Circumscribed circle

As explained in Geometry for Programmers

A circumcircle is the smallest circle containing a triangle and passing through all three of its vertices.

A circumscribed circle, or circumcircle, is the smallest circle containing a triangle and passing through all three vertices. Its radius forms the outer part of the Rin/Rout triangle-quality metric. section 2.2.2

Use the circumcircle to interpret triangle shape:

• Compare its radius with the incircle radius: Rin/Rout divides the inner-circle radius by the circumcircle radius. section 2.2.2
• Treat equilateral triangles as having the best proportion; thinner triangles drive the ratio toward zero. section 2.2.2
• Filter strictly degenerate and near-degenerate triangles when Rin/Rout falls below a chosen small threshold. section 2.2.2

From well-shaped to near-degenerate

An equilateral triangle has the strongest incircle-to-circumcircle proportion. As the triangle becomes extremely thin, the proportion declines toward zero, signaling increasingly error-prone geometric operations. section 2.2.2

Figure 2.14 The Rin/Rout metric gets smaller as a triangle heads to the degenerate state.

The figure shows how the Rin/Rout metric shrinks as a triangle approaches degeneracy. section 2.2.2

How Geometry for Programmers covers it

How deep it goes

Defines the circumcircle and its radius ratio, but does not provide construction formulas or computational procedures.

Examples

Contrasts equilateral, thin, near-degenerate, and strictly degenerate triangles through the Rin/Rout metric.

What it assumes

Assumes familiarity with triangles and introduces the incircle and circumcircle definitions directly.

What you can do after

Diagnose poor triangle quality by computing or applying a threshold to the Rin/Rout ratio.

How it connects

Connects circumcircles to triangles, incircles, triangle quality, degeneracy, and triangle meshes.

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Related concepts

• circlekind of · A circle is a plane curve consisting of points at a fixed distance from a common center.
• incirclecontrasts with · An incircle is the largest circle contained within a triangle and tangent to all three of its edges.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.