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 In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders is ...

. 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 alternatin ...

, along with

Steinmetz's equation Steinmetz's equation, sometimes called the power equation, is an empirical equation used to calculate the total power loss (core losses) per unit volume in magnetic materials when subjected to external sinusoidally varying magnetic flux. The equati ...

, Steinmetz solids, and Steinmetz equivalent circuit theory. In the case when the two cylinders have equal radii the curve degenerates to two intersecting ellipses.

References

Curves Euclidean geometry {{algebraic-geometry-stub

See also

References