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# Spiric Section (work in progress)

## History

Differential Equations, Mechanics, and Computation

Studied by Greek mathematician Perseus around 150 BC.

## Description

Spiric Section is the name given to a family of curves formed by the intersection of a torus and a plane that is parallel to the axis of the torus. That is any cross-sections of a donut sliced by a knife straight from top.

## Formulas

```Cartesian equation:
-a^2 + b^2 + c^2 + x^2 - 2*b*Sqrt[c^2 + x^2] + y^2 == 0
where a is the radius of the generating circle.
b is the axis to the center of the generating circle.
c is the distance from the plane to axis.

We want to elimate redundant parameters
so that remaining parameter still generate all
possible shapes of the curve.
Since the shape of a torus is defined by the ratio of
a and b, we can set b to 1, and let a be from 0 to b.
Thus, the new equation for spiric section is:
1 - a^2 + c^2 + x^2 - 2*Sqrt[c^2 + x^2] + y^2 == 0
with parameter limited to 0 to 1.

Now, how we want eliminate this parameter boundary.
How do we find a mapping from interval [0,Infinity]
to [0,1]? (preferabbly linear and algebraic,
but that's not possible.)

```

spiric_sections.gcf

Cassinian ovals are special cases of Spiric sections.

## Related Web Sites

MacTutor Famous Curve Index

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2006-05