For a sphere, ϵ = A/R² and ϵ = κA, so κ = 1/R² is the Gaussian curvature of a sphere.
For hyperbolic geometry, κ = -1/R² is the Gaussian curvature.
Gauss, 1827, "General Investigations of Curved Surfaces".

Gaussian = total, intrinsic curvature

ϵ =κA => κ = ϵ/A = angular excess per unit area at a given point

This definition requires imagining an arbitrarily small triangle centred at the point.

There will be other interpretations of curvature.

For now, it can be extended to n-gons. Since the angles of an n-gon in the plane sum (n-2)π, we can define the angle excess as ϵ = (angle sum) - (n-2)π.

My way of thinking the sign of κ:
- think κ as κ = κ_1κ_2
- the signs of either κ_1 or κ_2 mean:
	- κ_i > 0 => κ_1 is hugged by 0 => the point is hugged by the surface => negative, concave
	- κ_i < 0 => κ_1 hugs 0 => the point hugs the surface => positive, convex
	- κ_1 and κ_2 > 0 => minimum => κ > 0
	- κ_1 and κ_2 < 0 => maximum => κ > 0
	- κ_1 and κ_2 with different sign => saddle point => κ < 0
	- κ_1 and/or κ_2 = 0 => intrinsically flat => κ = 0

We could choose, instead of using triangles, the perimeter of a circle, and see how it departures from 2πr. In a sphere, a circle has length C(r), so

2πr - C(r) ≍ πr³/(3R²)

where r is the radius of the circle and R the radius of the sphere. Then,

κ ≍ (3/π)(2πr - C(r))/r³

which is true for every surface.

We could also choose the area of that circle, A(r), and then

κ ≍ (12/π)(πr² - A(r))/r⁴

which is also a universal formula.

Are there surfaces with constant and positive curvature besides the sphere?
Yes, but they must have either spikes or edges.
Intrinsically, they are identical to a sphere.

Are there surfaces with constant negative curvature? Yes.
They are called pseudospherical surfaces.
The simplest example is the pseudosphere.

If several surfaces have constant (positive or negative) curvature, they all have the same intrinsic geometry. However, surfaces with the same but not constant curvature can have different intrinsic geometry.

The local Gauss-Bonet theorem for a geodesic triangle on a general curved surface: the angular excess of that triangle is the total curvature inside it.

ϵ = α + β + γ - π = ∬κdA (∬ over the triangle)

Angular excess is additive. If we subdivide our geodesic triangle in sub-triangles, the sum of the excesses for each triangle would give the excess for the whole triangle:

ϵ(Δ) = ϵ(Δ_1) + ϵ(Δ_2)