Given a curve or surface, its curvature describes how sharply it is bending at a given point. It is defined as the inverse of the radius of curvature , which is the radius of the tangent circle that osculates (i.e. best approximates) the curve/surface at that point:
Typically, is positive for convex curves/surfaces, and negative for concave ones, although this distinction is somewhat arbitrary. Below, we calculate the curvature in several general cases.
2D height functions
We start with a specialized case: height functions, where one coordinate is a function of the other one (2D) or two (3D). In this case, we can use the calculus of variations to find the curvature.
This approach relies on the fact that a circle has the highest area-perimeter ratio of any 2D shape, and a sphere has the highest volume-surface ratio of any 3D body. By the definition of curvature, these shapes have constant .
We will thus minimize the perimeter/surface while keeping the area/volume fixed, which will give us a shape with constant curvature, and from that we can extrapolate an expression for .
In 2D, for a single-variable height function , the length of a small segment of the curve is:
Which leads us to define the following Lagrangian describing the “energy cost” of the curve:
Furthermore, we demand that the area under the curve (i.e. the “volume”) is constant:
By putting these things together, we arrive at the following energy functional , where is an ominously-named Lagrange multiplier:
Minimizing this functional leads to the following Lagrange equation of the first kind:
We evaluate the terms of this equation to arrive at an expression for the curvature :
In this optimization problem, is a constant, but in fact the statement above is valid for variable curvatures too, in which case is a function of .
2D in general
We can parametrically describe an arbitrary plane curve as a function of the arc length :
If we choose the horizontal -axis as a reference, we can furthermore define the elevation angle as the angle between the reference and the curve’s tangent vector :
Where . The curvature is defined as the -derivative of this elevation angle:
We have two ways of writing : using the derivatives and , or the elevation angle . Now, let us take the -derivative of both expressions, and equate them:
We multiply these equation by and , respectively, and subtract the first from the last:
Isolating this for and using the fact that thanks to being the arc length:
While this result is correct, we would like to generalize it to cases where the curve is parametrized by some other , not necessarily the arc length. Let prime denote the -derivative:
By inserting these expression into the earlier formula for , we find:
Since , we know that , which leads us to the following general expression for the curvature of a plane curve:
If the curve happens to be a height function, i.e. , then and , and we arrive at our previous result again.
3D height functions
The generalization to a 3D height function is straightforward: the cost of an infinitesimal portion of the surface is as follows, using the same reasoning as before:
Keeping the volume constant, we get the following energy functional to minimize:
Which gives us an Euler-Lagrange equation involving the Lagrange multiplier :
Inserting into this and evaluating all the derivatives yields a result for the (variable) curvature:
What are and ? Well, the problem in 3D is that the curvature of an osculating circle depends on the orientation of that circle. The principal curvatures and are the largest and smallest curvatures at a given point, but finding their values and the corresponding principal directions is not so easy. Fortunately, in practice, we are often only interested in their sum:
These principal radii and are important for e.g. the Young-Laplace law.
3D in general
To find a general expression for the mean curvature of an arbitrary surface, we “cut off” a small part of the surface that we can regard as a height function. We call the “cutting” reference plane , and the surface it describes . We then define the unit tangent vectors and to be parallel to the -axis and -axis, respectively:
Since they were chosen to lie along the axes, these vectors are not necessarily orthogonal, so we need to normalize the resulting normal vector :
Let us take a look at the divergence of , or to be precise, its projection onto the reference plane (although this distinction is not really important for our purposes):
Compare this with the expression for we found earlier, with the help of variational calculus:
The similarity is clearly visible. This leads us to the following general expression:
A useful property is that the principal directions of curvature are always orthogonal. To show this, consider the most general second-order approximating surface, in polar coordinates:
Sufficiently close to the extremum, where and are negligible, the curvature along a certain direction is given by our earlier formula for a 2D height function:
To find the extremes of , we differentiate with respect to and demand that it is zero:
After rearranging this a bit, we arrive at the following transcendental equation:
Since the function is -periodic, this has two solutions, and , which are clearly orthogonal, hence the principal directions are at an angle of .
Finally, it is also worth mentioning that the principal directions always lie in planes containing the normal of the surface.
- T. Bohr, Curvature of plane curves and surfaces, 2020, unpublished.
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.