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ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28
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The chordal distance has the standard coordinate formula on the finite plane

Example

For finite points z,w∈C, the chordal metric is χ(z,w)=2∣z−w∣(1+∣z∣2)(1+∣w∣2), while χ(z,∞)=21+∣z∣2.

Facts & Assumptions

Given: The chordal metric is Euclidean distance after stereographic projection.

[L1]

Stereographic projection and the chordal metric are given by Stereographic projection identifies the Riemann sphere with the unit two-sphere and The chordal metric on the Riemann sphere.

Verification

technique · direct
1.1

Substituting the stereographic coordinates of z and w into the Euclidean distance formula on S2 simplifies to χ(z,w)2=4∣z−w∣2/((1+∣z∣2)(1+∣w∣2)), and taking square roots gives the finite-point formula.

L1givenalgebra
2.1

Using Σ(∞)=(0,0,1) in the same calculation gives χ(z,∞)2=4/(1+∣z∣2), so χ(z,∞)=2/1+∣z∣2.

L1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources