# More definitions for associated curves

**Algebraic curve :**A curve whose cartesian equation can be expressed in terms of powers of

*x*and

*y*together with the operations of addition, subtraction, multiplication and division.

For example the

*astroid*,

*x*

^{2/3}+

*y*

^{2/3}=

*a*

^{2/3}, is an algebraic curve. The term is due to Leibniz.

**Anallagmatic curve :** A curve which is invariant under inversion.

The property was first discussed by Moutard in 1864.

**Asymptote :** A line which is the limit of the tangent to a curve as the point of contact of the tangent tends to infinity.

**Bipolar coordinates :** Let *O* and *O*' be two fixed points. A point *P* may be specified by giving its distances *r* and *r*' from *O* and *O*' respectively. These are called the *bipolar coordinates* of *P*. A curve may be defined by an equation, called the *bipolar equation*, connecting *r* and *r*'.

For example an ellipse is defined by *r* + *r*' = 2*a*.

**Brachistochrone curve :** A curve along which a particle will move from one point to another under the action of an accelerating force in the least possible time.

In 1696 Johann Bernoulli put out a challenge to find such a curve where the accelerating force is gravity.

**Caustic curves :** When light reflects off a curve then the envelope of the reflected rays is a caustic by reflection or a *catacaustic*. When light is refracted by a curve then the envelope of the refracted rays is a caustic by refraction or a *diacaustic*.

They were first studied by Huygens and Tschirnhaus around 1678. Johann Bernoulli, Jacob Bernoulli, de l'HÃ´pital and Lagrange all studied caustic curves.

**Cissoid :** Given two curves *C*_{1} and *C*_{2} and a fixed point *O*, let a line from *O* cut *C*_{1} at *Q* and *C*_{2} at *R*. Then the cissoid is the locus of a point *P* such that *OP* = *QR*.

The *cissoid of Diocles* is a cissoid where *C*_{1} is a circle, *C*_{2} is a tangent to *C*_{1} and *P* is the point on *C*_{1} diametrically opposite the point of contact of the tangent.

**Conchoid :** Let *C* be a curve and *O* a fixed point. Let *P* and *P*' be points on a line from *O* to *C* meeting it at *Q* where *P*'*Q* = *QP* = *k*, where *k* is a given constant.

If *C* is a circle and *O* is on *C* then the conchoid is a *limacon*, while in the special case that *k* is the diameter of *C*, then the conchoid is a *cardioid*.

**Curvature :** Let *C* be a curve and let *P* be a point on *C*. Let *N* be the normal at *P* and let *O* be the point on *N* which is the limit of where the normal to *C* at *P*' intersects *N* as *P*' tends to *P*. *O* is the *centre of curvature* at *P* and *PO* is the *radius of curvature* at that point.

**Cusp :** A point on a curve *C* where the gradient of the tangent to *C* has a discontinuity.

**Envelope :** A curve which touches every member of a family of curves or lines.

For example the axes are the envelope of the system of circles (*x*-*a*)^{2} + (*y*-*a*)^{2} = *a*^{2}.

**Evolute :** The envelope of the normals to a given curve.

This can also be thought of as the locus of the centres of curvature.

The idea appears in an early form in Apollonius's *Conics *Book V. It appears in its present form in Huygens' work from around 1673.

**Glissette :** The locus of a point *P* (or the envelope of a line) fixed in relation to a curve *C* which slides between fixed curves.

For example if *C* is a line segment and *P* a point on the line segment then *P* describes an ellipse when *C* slides so as to touch two orthogonal straight lines. The glissette of the line segment *C* itself is, in this case, an *astroid*.

**Inverse curves :** Given a circle *C* centre *O* radius *r* then two points *P* and *Q* are *inverse with respect to C* if *OP*.*OQ* = *r*^{2}. If *P* describes a curve *C*_{1} then *Q* describes a curve *C*_{2} called the *inverse* of *C*_{1} with respect to the circle *C*.

