Degenerate triangle

As explained in Geometry for Programmers

A degenerate triangle is a triangle with zero area because its vertices coincide or are collinear.

A degenerate triangle has zero area. Its three vertices still appear as a triangle in data, but its geometry collapses into a segment, making it a troublesome special case in programming and geometric modeling. section 2.2.2

Recognize the two common forms:

• Needle: two vertices coincide, producing a sharp end and an overlapping pair of vertices. section 2.2.2
• Cap: three distinct vertices lie on one line, so the triangle is flattened into a segment-like shape. section 2.2.2
• Both forms can damage a triangle mesh’s topology: the intended relationships among triangles, edges, and neighbors no longer behave normally. section 2.2.2

Checking triangle quality

A useful quality metric should report a degenerate triangle as horrible and an equilateral triangle as excellent. The Rin/Rout metric decreases as a triangle approaches degeneracy, providing a practical signal for identifying poor geometry. section 2.7 section 2.2.2

Figure 2.10 Not all degenerate triangles are the same. Some are needles, and others are caps.

Compare needles and caps visually before reasoning about their different vertex configurations. section 2.2.2

How Geometry for Programmers covers it

How deep it goes

Defines needles and caps, explains adjacency damage, and relates degeneracy to a quality metric without deriving its formula.

Examples

Uses needles, caps, triangle-neighbor diagrams, and Rin/Rout behavior approaching the degenerate state.

What you can do after

Evaluate triangle quality and recognize mesh-topology risks caused by coincident or collinear vertices.

How it connects

Connects degenerate triangles to triangle meshes, manifold meshes, topology, edges, neighbors, and triangle-quality metrics.

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Related concepts

• trianglekind of · A triangle is a polygon bounded by three line segments joining three noncollinear vertices.
• near-degenerate trianglecontrasts with · A nonzero-area triangle whose extreme thinness makes geometric computations numerically error-prone.

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.