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Deposit #54

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Osculating Circles for Curves in a Plane
Osculating circle illustration

The word "osculate" means "to kiss."  A succession of circles that converge to snuggly "kiss" a curve are said to be osculating circles.  They are also known as the Circle of Curvature.

Circle of curvature illustrationThe tangent to a point on a curve was applied  by the earliest investigators to study the derivative and thus Calculus.  Just as the tangent line approximates a curve at a point, the osculating circle is an even better approximation by capturing the curvature.

Osculating circle animation

Note that at the point of inflection (second derivative), the tangent must jump across the curve; thus, the circle also jumps across this curve.  Replay the animation to watch the jump at (0,0).

In our MATHEMATICA® animations, both the tangent and normal lines are drawn at the point of contact of the curve and the circle.
Our animations feature several

Classic Curves
with their osculating circles.

We ask you to recall the construction for circumscribing a circle about a triangle.

Circle circumscribed about triangle

A minimum of three points on the curve are needed to determine, first the perpendicular bisectors, and then the center and radius of the  osculating circle.  As the three points on a given curve move closer together - converge to a single point - the radius of the circle becomes the normal to the tangent at the point.

Animation: Point on a parabola converging
Play this animation.
As the three points on the parabola converge to the vertex, the circle of curvature takes on a radius and center containing the normal to the point of tangency.
Normal animation
Osculating sine curve animation
y = sin (x), a tangent, normal, and osculating circle.

In our MATHEMATICA® animations, both the tangent and normal lines are drawn at the point of contact of the curve and the circle.
Animation  y = 1/x
Replay this animation.

Animation y = x^2
Replay this animation.
y = sin(x)
Replay this animation.
y = cos(x)
Replay this animation.

y = tan (x)

Suggestions for the MATHEMATICA® code are on this link.
[ You may need to enlarge the new screen or open the link in another browser to read the print. ]

Useful Links and Books
Leibniz statue in London
Eves, Howard, AN INTRODUCTION TO THE HISTORY OF MATHEMATICS,  6th ed., Saunders College Publishing, 1992, p. 405.

Leibniz "defined the osculating circle and showed its importance in the study of curves."

A large statue of Leibniz is at the Royal Academy of Arts in the heart of fashionable London.  Note the English chose to spell his name as  . . . .
Leibniz on statue in London.
Gray, Alfred,  Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, 2nd ed., CRC Press, 1998, pp. 111-115.
Wagon, Stan, MATHEMATICA®IN ACTION, W. H. Freeman and Co., 1991.    ISBN 0-7167-2229-1  or  ISBN 0-7167-2202-X (pbk.)
Wagon, Stan, MATHEMATICA® IN ACTION, 2nd ed., Springer-Verlag, 2000.  ISBN 0-387-98684-7 for other animations.
Yates, Robert C., Curves and Their Properties,  NCTM, 1952, pp. 60-63.
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MATHEMATICA® Code and animation contributed by

  Gustavo Gordillo