Although it does not make much geometric sense to take the circle *C* having negative radius, it makes no difference to the definition of the inverse of a point, except in this case *P* and *Q* are on opposite sides of *O* whereas when *r* is positive *P* and *Q* are on the same side of *O*.

**Involute :** If *C* is a curve and *C*' is its evolute, then *C* is called an involute of *C*'.

Any parallel curve to *C* is also an involute of *C*'. Hence a curve has a unique evolute but infinitely many involutes.

Alternatively an involute can be thought of as any curve orthogonal to all the tangents to a given curve.

**Isoptic curve :** For a given curve *C* consider the locus of the point *P* from where the tangents from *P* to *C* meet at a fixed given angle. This is called an *isoptic curve* of the given curve.

**Negative pedal :** Given a curve *C* and *O* a fixed point then for a point *P* on *C* draw a line perpendicular to *OP*. The envelope of these lines as *P* describes the curve *C* is the *negative pedal* of *C*.

The ellipse is the negative pedal of a circle if the fixed point is inside the circle while the negative pedal of a circle from a point outside is a hyperbola.

**Node :** Point at which two branches of a curve cross.

**Normal :** The normal at the point *P* of a curve *C* is the line through *P* perpendicular to the tangent at *P*.

**Orthogonal :** Two curves are orthogonal at a point point *P* where they cross if the tangents at *P* are perpendicular.

**Orthoptic curve :** An *isoptic curve* formed from the locus of tangents meeting at right angles.

The orthoptic of a parabola is its directrix, the orthoptic of a central conic is a circle concentric with the conic which was investigated by Monge.

The orthoptic of a tricuspoid is a circle.

**Parallel curves :** Two curves are parallel if every normal to one curve is a normal to the other curve and the distance between where the normals cut the two curves is a constant.

Although parallel curves are at a fixed distance apart they can look rather different. For example Cayley's sextic and the nephroid are parallel.

Leibniz was the first to consider parallel curves.

**Pedal curve :** Given a curve *C* then the pedal curve of *C* with respect to a fixed point *O* (called the *pedal point*) is the locus of the point *P* of intersection of the perpendicular from *O* to a tangent to *C*.

**Radial curve :** Let *C* be a curve and let *O* be a fixed point. Let *P* be on *C* and let *Q* be the centre of curvature at *P*. Let *P*_{1} be the point with *P*_{1}*O* a line segment parallel and of equal length to *PQ*. Then the curve traced by *P*_{1} is the radial curve of *C*.

It was studied by Robert Tucker in 1864.

The radial of a cycloid is a circle.

**Roulette :** Let *C*_{1} be a curve and *C*_{2} a second curve. Then if *P* is a point on *C*_{2}, a roulette is the curve traced out by *P* as *C*_{2} rolls on *C*_{1}.

A cycloid is the roulette of a point on a circle rolling along a straight line.

Epicycloids, hypocycloids, epitrochoids and hypotrochoids are all roulettes of a circle rolling on another circle.

**Spiral :** The locus of a point *P* which winds around a fixed point *O* (called the *pole*) in such a way that *OP* is monotonically decreasing.

Sinusoidal spirals are not true spirals.

**Strophoid :** Let *C* be a curve, let *O* be a fixed point called the pole and let *O*' be a second fixed point. Let *P* and *P*' be points on a line through *O* meeting *C* at *Q* such that *P*'*Q* = *QP* = *QO*'. The locus of *P* and *P*' is called the *strophoid* of *C* with respect to the pole *O* and fixed point *O*'.

A *right strophoid* is the strophoid of a line *L* with pole *O* not on *L* and fixed point *O*' being the point where the perpendicular from *O* to *L* cuts *L*.

**Tautochrone :** A curve down which a particle acted on by a force will traverse the distance to the lowest point in the curve in a fixed time independent of the starting position.

**Transcendental curve :** A curve of the form *f(x*,*y*) = 0 where *f(x*,*y*) is not a polynomial in *x* and *y*.

For example the *cycloid* is a transcendental curve.

The term is due to Leibniz.