# 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, x2/3 + y2/3 = a2/3, 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' = 2a.

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'Hôpital and Lagrange all studied caustic curves.

Cissoid : Given two curves C1 and C2 and a fixed point O, let a line from O cut C1 at Q and C2 at R. Then the cissoid is the locus of a point P such that OP = QR.
The cissoid of Diocles is a cissoid where C1 is a circle, C2 is a tangent to C1 and P is the point on C1 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 = a2.

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 = r2. If P describes a curve C1 then Q describes a curve C2 called the inverse of C1 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 P1 be the point with P1O a line segment parallel and of equal length to PQ. Then the curve traced by P1 is the radial curve of C.
It was studied by Robert Tucker in 1864.
The radial of a cycloid is a circle.

Roulette : Let C1 be a curve and C2 a second curve. Then if P is a point on C2, a roulette is the curve traced out by P as C2 rolls on C1.
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.