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 ...
:
.
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
and at the same time by an angle in constant proportion to
, say
for some constant
.
The intersections of the tangent lines to the circle at these new points rotated from that single point for every
would trace out an epispiral.
The polar equation can be derived through simple geometry as follows:
To determine the polar coordinates
of the intersection of the tangent lines in question for some
and