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[help-3dldf] Reg_Cl_Plane_Curves

 From: Martijn van Manen Subject: [help-3dldf] Reg_Cl_Plane_Curves Date: Fri, 04 Feb 2005 07:16:27 -0500

From the manual:

"At present, I have no fixed definition of what constitutes "regularity" as far
as Reg_Cl_Plane_Curves are concerned"

In algebraic geometry one calls a curve regular when it is smooth.
To illustrate the concept consider the cusp in the plane.
Its implicit equation is

F(x,y)=y^2-x^3=0

Its parameterization is

gamma(t)=(t^2,t^3)

The cusp is almost everywhere smooth, except at the cusp point (t=0) or
(x,y)=(0,0).
The cusp at the origin is a so-called singularity.
To again illustrate the concept consider a hyperbola.

F(x,y)=x^2-y^2 - 1 = 0

Its parameterization is

gamma(t)= ( cosh(t), sinh(t) )

The hyperbola is everywhere smooth. It has no singularities.

To test whether a curve has a singularity at a point test look where

F= \frac{\partial F}{\partial x} = \frac{\partial F}{\partial y} = 0

Or, if given a parameterization, look for values of t where the
derivative of gamma wrt. to t is zero.
A curve without singularities is called regular.
There is though a subtle difference between regular and smooth.
Some people will also call y^2-x^2=0 smooth in the origin. After
all it only has a self-intersection. But everybody will say that
this self-intersection is a "double-point" singularity.

Great fun with curves can be found in
"Ebene algebraische Kurven" by Brieskorn & Knoerrer.
There is also an english translation of that book.

Martijn

Mieulx est de ris que de larmes escripre,
Pour ce que rire est le propre de l'homme.

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