Formula for the Curvative of a CurveDate: 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/ |
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