange who first shed light on that problem. Plateau’s problem went unsolved for some time. Mathematicians, specifically Weierstrass, Riemann, and Schwarz, all contributed to solving Plateau’s problem, but it wasn’t until Douglas and Radó individually published their solutions of the problem around 1930 that it was completely solved (O'Connor & Robertson).
Surface Parameterizations
Before I begin explaining curvature of surfaces, it is essential to have a thorough understanding of surface parameterizations. Sometimes it is difficult to determine the curvature of a surface at a point when the equation of a surface is given in its implicit form; therefore, finding a parameterization of the equation allows for the curvature to be calculated more easily. Because a surface in R3 is being parameterized, two parameters are needed to define it (Schreck, 2019a).
A surface in R3 can be parameterized as r(u,v)=< x(u,v), y(u,v), z(u,v) > with u and v being the two parameters. As the parameters vary over the rectangle R={(u,v): a ≤ u ≤ b, c ≤ v ≤ d}, the vector r(u,v) generates a surface in R3 (Schreck, 2019a).
By allowing v to equal v0, we are fixing v and allowing u to vary, so r(u,v0) depends on one variable, making it a curve. This curve is called the u-parameter curve. Likewise, when u=u0 and v varies, r(u0,v) is a curve called the v-parameter curve. In order to find the tangent vectors of the u and v-parameter curves, the partial derivative of r(u,v) must be taken with respect to u and v. The partial derivatives are:
A surface of revolution is a surface that is generated when a curve is revolved around the x or y axis or a straight line. For example, the catenoid is a surface of revolution generated by rotating a catenary curve about its directrix. Consider the surface created by the curve y=f (x) on a ≤ x ≤ b rotated about the x-axis. If v is the angle of rotation and x=u, then the surface of revolution can be parameterized by r(u,v)=(u, f(u)cos(v), f(u)sin(v)) (Schreck, 2019a). This parameterization was seen in my goblet post, except that I revolved the goblet about the z-axis rather than the x-axis.
A surface in R3 can be parameterized as r(u,v)=< x(u,v), y(u,v), z(u,v) > with u and v being the two parameters. As the parameters vary over the rectangle R={(u,v): a ≤ u ≤ b, c ≤ v ≤ d}, the vector r(u,v) generates a surface in R3 (Schreck, 2019a).
By allowing v to equal v0, we are fixing v and allowing u to vary, so r(u,v0) depends on one variable, making it a curve. This curve is called the u-parameter curve. Likewise, when u=u0 and v varies, r(u0,v) is a curve called the v-parameter curve. In order to find the tangent vectors of the u and v-parameter curves, the partial derivative of r(u,v) must be taken with respect to u and v. The partial derivatives are:
Evaluating the partial derivatives at point (u0,v0) gives us the tangent vectors of the parameter curves, ru(u0,v0) and rv(u0,v0). If C is any curve on the surface S passing through point P, we differentiate the curve r(t)=< u(t), v(t) > with respect to t in the following manner:
Note that the Chain Rule was applied.
Example 1:
Example 2:
Various Types of Curvature with Examples
Prior to looking at curvature of surfaces, it is beneficial to study curvature of 2D curves first. Curvature is the rate of change in which a curve changes direction. It is also the rate at which the direction of the unit tangent vector changes (Schreck, 2019b).
Given r(t)=< x(t), y(t), z(t) >, the unit tangent vector is
This vector points in the direction of r’(t).
The magnitude of the rate at which T changes with respect to the arc length is the curvature of the curve. The curve has a large curvature if a small increment in Δs results in a big change in the direction of T, meaning that the curve is bending quickly. On the other hand, the curve has a small curvature if a small increment in Δs results in a small change in the direction of T; thus, the curve is bending slowly (Schreck, 2019b). |
| The curve on the left is a depiction of large curvature as T changes greatly when there is a small increase of the arc length. The curve of the right shows small curvature because there is very little change in T as the arc length changes (Schreck, 2019b). |
The formal definition of curvature is as follows:
Let r(t)=< x(t), y(t), z(t) > be a smooth parameterized curve, then the curvature defined as
where s denotes arc length and T is defined the same as above.
κ is a scalar-valued function such that κ ≥ 0. If κ is large, the curve is tight and changing direction quickly. If κ is small, the curve is flatter and is changing direction slowly. Lastly, if κ = 0, the curve is a straight line and T is not changing direction (Schreck, 2019b).
Furthermore, the formula for curvature is
Example:
The curve r(t) is a circle with curvature of 1/10.
In mathematics, we know that most concepts have numerous equations that represent them. Curvature is one of those concepts as its alternative formula is the following:
Let r(t)=< x(t), y(t), z(t) > be a smooth parameterized curve, then the curvature defined as
κ is a scalar-valued function such that κ ≥ 0. If κ is large, the curve is tight and changing direction quickly. If κ is small, the curve is flatter and is changing direction slowly. Lastly, if κ = 0, the curve is a straight line and T is not changing direction (Schreck, 2019b).
Furthermore, the formula for curvature is
when r(t) describes a smooth parameterized curve and t is any parameter (Schreck, 2019b).
Proof:
In mathematics, we know that most concepts have numerous equations that represent them. Curvature is one of those concepts as its alternative formula is the following:
where r(t) is the position function, v(t)=r'(t) is the velocity, and a(t)=v'(t) is the acceleration (Schreck, 2019b).
Example:
Thus far, curvature has only told us about how fast a curve turns, not the direction in which it is turning. The principal unit normal vector tells us the direction the curve is turning as it points to the inside of the curve (Schreck, 2019b).
