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 **t**^{2} = 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’** = K**n **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**

### Intrinsic Geometry

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/a^{2}, 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.

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