Steinmetz curve

A Steinmetz curve is the curve of intersection of two right circular cylinders of radii $a$ 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 $a\leq b$ , then the Steinmetz curves are given by the parametric equations:

{\begin{aligned}x(t)&=a\cos t\\y(t)&=\pm {\sqrt {b^{2}-a^{2}\sin ^{2}t}}\\z(t)&=a\sin t\end{aligned}}

It is named after mathematician Charles Proteus Steinmetz, 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 ellipses.

References

^[1]^[2]^[3]

^ Abbena, Elsa; Salamon, Simon; Gray, Alfred (2006). Modern Differential Geometry of Curves and Surfaces with Mathematica (3rd ed.). Chapman and Hall/CRC. ISBN 978-1584884484.
^ Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.
^ Weisstein, Eric W. "Steinmetz Curve". Wolfram MathWorld. Retrieved October 28, 2018.

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[1] Abbena, Elsa; Salamon, Simon; Gray, Alfred (2006). Modern Differential Geometry of Curves and Surfaces with Mathematica (3rd ed.). Chapman and Hall/CRC. ISBN 978-1584884484.

[2] Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.

[3] Weisstein, Eric W. "Steinmetz Curve". Wolfram MathWorld. Retrieved October 28, 2018.

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See also

References