Incircle

As explained in Geometry for Programmers

An incircle is the largest circle contained within a triangle and tangent to all three of its edges.

An incircle is the largest circle contained within a triangle, touching all three edges. Its radius forms the inner measure in the Rin/Rout triangle-quality metric. section 2.2.2

Use the incircle to assess triangle shape:

• Divide the incircle radius by the circumcircle radius to obtain Rin/Rout. section 2.2.2
• Treat larger ratios as characteristic of better-shaped triangles; thin triangles drive the ratio toward zero. section 2.2.2
• Filter triangles whose ratio falls below a chosen small threshold, removing degenerate and near-degenerate cases. section 2.2.2

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

Visualizes how the Rin/Rout metric decreases as a triangle approaches degeneracy. section 2.2.2

How Geometry for Programmers covers it

How deep it goes

Defines the incircle and its ratio with the circumcircle, but defers advanced quality-measure treatment.

Examples

Uses equilateral and increasingly thin triangles to illustrate changing Rin/Rout values.

What you can do after

Diagnose poor triangle shapes by computing Rin/Rout and applying a small rejection threshold.

How it connects

Connects incircles with circumcircles, triangle quality, degenerate triangles, and triangle meshes.

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Related concepts

• circumscribed circlecontrasts with · A circumscribed circle is a circle that passes through all vertices of a polygon.

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.