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The epispiral is a
plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...
with
polar equation In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from ...
:\ r=a \sec. There are ''n'' sections if ''n'' is odd and 2''n'' if ''n'' is even. It is the polar or circle
inversion Inversion or inversions may refer to: Arts * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ''Inversions'' (novel) by Iain M. Bank ...
of the
rose A rose is either a woody perennial plant, perennial flowering plant of the genus ''Rosa'' (), in the family Rosaceae (), or the flower it bears. There are over three hundred Rose species, species and Garden roses, tens of thousands of cultivar ...
curve. In
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
the epispiral is related to the equations that explain planets' orbits.


Alternative definition

There is another definition of the epispiral that has to do with tangents to circles: Begin with a circle. Rotate some single point on the circle around the circle by some angle \theta and at the same time by an angle in constant proportion to \theta, say c\theta for some constant c. The intersections of the tangent lines to the circle at these new points rotated from that single point for every \theta would trace out an epispiral. The polar equation can be derived through simple geometry as follows: To determine the polar coordinates (\rho,\phi) of the intersection of the tangent lines in question for some \theta and -1, note that \phi is halfway between \theta and c\theta by congruence of triangles, so it is \frac. Moreover, if the radius of the circle generating the curve is r, then since there is a right-angled triangle (it's right-angled as a tangent to a circle meets the radius at a right angle at the point of tangency) with hypotenuse \rho and an angle \frac to which the adjacent leg of the triangle is r, the radius \rho at the intersection point of the relevant tangents is r\sec(\frac). This gives the polar equation of the curve, \rho=r\sec(\frac) for all points (\rho,\phi) on it.


See also

*
Logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewi ...
*
Rose (mathematics) In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied ...


References

* * https://www.mathcurve.com/courbes2d.gb/epi/epi.shtml {{geometry-stub Plane curves