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Curves that can be given in implicit form as *f*(*x*,*y*)=0, where *f* is
a polynomial, are called **algebraic**. The degree of *f* is called
the degree or **order** of the curve. Thus conics
(Section 7) are algebraic curves of degree two. Curves of
degree three already have a great variety of shapes, and only a few
common ones will be given here.

The simplest case is when the curve is the graph of a polynomial of
degree three: *y*=*ax*+*bx*+*cx*+*d*, with *a*0. This curve is a
(general) **cubic parabola** (Figure 1). It is
symmetric with respect to the point *B* where *x*=-*b*/3*a*.

**Figure 1:** The general cubic parabola for *a*>0. For *a*<0, reflect in
a horizontal line.

The **semicubic parabola** (Figure 2, left) has
equation *y*=*kx*; by proportional scaling one can take *k*=1.

**Figure 2:** The semicubic parabola, the cissoid of Diocles, and the witch
of Agnesi

This curve should not be confused with the **cissoid of Diocles**
(Figure 2, middle), which has equation
(*a*-*x*)*y*=*x*

with *a*0. The latter is asymptotic to the line *x=a*, while the
semicubic parabola has no asymptotes. The cissoid's points are
characterized by the equality *OP=AB* in Figure 2,
right. One can take *a*=1 by proportional scaling.

More generally, any curve of degree three with equation
(*x*-*x*)*y*=*f*(*x*), where *f* is a polynomial, is symmetric with
respect to the *x*-axis and asymptotic to the line *x*=*x*. In
addition to the cissoid, the following particular cases are important:

- The
**witch of Agnesi**has equation*xy*=*a*(*a*-*x*), with*a*0, and is characterized by the geometric property shown in Figure 2, right. The same property provides the parametric representation*x*=*a*(1+sin 2),*y*=*a*tan . Once more, proportional scaling reduces to the case*a*=1. - The
**folium of Descartes**(Figure 3, left) has equation (*x*-*a*)*y*=-*x*(*x*+*a*), with*a*0 (reducible to*a*=1 by proportional scaling). By rotating 135°(right) we get the alternative and more familiar equation*x*+*y*=*cxy*, where*c*=*a*. The folium of Descartes is a**rational curve**, this is, it has a parametric representation by rational functions. In the tilted position such a representation is*x*=*ct*/(1+*t*),*y*=*ct*/(1+*t*) (so that*t*=*y*/*x*).**Figure 3:**The folium of Descartes in two positions, and the strophoid.

- The
**strophoid**has equation (*x*-*a*)*y*=-*x*(*x*+*a*), with*a*0 (reducible to*a*=1 by proportional scaling). It satisfies the property*AP*=*AP'*=*OA*in Figure 3, right; this means that*POP'*is a right angle. The strophoid has the polar representation*r*=-*a*cos 2 sec , and the rational parametric representation*x*=*a*(*t*-1)/(*t*+1),*y*=*at*(*t*-1)/(*t*+1) (so that*t*=*y*/*x*).

Among the important curves of degree four are the following:

- A
**Cassini's oval**is characterized by the following condition: given two**foci***F*and*F'*, a distance 2*a*apart, a point*P*belongs to the curve if the product of the distances*PF*and*PF'*is a constant*k*. If the foci are on the*x*-axis and equidistant from the origin, the curve's equation is (*x*+*y*+*a*)-4*a**x*=*k*. Changes in*a*correspond to rescaling, while the value of*k*/*a*controls the shape: the curve has one smooth piece, one piece with a self-intersection, or two pieces depending on whether*k*is greater than, equal to, or smaller than*a*(Figure 4). The case*k=a*is also known as the**lemniscate**(of Jakob Bernoulli); the equation reduces to (*x*+*y*)=*a*(*x*-*y*), and upon a 45° rotation to (*x*+*y*)=2*a**xy*. Each Cassini's oval is the section of a torus of revolution by a plane parallel to the axis of revolution.**Figure 4:**Cassini's ovals for*k*=.5*a*, .9*a*, 1.1*a*and 1.5*a*(from the inside to the outside). The foci (dots) are at*x=a*and*x=-a*. The black curve,*k=a*, is also called Bernoulli's lemniscate.

- A
**conchoid of Nichomedes**is the set of points such that the signed distance*AP*in Figure 5, left, equals a fixed real number*k*(the line*L*and the origin*O*being fixed). If*L*is the line*x=a*, the conchoid has polar equation*r*=*a*sec +*k*.Once more,

*a*is a scaling parameter, and the value of*k*/*a*controls the shape: when*k*>-*a*the curve is smooth, when*k*=*-a*there is a cusp, and when*k*<*-a*there is a self-intersection. The curves for*k*and*-k*can also be considered two leaves of the same conchoid, with cartesian equation (*x*-*a*)(*x*+*y*)=*k**x*.**Figure 5:**Defining property of the conchoid of Nichomedes (left), and curves for*k*=±.5*a*,*k*=±*a*, and*k*=±1.5*a*(right).

- A
**limaçon of Pascal**is the set of points such that the distance*AP*in Figure 6, left, equals a fixed positive number*k*measured on either side (the circle*C*and the origin*O*being fixed). If*C*has diameter*a*and center at (0,½*a*), the limaçon has polar equation*r*=*a*cos +*k*,and cartesian equation (

*x*+*y*-*ax*)=*k*(*x*+*y*). The value of*k*/*a*controls the shape, and there are two particularly interesting cases. For*k=a*we get a**cardioid**(see also Figure 8.2.2 , right). For*a*=½*k*we get a curve that can be used to**trisect**an arbitrary angle : if we draw a line*L*through the center of the circle*C*and making an anglewith the positive

*x*-axis, and if we call*P*the intersection of*L*with the limaçon*a*=2*k*, the line from*O*to*P*makes an angle with*L*.

**Figure 6:** Defining property of the limaçon of Pascal (left), and
curves for *k*=1.5*a*, *k=a*, and *k*=.5*a* (right).
The middle curve is the cardioid, the one on the right a trisectrix.

Hypocycloids and epicycloids with rational ratio (see next section) are also algebraic curves, generally of higher degree.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.