1a. Families of Curves
In this optional section we present several important examples of curve families: the cardiods, the cycloids, the epicycloids, the catenaries, and the conic sections.5The previous exercises concern families of curves that depend a choice of one or two positive integers. Computer graphics displays are well suited for presenting one-parameter families of curves.
We define Xc(t) =[(c+cos(t))cos(t), (c+cos(t) )sin(t)] over the domain (-pi, pi). For which values of c will the curve Xc have a cusp? Which curves will have double points (points where X(t) = X(t + d) for some non-zero d such that t + d is still in the domain)? For which c will the curve have inflection points?
Consider the path of a point on the circumference of a wheel of radius a rolling without slipping along the x-axis so that the center has coordinates (t, a) and when t=0, the point is at the origin. The parametric equation for the curve is X(t)=a(t-sin(t), 1-cos(t)). Draw a diagram that indicates why this is a reasonable representation of the curve. Where does the curve have cusps? How far does the point go on the x-axis?
In the following demo, you can also vary c, the length of the stick, so the point is not necessarily drawn at the circumference of the wheel.
As shown in the first demo, these are curves
traced out by the path of a point on the circumference of a wheel of
radius b rolling without slipping on a circle of radius a, so that the
path of the center has coordinates ((a+b)cos(t),(a+b)sin(t))
and the
point is at ((a+b), 0)
when t = 0. The parametric equation for the
curve is
X(t)
=()(a+b)cos(
t)-bcos(
The Catenary Series
The family of curves given by X(t)
=
(t,
In addition to circles and parabolas, there are two other families of conic sections of great interest in differential geometry of curves, the ellipses and the hyperbolas. The family of ellipses with axes parallel to the coordinate axes is given by X(t)=[acos(t), b sin(t)], while the corresponding family of hyperbolas is given by X(t)=[acosh(t), bsinh(t)].
