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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 


Its parameterization is


The cusp is almost everywhere smooth, except at the cusp point (t=0) or 
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.


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

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