Chapter 4: Geometry

FIGURE 4.21
The semi-cubic parabola, the cissoid of Diocles, and the witch of Agnesi.
È
È
Ç
diameter
Ç
diameter
More generally, any curve of degree three with equation ´Ü Ü ¼ µÝ ¾ ´Üµ,
where is a polynomial, is symmetric with respect to the Ü-axis and asymptotic
to the line Ü Ü ¼ . In addition to the cissoid, the following particular cases are
important:
1. The witch of Agnesi has equation ÜÝ ¾ ¾´ ܵ, with ¼, and is
characterized by the geometric property shown in Figure 4.21, right. The same
property provides the parametric representation Ü Ó× ¾ , Ý ØÒ .
Once more, proportional scaling reduces to the case ½.
2. The folium of Descartes (Figure 4.22, left) is described by equation ´Ü
µÝ ¾ Ü ¾´ ½ Ü · µ, with ¼(reducible to ½by proportional scaling).
By rotating 135 Æ (right) Ôwe get the alternative and more familiar equation
¿
Ü ¿ ·Ý ¿ ÜÝ, where ½ ¾. The folium of Descartes is a rational curve,
¿
that is, it is parametrically represented by rational functions. In the tilted position,
the equation is Ü Ø´½ · Ø ¿ µ, Ý Ø ¾ ´½ · Ø ¿ µ (so that Ø ÝÜ).
3. The strophoid’s equation is ´Ü µÝ ¾ Ü ¾´Ü · µ, with ¼(reducible
to ½by proportional scaling). It satis es the property È È ¼ Ç
in Figure 4.22, right; this means that ÈÇÈ ¼ is a right angle. The strophoid’s
polar representation is Ö Ó× ¾ × , and the rational parametric representation
is Ü ´Ø ¾ ½µ´Ø ¾ ·½µ, Ý Ø´Ø ¾ ½µ´Ø ¾ ·½µ(so that Ø ÝÜ).
Among the important curves of degree four are the following:
1. A Cassini’s oval is characterized by the following condition: Given two foci
and ¼ , a distance ¾ apart, a point È belongs to the curve if the product of
the distances È and È ¼ is a constant ¾ . If the foci are on the Ü-axis and
equidistant from the origin, the curve’s equation is ´Ü ¾ · Ý ¾ · ¾ µ ¾ ¾ Ü ¾
. Changes in correspond to rescaling, while the value of controls
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