Curvature of Surfaces
How curved is a curve? How curved is a surface? When is a ‘curved surface’ flat? We shall only briefly mention curves in the plane and then move on to discuss positive and negative curvature of surfaces.
Imagine tracing out the ellipse shown in the diagram. How does the curvature change as you go around the ellipse? Without applying any mathematics everyone would agree that the tightest bends are at the ends and the least curvature on the track around the ellipse is halfway between these points .
To measure the curvature at a point you have to find the circle of best fit at that point. This is called the osculating circle. The curvature of the curve at that point is defined to be the reciprocal of the radius of the osculating circle. We shall not discuss in this article the method for finding this radius accurately which needs calculus. Suffice it to say that this circle has not only the same tangent at the point (the first derivatives being the same) but also the curve and the osculating circle have the same second derivatives at the point.
Why use the reciprocal in defining curvature? It is natural for the curvature of a straight line to be zero. Imagine straightening out a curve making it into a straight line. In the limit the circle of best fit has infinite radius giving zero curvature.
The diagram shows osculating circles to the ellipse at points A, B and C. At A the curvature is 23
The equation of this ellipse is
x236+y29=1.
At the point on the ellipse (x,y)=(acosθ,bsinθ) with (a=6, b=3), the curvature is given by
ab(a2sin2θ+b2cos2θ)3/2.
A perfect sphere has constant curvature everywhere on the surface whereas the curvature on other surfaces is variable. For example on a rubgy ball the curvature is greatest at the ends and least in the middle. Measuring curvature at a point using curves through that point on the surface will not work. Which plane curve should we use? At the ‘2’ on the rugby ball, the curve in one direction, going between the B and the E, has greater curvature than the curve along the length of the ball.
Gauss proved that, taking the curvatures in all directions at a point on a surface, the product of the maximum and minimum curvatures at the point is constant when the surface is distorted provided that lengths in the surface are unchanged.
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One method used to measure the Gaussian curvature of a surface at a point is to take a small circle of radius r on the surface with centre at that point and to calculate the circumference or area of the circle. If the circumference is 2πr and the area is πr2 the surface is flat and is said to have zero curvature. If the circumference is less than 2πr and the area is less than πr2 the surface has positive curvature; if the circumference is greater than 2πr and the area is greater than πr2 the surface has negative curvature. There are three distinct geometries for the three types of surfaces and each of these geometries has its own trigonometry with the results in each trigonometry having counterparts in the other two trigonometries. Euclidean geometry is the geometry of surfaces with zero curvature. Spherical geometry, also known as elliptic geometry, is the geometry of surfaces with positive curvature. Hyperbolic geometry is the geometry of surfaces with negative curvature.