curves.html

A History Thread on Curves

The notion of a curve has developed over time along with mathematics. In Early Classical Greek mathematics, there were really only two curves: the circle, which was drawn with compasses and the line, which was drawn with a straight edge. In later Greek mathematics, certain curves were defined as intersections of surfaces : thus the conic sections were intersections of a plane with a right regular double cone that were used to solve 'point' construction problems that were not easily solved with staight edge and compasses alone. Also, certain other curves were defined as the locus of a moving point: examples of such were the quadratrix of Hippias and the pursuit curves of Appolonius. The notion of a curve did not change much until the time of Fermat and Descartes, with the invention of analytic geometry. From that time, a plane curve could be defined as the graph of an equation in two variables, and a space curve as the intersection of the graphs of two equations in three variables. After this time too, many locus problems such as the Appolonius/Pappus m+n lines locus problem could be solved with algebraic methods.

Derived curves

With the developement of calculus came the tools and terminology to define some interesting curves which are derived from a given curve. There are several such curves:

Evolutes

The evolute of a base curve was first studied by Huygens in 1673. If C is a nice smooth curve with a circle of curvature at each point, then as we trace out C, the center of curvature traces out the evolute of C. For example, the evolute of a circle is a point, the center of the circle; whereas the evolute of a line is not defined since its curvature is 0. The evolute of a parabola is interesting to compute.

code

[Maple Plot]

Parametric equations for the evolute of the parabola y = x^2 are

> [x,y] = expand(evalm(ev(t)));

Problem: Work out the evolute of the hyperbola [Maple Math] (right branch).

Involutes

Huygens also worked with an involute of a base curve. Take any tangent line to a curve C and any point P on the tangent line. The locus of P as the line rolls along G with out slipping is called an involute of C. If the point P starts off as the point of tangency, the involute is a principal involute of C. It is interesting to compute the principal involute of a circle. You can draw it by attaching a pen to the end of a thread wound around the circle and then unwinding the thread holding it taut and using the pen to trace the path of the end of the thread. It traces out a spiral with equispaced whorls.

Parallel Curves

Parallel curves to a given curve were first studied by Leibnitz in 1693. Take a curve C and a point P not on C. P has a certain distance from C, say r. The set of all points traced out as P moves allway staying r units from C (shortest distance) is called a parallel curve to C. The parallell curves to a line are lines; the parallel curves to a circle are circles. What about parallel curves of an ellipse? are they ellipses? ( check here for answer

Modern Curve Theory

In the begining of the 19th century, more attention was paid to careful definition of the concept of a curve. The definition of continuous function was stated by Bolzano, Weirstrasse, and Cauchy, and it was realized that curves don't have to tangents everywhere. Bolzano in 1817 gave an example of a curve y = f(x) which had no tangents at all!

code

> clrs := [blue,red,black,tan,magenta]:
plots[display]([seq(bolzano(i,clrs[i]),i=1..5)],insequence=true,scaling=constrained);

[Maple Plot]

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The sequence of curves bolzano(n) as n gets large converges uniformly to a curve y = f(x) which has no tangent lines.

Jordan in the 1880s defined a plane curve as the range of a continuous mapping x = f(t), y = g(t) from an interval say [0,1] to the plane. He along with nearly every one else just assumed that this meant that plane curves were 1-dimensional objects (that is contained no disk). However, Peano and later Hilbert (in 1891) defined a continuous mapping from the unit interval [0,1] onto the unit square!. This was contrary to the intuition of the time, which was that the range of a path had to be 1-dimensional. As in Bolzano's case, Hilbert's space filling curve is the uniform limit of a sequence of curves.

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[Maple Plot]

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