A Steinmetz curve is the curve of intersection of two right circular cylinders of

radii In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

a

and

b,

whose axes intersect perpendicularly. In case of

a=b

the Steimetz curves are the edges of a Steinmetz solid. If the cylinder axes are the x- and y-axes and

a\le b

, then the Steinmetz curves are given by the

parametric equations Parametric may refer to: Mathematics * Parametric equation, a representation of a curve through equations, as functions of a variable *Parametric statistics, a branch of statistics that assumes data has come from a type of probability distribu ...

: :

\begin
x (t) & = a \cos  t \\
y (t) & = \pm \sqrt \\
z (t) & =  a \sin t
\end

It is named after mathematician

Charles Proteus Steinmetz Charles Proteus Steinmetz (born Karl August Rudolph Steinmetz, April 9, 1865 – October 26, 1923) was a German-born American mathematician and electrical engineer and professor at Union College. He fostered the development of alternati ...

, along with Steinmetz's equation, Steinmetz solids, and Steinmetz equivalent circuit theory. In the case when the two cylinders have equal radii the curve degenerates to two intersecting

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

References

Curves Euclidean geometry {{algebraic-geometry-stub

See also

References