Printer-friendly version

26 March 2004

More About Plane Curves

by George E. Hrabovsky, President, MAST

What We Did Last Time

We explored several ideas about plane curves.

Where We Will Go This Time

We will explore some aspects of curves related to integration, then we will discuss some specific plane curves.

Vector Integration

Last time we discussed curvature of a curve denoted α (t).  The curvature is based on differentiation.  It is natural to ask, now that we know how to differentiate a vector, how do we integrate it?  The answer is component-by-component,

∫a(t) t = (∫a_x(t) t, ∫a_y(t) t, ∫a_z(t) t) .

Arc Length

If our curve is dependent on a parameter, we can define the rate of change of the length of the curve with respect to the parameter as s/t, then we note that we can separate the variables

s = t

and integrate from some point a to some point b,

s = ∫_a^bs

    = ∫_a^bs/tt

If we define a radius vector from the origin to any given point as r (t), then we can write r ' (t) = s/t, so

s = ∫_a^b || r ' (t) || t .

This is called the arc length of our curve.

Mathematics Challenge from Last Time

How did we get the second expression for curvature,

κ = (x ' (t) y'' (t) - x'' (t) y ' (t))/[x ' (t)^2 + y ' (t)^2]^(3/2)

from

κ = (α'' (t) (α ' (t)))/(|| α ' (t) ||^3)

for the parameterized curve

α(t) = (x(t), y(t)) ?

    We begin with the curvature,

κ = (α'' (t) (α ' (t)))/(|| α ' (t) ||^3) .

By definition, we know that

α ' (t) = (x ' (t), y ' (t)),

α'' (t) = (x'' (t), y'' (t)),

and,

(α ' (t)) = (-y ' (t), x ' (t)) .

So,

α'' (t) (α ' (t)) = (x'' (t)) (-y ' (t)) + (y'' (t)) (x ' (t))

               &nbs ... ;           = -y ' (t) x'' (t) + y'' (t) x ' (t)

               &nbs ...            = y'' (t) x ' (t) - y ' (t) x'' (t) .

We also have

|| α ' (t) ||^3 = ((α ' (t) · α ' (t))^(1/2))^3

                       = ((x ' (t)^2 + y ' (t)^2)^(1/2))^3

                       = [x ' (t)^2 + y ' (t)^2]^(3/2) .

So,

κ = (α'' (t) (α ' (t)))/(|| α ' (t) ||^3)

   = (y'' (t) x ' (t) - y ' (t) x'' (t))/( [x ' (t)^2 + y ' (t)^2]^(3/2)) .

Mathematics Challenge

How does this relate to differential equations?

Sources About Differential Geometry of Curves

On-Line:

A nice set of lectures is located here,

http://noodle.med.yale.edu/seminar/shi/lecture1.pdf

Wikipedia is very nice and free,

http://en.wikipedia.org/wiki/Differential_geometry

Here is a free textbook online

http://people.uncw.edu/lugo/COURSES/DiffGeom/dg1.htm

Books:

Heinrich W. Guggenheimer (1963), Differential Geometry, McGraw-Hill Book Company (reprinted in 1977 by Dover Publications).  This is a very complete book, with lots of useful information.

Dirk J. Struik (1961), Lectures on Classical Differential Geometry, Addison Wesley (reprinted in 1988 by Dover Publications).  This is a nice set of lectures on classical differential geometry.

Alfred Gray (1998), Modern Differential Geometry of Curves and Surfaces with Mathematica 2nd Edition, CRC Press. This is a huge tome that not only covers the basic theory of differential geometry, it also covers how to develop Mathematica programs for it.

Created by Mathematica  (March 26, 2004)