Back to . . .  Curve Bank Home Classics Index Page Deposit #54 Osculating Circles for Curves in a Plane

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