1.3.2 Estimating length via line segments
- Author:
- Sarah Harrelson
The goal of today's lesson is to understand what it means to compute the length of a curve. The GeoGebra applet below demonstrates our approach to approximating length.
We've parameterized a curve via a path . Then we form a regular partition of the domain. A regular partition of an interval of real numbers is a subdivision of the interval into smaller, equally sized sub-intervals. The length of each sub-interval (also referred to as the mesh of the partition) is notated as .
The endpoints of the sub-intervals are then mapped onto the curve via the path . The straight-line distance between successive points is computed by defining something I'll call a secant vector (this is the same usage of the word secant as in Geometry (a line that cuts through a circle in two points) and AP Calculus (a line connecting two points on a graph)). We'll define the secant vector this way:
. Our approximation for the length of the curve is then the sum of the lengths of these secant vectors.
There are several subtleties to consider.
A regular partition of the interval does not translate to a regular partition of the image curve. In other words, even though the sub-intervals of are of equal length, the corresponding sections of the curve are often of varying lengths (and correspondingly the secant vectors are of varying lengths). Why is that? Can you find a curve and a parameterization for which a regular partition of the creates a regular partition of the image curve?
There are parameterizations for which this approach will not yield a good approximation of the length of the curve. Why?