##  [Cone Point](/cone-point-0) 

 Definition

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

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

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

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

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

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

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

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

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.