How 3 points uniquely determine a circle |
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A circle through 3 points on a curve |
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Since the curvature of a circle of radius R is the constant value 1/R, it is natural to use circles to help define curvature. Just as two distinct points determine a line, three non-collinear points determine a circle. Recall that the tangent line is the limit of lines through two points on a curve. Similarly, the "osculating circle" 1 is the limit of the circles through three points on the curve as the points approach one another. The radius of the osculating circle is called the "radius of curvature."
Dangerous Bend: The curvature is the reciprocal of the radius of curvature, so the radius of curvature is a local maximum when the curvature is a local minimum, and vice versa.
Explore the process of using circles to define curvature on the discover page.
These properties tell us more about the shape of the curve. Try using these ideas on some practice problems.
Calculating limits of circles through three points is tedious, but fortunately there are formulas for the radius of curvature in terms of derivatives. 2 If the curve is r(t) = < f(t), g(t) &rt; then
The radius of curvature is:
The curvature is:
The curvature and radius of curvature exist only when f and g are twice differentiable and the respective denominator is non-zero when |f'(t)g''(t) - f''(t)g'(t)| = 0. This corresponds to a point where the curve is very flat. These points occur when the curve is a straight line, or at an inflection, or sometimes at other places. At these points the osculating circle actually becomes a straight line and so the radius of curvature is undefined.
Footnote 1: Osculate is the Latin word for "to kiss."
Footnote 2: The derivation of these formulas is lengthy and we will not pursue them here.