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Surfaces and Gaussian curvature

Principal curvatures, Gauss's Theorema Egregium, geodesics.

At each point a surface bends most and least in two perpendicular directions; their product is the Gaussian curvature: positive on a sphere, zero on a cylinder, negative on a saddle. Gauss proved it can be measured from inside the surface. Picture it: the saddle z = x² − y², curving up one way and down the other. Think it: the same curvature, in four dimensions, is what gravity is in general relativity.

Worked example · x^2 - y^2

Algebra · Expressions

Analyse x^2 - y^2

x^{2} - y^{2}

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  1. x^{2} - y^{2}

    An expression in x, y. Here is what it does.

  2. \left(x - y\right) \left(x + y\right)

    Factored form.

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x^{2} - y^{2}

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How to: Surfaces and Gaussian curvature

  1. An expression in x, y. Here is what it does.
  2. Factored form.

Questions people ask

What is curvature?

For a curve, how fast its direction turns per unit length; for a surface, Gauss's combination of the two principal bendings, and Gauss's theorem says it can be measured from inside the surface without leaving it.

More in Differential Geometry

Curves, arc length and curvature