Radius of Curvature: Math Homework Help

[LareeRudi]

Homework Statement

I had my college math courses in 1955-1957, so I'm rusty. Lately interested in Radius of Circle of Curvature. I don't have a math typing program, so I'll try to describe the equation that I found recently, but it's complexity [though so far, I can handle any common equations that I'm involved with now] baffles me. I THOUGHT that we had learned this in Algebra, perhaps Intermediate algebra. Well, MAYBE it was Analytic Geometry, and well, maybe Calculus.

In either case, my subconscious seems to remember something not involving differentials. Is my memory fooling me, or am I asking the wrong question? Seemed to me we were given an equation, then asked to find the "radius of curvature" AND it's center point. And again, I thought it was so much simpler. Was there something you can think of that I MGIHT be remembering, similar to this?

Homework Equations

radius curvature = numerator and a denominator

numerator is [ 1 + (dy/dx)^2 ]^(3/2)

denominator is the second differential, or D^2Y/DX^2

The Attempt at a Solution

I can handle most equations, so my question is more about the CONCEPT of what I'm asking than a problem per se.

As to finding the center point of the circle that is found, I'd take the first differential, insert the point of interest on the curve, find it's "slope", then find a line perpendicular to that that PASSES THRU the point of interst, then go out the distance of the radius previously found, then Pythagorean Theorem to see which POINT would have the radius desired using the X and Y coordinates of point.

thx,

LarryR : )

 

[hotvette]

Maybe you are remembering analytic geometry involving circles (that have a constant radius) and various ways to find the center and/or radius based on known points.

The equation you show is for a general (twice differentiable) function and is differential in nature because the curvature is (in general) continuously variable along the curve. A simple example is a parabola y = ax² + bx + c. The second derivative y'' = 2a is fixed but the first derivative y' = 2ax + b continuously varies with the value of x, thus the curvature is a function of x.