# Singular Sets of Algebraic Curves and Surfaces

If we are given a curve f(x,y)=0 in the plane, or a surface
f(t,x,y)=0 in 3-space, there are many points on the curve
that have a well-defined, unique tangent line, and many points
on the surface that have a well-defined unique tangent plane.
See the pictures below for some examples

### The nephroid with one of its tangent lines, and
the surface F(t,x,y)=0 with one of its tangent planes

If we want to compute the equations for these tangent lines
or planes, we can use a fact we learned in multivariable
calculus. Since the curve is the level set of the value 0
for the function f(x,y) or f(x,y,z), the tangent line/plane
is perpendicular to the gradient vector (df/dx,df/dy)
or (df/dx,df/dy,df/dz) evaluated at that point. Since we
already know one point that we want the tangent line/plane
to go through, this gives enough information to compute
the tangent line or plane's equation, as long as the
*gradient is not the zero vector* at that point.
On the other hand, points on the curve/surface where the
gradient *is* the zero vector do not have a well-defined
tangent line/plane. These points are highlighted in the
pictures below

### Singular points on the nephroid, and on
the surface F(t,x,y)=0

These points are called the *singular set* of the curve/surface,
and are defined by the (polynomial) equations
f(x,y)=0

d/dx f(x,y)=0

d/dy f(x,y)=0

for an algebraic curve f(x,y)=0, and by the equations

F(t,x,y)=0

d/dx F(t,x,y)=0

d/dy F(t,x,y)=0

d/dt F(t,x,y)=0

for an algebraic surface F(t,x,y)=0.
If one has a method for solving polynomial equations, then
one can find the solutions to the above equations and compute
the singular set of your curve/surface. For example, the Maple
computations shown below give the singularities of the nephroid
and the graph surface F(t,x,y)=0.

4 2 2 2 2 2 4 2 4 6
nephroid := 12 x y - 4 - 15 y + 12 x - 24 y x + 12 y x - 12 y + 4 y
4 6
- 12 x + 4 x
2 2 2 2 2 2
F := ((y - t) (2 - t ) + t (4 - t )) - t x (4 - t )
> solve({nephroid=0,diff(nephroid,x)=0,diff(nephroid,y)=0},{x,y});
2
{y = 0, x = 1}, {y = 0, x = -1}, {x = 0, y = RootOf(1 + 2 _Z )}
> solve({F=0,diff(F,x)=0,diff(F,y)=0,diff(F,t)=0},{t,x,y});
t
{x = x, y = 0, t = 0}, {t = t, x = 0, y = 2 --------},
2
- 2 + t
2 2
{x = 0, y = - 1/2 RootOf(2 + _Z ), t = RootOf(2 + _Z )}

## Questions to think about

- Do the singularities of the nephroid computed by Maple agree
with what you see in the picture of the nephroid? Why is
there a singularity computed by Maple which you can't see?
- Do the singularities of the graph surface F(t,x,y)=0 computed
by Maple agree with the singular curves in the picture?
Why is there a singular curve computed by Maple that you can't see?

For a more thorough treatment of singularities, see
Cox, Little and O'Shea Section 3.4 and 9.6.

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Conclusions

**Up: **Introduction

Vic Reiner <reiner@math.umn.edu>
Frederick J. Wicklin <fjw@geom.umn.edu>
Last modified: Thu Apr 25 08:19:17 1996