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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 a0. This curve is a (general) cubic parabola (Figure 1). It is symmetric with respect to the point B where x=-b/3a.
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 a0. 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:
Among the important curves of degree four are the following:
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)=kx.
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=½k 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 angle
with the positive x-axis, and if we call P the intersection of L with the limaçon a=2k, 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=1.5a, k=a, and k=.5a (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.
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.