Unified Pythagorean Theorem

A few days ago I wrote that the spherical counterpart of the Pythagorean theorem is

cos(c) = cos(a) cos(b)

where sides a and b are measured in radians. If we’re on a sphere of radius R and we measure the sides in terms of arc length rather than in radians, the formula becomes

cos(c/R) = cos(a/R) cos(b/R)

because an of length x has angular measure x/R.

How does this relate to the more familiar Pythagorean theorem on the plane? If a, b, and c are small relative to R, then the plane Pythagorean theorem holds approximately:

c² ≈ a² + b².

Unified Pythagorean Theorem

In this post I’ll present a version of the Pythagorean theorem that holds exactly on the sphere and the plane, and on a pseudosphere (hyperbolic space) as well. This is the Unified Pythagorean Theorem [1].

A sphere of radius R has curvature 1/R². A plane has curvature 0. A hyperbolic plane can have curvature k for any negative value of k.

Let A(r) be the area of a circle of radius r as measured on a surface of curvature k. Here area and radius are measured intrinsic to the surface. Then the Unified Pythagorean Theorem says

A(c) = A(a) + A(b) – k A(a) A(b) / 2π.

Plane

If k = 0, the final term on the right drops out, and we’re left with the ordinary Pythagorean theorem with both sides of the equation multiplied by π.

Sphere

On a sphere of radius R, the area of a circle of radius r is

A(r) = 2πR²(1 – cos(r/R)).

Note that for small x,

1 – cos(x) ≈ x²/2,

and so A(r) ≈ πr² when Rr.  (Notation explained here.)

When you substitute the above definition for A in the unified theorem and plug in k = 1/R² you get

cos(c/R) = cos(a/R) cos(b/R)

as before.

Pseudosphere

In a hyperbolic space of curvature k < 0, let R = 1/√(-k). Then the area of a circle of radius r is

A(r) = 2π(cosh(r/R) – 1)

As with the spherical case, this is approximately the plane area when Rr because

cosh(x) – 1 ≈ x²/2

for small x. Substituting the definition of A for hyperbolic space into the Universal Pythagorean Theorem reduces to

cosh(c/R) = cosh(a/R) cosh(b/R),

which is the hyperbolic analog of the Pythagorean theorem. Note that this is the spherical Pythagorean theorem with cosines replaced with hyperbolic cosines.

[1] Michael P. Hitchman. Geometry with an Introduction to Cosmic Topology. Theorem 7.4.7. Available here.

 

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