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Osculating circle
 
 
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In differential geometry of curves
Differential geometry of curves

Differential geometry of curves is the branch of geometry that dealswith smooth curve in the Euclidean plane and in the Euclidean space by methods of differential calculus and integral calculus....
, the osculating circle of a sufficiently smooth plane curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 at a given point on the curve is the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 whose center lies on the inner normal line and whose curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 is the same as that of the given curve at that point. This circle, which is the one among all tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 circles at the given point that approaches the curve most tightly, was named circulum osculans (Latin
Latin

Latin is an Italic language, historically spoken in Latium and Ancient Rome. Through the Military history of the Roman Empire, Latin spread throughout the Mediterranean and a large part of Europe....
 for "kissing circle") by Leibniz.

The center and radius of the osculating circle at a given point are called radius of curvature and center of curvature of the curve at that point.






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Osculatingcircle2
In differential geometry of curves
Differential geometry of curves

Differential geometry of curves is the branch of geometry that dealswith smooth curve in the Euclidean plane and in the Euclidean space by methods of differential calculus and integral calculus....
, the osculating circle of a sufficiently smooth plane curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 at a given point on the curve is the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 whose center lies on the inner normal line and whose curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 is the same as that of the given curve at that point. This circle, which is the one among all tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 circles at the given point that approaches the curve most tightly, was named circulum osculans (Latin
Latin

Latin is an Italic language, historically spoken in Latium and Ancient Rome. Through the Military history of the Roman Empire, Latin spread throughout the Mediterranean and a large part of Europe....
 for "kissing circle") by Leibniz.

The center and radius of the osculating circle at a given point are called radius of curvature and center of curvature of the curve at that point. A geometric construction was described by Isaac Newton
Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
 in his :

Description in lay terms


Imagine a car moving along a curved road on a vast flat plane. Suddenly, at one point along the road, the steering wheel locks in its present position. Thereafter, the car moves in a circle that "kisses" the road at the point of locking. The curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 of the circle is equal to that of the road at that point. That circle is the osculating circle of the road curve at that point.

Mathematical description


Let γ(s) be a regular parametric curve, where s is the arc length
Arc length

Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases....
, or natural parameter. This determines the unit tangent vector T, the unit normal vector N, the signed curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 k(s) and the radius of curvature at each point:



Suppose that P is a point on C where k ≠ 0. The corresponding center of curvature is the point Q at distance R along N, in the same direction if k is positive and in the opposite direction if k is negative. The circle with center at Q and with radius R is called the osculating circle to the curve C at the point P.

If C is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector N. It lies in the osculating plane
Osculating plane

In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point....
, the plane spanned by the tangent and principal normal vectors T and N at the point P.

Properties


For a curve C given by a sufficiently smooth parametric equations (twice continuously differentiable), the osculating circle may be obtained by a limiting procedure: it is the limit of the circles passing through three distinct points on C as these points approach P. This is entirely analogous to the construction of the tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 to a curve as a limit of the secant lines through pairs of distinct points on C approaching P.

The osculating circle S to a plane curve C at a regular point P can be characterized by the following properties:
  • The circle S passes through P.
  • The circle S and the curve C have the common tangent
    Tangent lines to circles

    In Euclidean geometry, tangent lines to circles form the subject of several theorems, and play an important role in many geometrical Compass and straightedge constructionss and Mathematical proof....
     line at P, and therefore the common normal line.
  • Close to P, the distance between the points of the curve C and the circle S in the normal direction decays as the cube or a higher power of the distance to P in the tangential direction.
This is usually expressed as "the curve and its osculating circle have the third or higher order contact" at P. Loosely speaking, the vector functions representing C and S agree together with their first and second derivatives at P.
Vertexofacurve
If the derivative of the curvature with respect to s is nonzero at P then the osculating circle crosses the curve C at P. Points P at which the derivative of the curvature is zero are called vertices
Vertex (curve)

In the geometry of curves a vertex is a point of where the first derivative of curvature is zero. This is typically a local Maxima and minima of curvature....
. If P is a vertex then C and its osculating circle have contact of order at least four. If, moreover, the curvature has a non-zero local maximum or minimum at P then the osculating circle touches the curve C at P but does not cross it.

The curve C may be obtained as the envelope
Envelope (mathematics)

In mathematics, an envelope of a index set#Families of manifolds is a manifold that is tangent to each member of the family at some point....
 of the one-parameter family of its osculating circles. Their centers, i.e. the centers of curvature, form another curve, called the evolute
Evolute

In the differential geometry of curves, the evolute of a curve is the locus of all its Osculating circle. Equivalently, it is the envelope of the perpendicular to a curve....
 of C. Vertices of C correspond to singular points on its evolute.

Further reading

For some historical notes on the study of curvature, see* For application to maneuvering vehicles see
  • JC Alexander and JH Maddocks: [https://drum.umd.edu/dspace/bitstream/1903/4630/1/TR_87-122.pdf On the maneuvering of vehicles]

External links



See also

  • Contact (mathematics)
    Contact (mathematics)

    In mathematics, contact of order k of function s is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k....
  • Osculating curve
    Osculating curve

    In mathematics and geometry, an osculating curve is an extension of the concept of tangent. A tangent line to a curve is the straight line that shares the location and Direction of the curve, while an osculating circle to the same curve shares the location, direction, and curvature....