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Let be a curve, let
be a fixed point (the Pole), and let
be a second fixed point. Let
and
be
points on a line through
meeting
at
such that
. The Locus of
and
is called the
strophoid of
with respect to the Pole
and fixed point
. Let
be represented parametrically by
, and let
and
. Then the equation of the strophoid is
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(1) |
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(2) |
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(3) |
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(4) |
Curve | Pole | Fixed Point | Strophoid |
line | not on line | on line | oblique strophoid |
line | not on line | foot of Perpendicular origin to line | Right Strophoid |
Circle | center | on the circumference | Freeth's Nephroid |
See also Right Strophoid
References
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 51-53 and 205, 1972.
Lockwood, E. H. ``Strophoids.'' Ch. 16 in A Book of Curves. Cambridge, England: Cambridge University Press,
pp. 134-137, 1967.
MacTutor History of Mathematics Archive. ``Right.''
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Right.html.
Yates, R. C. ``Strophoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 217-220, 1952.
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© 1996-9 Eric W. Weisstein