If r is a smooth curve parameterized by arc length, s, the principal unit normal vector at point P is
However, if r=r(t), meaning it is not parameterized by arc length, then the following formula can be used:
 |
| I added a couple more unit tangent vectors and principal unit normal vectors to the curve above to highlight the movement of the vectors as they travel along the curve. Image from (Schreck, 2019b). |
Example:
Torsion is the rate at which the curves moves out of the osculating plane, which is also the rate at which B changes as it moves along the curve (Schreck, 2019c). The formula for torsion is
Normal curvature, also denoted by κ, is the curvature of the normal section.
For a given unit tangent vector T, the normal curvature in the direction of T is defined as
where r(t) is the curve of intersection and r(t) needs to be parameterized first with respect to the arc length to make calculation easier (Schreck, 2019a). To parameterize r(t) with respect to arc length, the following four steps are necessary:
Example:
Proof:
Because r’(s) is a tangent vector and N is normal to the tangent plane, then the dot product of r’ and N is zero. By taking the derivative of both sides of the equation, we obtain:
Recall that the derivative of r is taken with respect to arc length; thus, the derivative of N must also be taken with respect to arc length and using the Chain Rule, we get:
where e, f, and g are coefficients of the Second Fundamental Form. The Second Fundamental Form gives extrinsic information about the surface, while the First Fundamental Form gives intrinsic information of a surface. The First Fundamental Form is computed by taking the dot product of the tangent vector with itself, giving us the following coefficients:
We will see that the two fundamental forms are crucial when calculating the Gaussian and mean curvature.
As stated earlier, Euler presented his formula for normal curvature in 1760. His formula is given by
where k1 and k2 are the principal curvatures.
Given unit vectors w1 and w2, the maximum and minimum values of the normal curvature are given by
w1 and w2 are called the principal vectors or directions (Schreck, 2019a).
Euler’s formula is important because it proves the existence of principal curvature and principal direction, leading us to study two other types of curvature: Gaussian and mean curvature. |
| On this saddle surface, the minimum curvature is when theta equals zero. |
 |
| The maximum curvature is when theta equals 1.5708. |
Gaussian curvature is given by
where k1 and k2 are the principal curvatures (Schreck, 2019a).
It can also be defined using the coefficients of the fundamental forms:
However, in order to make computing the mean curvature simpler, it can be expressed in terms of the coefficients of the First and Second Fundamental Form, denoted as
Mean curvature has many applications, but the most notable is its importance in the analysis of minimal surfaces and soap films. As stated a couple times throughout this post, minimal surfaces are characterized as having zero mean curvature and the same applies to soap films. In the following example, I will show how to compute the Gaussian and mean curvature for Enneper’s surface.
Overall, the defining property of a minimal surface is having zero mean curvature. There is so much more to minimal surfaces, but all of the math that I learned in the course of eight weeks is included in this post. I was not able to further study the math behind minimal surfaces as it became more advanced; however, I enjoyed reading about the higher level math that goes into minimal surfaces, and I hope to continue studying it as I gain the mathematical skills in the next couple of years.
Schreck, K. R. (2016). Monge’s legacy of descriptive and differential geometry. Boston, MA: Docent Press.
We can determine characteristics and geometries of surfaces by looking at the sign of Gaussian curvature. For example, if the product of the principal curvatures is zero, the surface is a developable surface, meaning that the surface can be flattened out to the point where it is a plane. Cones and cylinders are examples of developable surfaces. Moreover, the geometry of a surface with zero Gaussian curvature is Euclidean geometry. If the Gaussian curvature is positive, the product of the principal curvatures is greater than zero and the surface is a sphere. The geometry of this surface is spherical geometry. Lastly, if the product of the principal curvatures is less than zero, the Gaussian curvature is negative; therefore, the surface is a pseudosphere and its geometry is hyperbolic geometry (Schreck, 2019a).
 |
| I was not familiar with the pseudosphere until now and I am intrigued by it. Image from Wolfram Mathworld |
The last type of curvature to be discussed is mean curvature. Germain formulated the following equation as the mean curvature of a surface:
 |
| Different perspectives of Enneper's surface that I plotted in Maple. |
Overall, the defining property of a minimal surface is having zero mean curvature. There is so much more to minimal surfaces, but all of the math that I learned in the course of eight weeks is included in this post. I was not able to further study the math behind minimal surfaces as it became more advanced; however, I enjoyed reading about the higher level math that goes into minimal surfaces, and I hope to continue studying it as I gain the mathematical skills in the next couple of years.
References:
Bradley, M. (2006). The foundations of mathematics: 1800 to 1900. New York, NY: Facts on File.
Dillen, F. J. E., & Verstraelen, L. C. A. (2000). Handbook of differential geometry, volume 1. Amsterdam, The Netherlands: North Holland.
McCleary, J. (2012). Geometry from a differentiable viewpoint (2nd ed.). New York, NY: Cambridge University Press.
McCleary, J. (2012). Geometry from a differentiable viewpoint (2nd ed.). New York, NY: Cambridge University Press.
O’Connor, J. J., & Robertson, E. F. Joseph Plateau. MacTutor History of Mathematics. Retrieved from http://www-history.mcs.st-and.ac.uk/Biographies/Plateau.html
Schreck, K. R. (2016). Monge’s legacy of descriptive and differential geometry. Boston, MA: Docent Press.
Schreck, K. R. (2019a). Curvature of Surfaces. Personal Collection of K. R. Schreck, Saint Xavier University, Chicago, IL.
Schreck, K. R. (2019b). Introduction to Curvature. Personal Collection of K. R. Schreck, Saint Xavier University, Chicago, IL.
Schreck, K. R. (2019c). TNB Frame. Personal Collection of K. R. Schreck, Saint Xavier University, Chicago, IL.
Weisstein, E. W. Minimal surface. Wolfram Mathworld. Retrieved from http://mathworld.wolfram.com/MinimalSurface.html
Comments
Post a Comment