##  [Cusp](/cusp) 

  ##  [Cusp](https://puremath.quantumdictionary.io/cusp-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A pointed singularity on a curve or boundary where smoothness fails because two tangent directions coalesce into a single tangent with higher contact; the curve is not regular there and curvature typically blows up.

 

 

 

 

 





 

 



 ##  [Cusp](https://algebra.quantumdictionary.io/cusp-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A cuspidal singularity (cusp) is a point where a single branch of a curve has a self-tangency producing a pointed, nontransverse local geometry; algebraically the quadratic part is a square and higher-order terms produce a single tangent direction with multiplicity.