Formula for the Curvative of a Curve

Date: 10/10/2002 at 21:10:15
From: Adele Champlin
Subject: Dimensions

I have an assignment in English class to convince someone who believes 
a wall has only one side that there is another side to it. I was 
wondering if there is a formula or theorem I can use for this?

Date: 11/17/2002 at 20:26:48
From: Doctor Nitrogen
Subject: Re: Dimensions

Hello, Adele:

Suppose both of you are in a very large room with a very, very long 
wall. 

1. Imagine both of you traveling along the long length of the wall. If 
the wall is perfectly flat, you might never be able to convince the 
other party there is another side to the wall.

2. If the very long wall is curved away from both of you as you travel 
alongside it inside the room, you could convince the party there must 
be another side to the wall, since it is curved and curving in some 
direction away from both of you and in some direction normal 
(perpendicular) to the wall. 

3. If the very long wall is curved inward toward the room and toward  
both of you, you could convince the other party there must be another 
side to the wall, since it is curved and curving in some direction 
toward both of you and in some direction normal (perpendicular) to the 
wall. 

If you take a magic marker and draw a long curve on the curved wall 
as both of you travel around the big room, you can calculate the 
radius of curvature for that curve, and that perpendicular radius of 
curvature would either point out from the big room (the wall curving 
away from you both) or into the big room (the wall curving inward 
toward you both).

The direction the radius of curvature would be along would be 
perpendicular to the direction in which you both are traveling, and 
perpendicular to the up or down direction of the room, so for a 
curved wall you could hypothesize there lies another side to the 
wall. 

In Calculus the following formula computes such curvature for a 
curve drawn on a curved surface:

     Curvature  K = y"/[1 + (y')^2]3/2.

y' is the derivative of y with respect to x, and y" is the second 
derivative of y with respect to x.

I hope this helped answer the questions you had concerning your 
mathematics problem. You are welcome to return to The Math 
Forum/Doctor Math whenever you have any math-related questions.

- Doctor Nitrogen, The Math Forum
  http://mathforum.org/dr.math/