Cubic forms

The classification of umbilics is closely linked to the classification of real cubic forms ${\displaystyle ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}}$. A cubic form will have a number of root lines ${\displaystyle \lambda (x,y)}$ such that the cubic form is zero for all real ${\displaystyle \lambda }$. There are a number of possibilities including:

• Three distinct lines: an elliptical cubic form, standard model ${\displaystyle x^{2}y-y^{3}}$.
• Three lines, two of which are coincident: a parabolic cubic form, standard model ${\displaystyle x^{2}y}$.
• A single real line: a hyperbolic cubic form, standard model ${\displaystyle x^{2}y+y^{3}}$.
• Three coincident lines, standard model ${\displaystyle x^{3}}$.^[4]

The equivalence classes of such cubics under uniform scaling form a three-dimensional real projective space and the subset of parabolic forms define a surface – called the umbilic bracelet by Christopher Zeeman.^[4] Taking equivalence classes under rotation of the coordinate system removes one further parameter and a cubic forms can be represent by the complex cubic form ${\displaystyle z^{3}+3{\overline {\beta }}z^{2}{\overline {z}}+3\beta z{\overline {z}}^{2}+{\overline {z}}^{3}}$ with a single complex parameter ${\displaystyle \beta }$. Parabolic forms occur when ${\displaystyle \beta ={\tfrac {1}{3}}(2e^{i\theta }+e^{-2i\theta })}$, the inner deltoid, elliptical forms are inside the deltoid and hyperbolic one outside. If ${\displaystyle \left|\beta \right|=1}$ and ${\displaystyle \beta }$ is not a cube root of unity then the cubic form is a right-angled cubic form which play a special role for umbilics. If ${\displaystyle \left|\beta \right|={\tfrac {1}{3}}}$ then two of the root lines are orthogonal.^[5]

A second cubic form, the Jacobian is formed by taking the Jacobian determinant of the vector valued function ${\displaystyle F:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}}$, ${\displaystyle F(x,y)=(x^{2}+y^{2},ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3})}$. Up to a constant multiple this is the cubic form ${\displaystyle bx^{3}+(2c-a)x^{2}y+(d-2b)xy^{2}-cy^{3}}$. Using complex numbers the Jacobian is a parabolic cubic form when ${\displaystyle \beta =-2e^{i\theta }-e^{-2i\theta }}$, the outer deltoid in the classification diagram.^[5]

Umbilic classification

Any surface with an isolated umbilic point at the origin can be expressed as a Monge form parameterisation ${\displaystyle z={\tfrac {1}{2}}\kappa (x^{2}+y^{2})+{\tfrac {1}{3}}(ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3})+\ldots }$, where ${\displaystyle \kappa }$ is the unique principal curvature. The type of umbilic is classified by the cubic form from the cubic part and corresponding Jacobian cubic form. Whilst principal directions are not uniquely defined at an umbilic the limits of the principal directions when following a ridge on the surface can be found and these correspond to the root-lines of the cubic form. The pattern of lines of curvature is determined by the Jacobian.^[5]

The classification of umbilic points is as follows:^[5]

• Inside inner deltoid - elliptical umbilics
  • On inner circle - two ridge lines tangent
• On inner deltoid - parabolic umbilics
• Outside inner deltoid - hyperbolic umbilics
  • Inside outer circle - star pattern
  • On outer circle - birth of umbilics
  • Between outer circle and outer deltoid - monstar pattern
  • Outside outer deltoid - lemon pattern
• Cusps of the inner deltoid - cubic (symbolic) umbilics
• On the diagonals and the horizontal line - symmetrical umbilics with mirror symmetry

In a generic family of surfaces umbilics can be created, or destroyed, in pairs: the birth of umbilics transition. Both umbilics will be hyperbolic, one with a star pattern and one with a monstar pattern. The outer circle in the diagram, a right angle cubic form, gives these transitional cases. Symbolic umbilics are a special case of this.^[5]