Peak (singularity theory)

The point is a tip of the curve . ${\displaystyle (0,0)}$ ${\displaystyle x^{3}-y^{2}=0}$

In mathematics are tips (also cusps , Eng .: cusps ), one type of singularities of curves . A point moving on the curve would have to change direction abruptly at the tip .

definition

A curve in the plane is defined by the equation ${\displaystyle C}$ ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle f(x,y)=0}$.

A point lying on the curve is a singularity if ${\displaystyle (x,y)}$

${\displaystyle {\frac {\partial f}{\partial x}}(x,y)={\frac {\partial f}{\partial y}}(x,y)=0}$,

and that singularity is a bit of a tip if additional

${\displaystyle \det \left({\begin{array}{cc}{\frac {\partial ^{2}f}{\partial x^{2}}}(x,y)&{\frac {\partial ^{2}f}{\partial x\,\partial y}}(x,y)\\{\frac {\partial ^{2}f}{\partial y\,\partial x}}(x,y)&{\frac {\partial ^{2}f}{\partial y^{2}}}(x,y)\end{array}}\right)=0}$

applies.

Classification and examples

Each tip can be changed into the shape by local reparameterization

${\displaystyle x^{2}-y^{2k+1}=0}$

to be brought with. In the classification of singularities , this peak corresponds to a -singularity. ${\displaystyle k\geq 1,k\in \mathbb {Z} }$ ${\displaystyle A_{k}}$

For you get the usual cuff ${\displaystyle k=1}$

${\displaystyle x^{2}-y^{3}=0}$.

For you get the rhamphoide kuspe ${\displaystyle k=2}$

${\displaystyle x^{2}-y^{5}=0}$.

Web links

• Cusp (MathWorld)