Today I wanted to discuss the geometry of curves and surfaces.
Curves, Curvature and Normals
First let us consider a curve r(s) which is parameterised by s, the arc length.
Now, t(s) = is a unit tangent vector and so t2 = 1, thus t.t = 1. If we differentiate this, we get that t.t‘ = 0, which specifies a direction normal to the curve, provided t‘ is not equal to zero. This is because if the dot product of two vectors is zero, then those two vectors are perpendicular to each other.
Let us define t’ = Kn where the unit vector n(s) is called the principal normal and K(s) is called the curvature. Note that we can always make K positive by choosing an appropriate direction for n.
Another interesting quantity is the radius of curvature, a, which is given by
a = 1/curvature
Now that we have n and t we can define a new vector b = t x n, which is orthonormal to both t and n. This is called the binormal. Using this, we can then examine the torsion of the curve, which is given by
T(s) = –b’.n
As the plane is rotated about n we can find a range
where and are the principal curvatures. Then
is called the Gaussian curvature.
Gauss’ Theorema Egregium (which literally translates to ‘Remarkable Theorem’!) says that K is intrinsic to the surface. This means that it can be expressed in terms of lengths, angles, etc. which are measured entirely on the surface!
For example, consider a geodesic triangle on a surface S.
Let θ1, θ2, θ3 be the interior angles. Then the Gauss-Bonnet theorem tells us that
which generalises the angle sum of a triangle to curved space.
Let us check this when S is a sphere of radius a, for which the geodesics are great circles. We can see that == 1/a, and so K = 1/a2, a constant. As shown below, we have a family of geodesic triangles D with θ1 = α, θ2 = θ3 = π/2.
Since K is constant over S,
Then θ1 + θ2 + θ3 = π + α, agreeing with the prediction of the theorem.