The epispiral is a plane curve with polar equation

- .

There are *n* sections if *n* is odd and 2*n* if *n* is even.

It is the polar or circle inversion of the rose curve.

In astronomy the epispiral is related to the equations that explain planets' orbits.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

A **logarithmic spiral**, **equiangular spiral**, or **growth spiral** is a self-similar spiral curve that often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it *Spira mirabilis*, "the marvelous spiral".

In geometry, a **coordinate system** is a system that uses one or more numbers, or **coordinates**, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the *x*-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and *vice versa*; this is the basis of analytic geometry.

In mathematics, the **winding number** of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.

A **hyperbolic spiral** is a plane curve, which can be described in polar coordinates by the equation

In geometry, a **limaçon** or **limacon**, also known as a **limaçon of Pascal**, is defined as a roulette formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The **cardioid** is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp.

In mathematics, a **rose** or **rhodonea curve** is a sinusoid plotted in polar coordinates.

In geometry, the **cissoid of Diocles** is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the curve family of cissoids is named for this example and some authors refer to it simply as *the* cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix.

The **conchoid(s) of de Sluze** is a family of plane curves studied in 1662 by René François Walter, baron de Sluze.

In mathematics, a **plane curve** is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves, and algebraic plane curves. Plane curves also include the Jordan curves and the graphs of continuous functions.

The **Kampyle of Eudoxus** is a curve with a Cartesian equation of

In geometry, a **strophoid** is a curve generated from a given curve *C* and points *A* and *O* as follows: Let *L* be a variable line passing through *O* and intersecting *C* at *K*. Now let *P*_{1} and *P*_{2} be the two points on *L* whose distance from *K* is the same as the distance from *A* to *K*. The locus of such points *P*_{1} and *P*_{2} is then the strophoid of C with respect to the pole *O* and fixed point *A*. Note that *AP*_{1} and *AP*_{2} are at right angles in this construction.

In geometry, the **trisectrix of Maclaurin** is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742.

The **quadrifolium** is a type of rose curve with n=2. It has the polar equation:

In projective geometry, a **dual curve** of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic then so is its dual and the degree of the dual is known as the *class* of the original curve. The equation of the dual of C, given in line coordinates, is known as the *tangential equation* of C.

A **quartic plane curve** is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:

In algebraic geometry, the **first polar**, or simply **polar** of an algebraic plane curve *C* of degree *n* with respect to a point *Q* is an algebraic curve of degree *n*−1 which contains every point of *C* whose tangent line passes through *Q*. It is used to investigate the relationship between the curve and its dual, for example in the derivation of the Plücker formulas.

In geometry, the **Tschirnhausen cubic**, or **Tschirnhaus' cubic** is a plane curve defined, in its left-opening form, by the polar equation

In inversive geometry, an **inverse curve** of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and *OP*·*OQ* = *k*^{2}. The inverse of the curve C is then the locus of P as Q runs over C. The point O in this construction is called the **center of inversion**, the circle the **circle of inversion**, and k the **radius of inversion**.

For a plane curve *C* and a given fixed point *O*, the **pedal equation** of the curve is a relation between *r* and *p* where *r* is the distance from *O* to a point on *C* and *p* is the perpendicular distance from *O* to the tangent line to *C* at the point. The point *O* is called the *pedal point* and the values *r* and *p* are sometimes called the *pedal coordinates* of a point relative to the curve and the pedal point. It is also useful to measure the distance of *O* to the normal even though it is not an independent quantity and it relates to as .

- J. Dennis Lawrence (1972).
*A catalog of special plane curves*. Dover Publications. p. 192. ISBN 0-486-60288-5. - https://www.mathcurve.com/courbes2d.gb/epi/epi.shtml

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