A point in a topological or geometric space whose neighborhood is homeomorphic (or isometric in metric cones) to a cone over some base space; it is a localized conical singularity where curvature or topology concentrates at the apex.

Principle

Model the local structure by taking a product of a base space with a radial coordinate and collapsing the base at zero radius to a single apex; the cone point captures singular behavior that is homogeneous in the angular directions.

Demonstration

The apex of the metric cone over a circle produces a 2D cone point with total angle not equal to 2π; in algebraic geometry, an ordinary double point can be topologically a cone over a projective curve, giving a cone singularity at the vertex.

Misapplication

Calling any isolated non-smooth point a cone point without verifying a genuine conical neighborhood; or assuming metric cone geometry (radial homogeneity) where the local model only topologically resembles a cone.

Consequence

Recognizing a cone point allows use of cone calculus, adapted coordinates, and analysis of singular curvature or holonomy; it informs index theorems and spectral behavior sensitive to the cone angle or base topology.

Reversal

The reverse is a smooth point where neighborhoods are Euclidean balls rather than cones; reversing removes concentrated angular defect and restores ordinary smooth differential structure.

Boundary

Applies where a neighborhood is (topologically or metrically) a cone over a well-defined base; excludes cusp singularities, branch points with different local models, and accumulation of cone points that destroy isolated conicality.

Semantic Tension

Tension exists between topological cone points (homeomorphic neighborhoods) and metric cone points (isometric radial structure); some singularities are topologically conical but lack the precise radial metric needed for analytic cone techniques.

Synthesis

A cone point is an isolated apex where local neighborhoods collapse to a cone over a base, concentrating angular or topological defect into a single point and serving as a canonical local singular model for geometric and analytic study.