 A circle with 4-point contact at a vertex of a curve In differential geometry, the osculating circle of a curve at a point, is a circle which: Image File history File links OsculatingCircle2. ...
Image File history File links OsculatingCircle2. ...
Image File history File links VertexOfACurve. ...
Image File history File links VertexOfACurve. ...
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...
- Touches the curve at that point
- Has its unit tangent vector
 , equal to the unit tangent of the curve at that point - Has its derivative of unit tangent (with respect to arc length),
 , equal to that of the curve at that point. Note that both the circle and the curve must be parameterised by arc length. In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ...
For other uses, see Curve (disambiguation). ...
For other uses, see Curve (disambiguation). ...
These three conditions define a unique circle for each point on the curve, provided that the derivative of the unit tangent is not zero.
The length of the vector is called the curvature, and is denoted Curvature refers to a number of loosely related concepts in different areas of geometry. ...
The radius of the osculating circle, r is called the radius of curvature and is given by Curvature is the amount by which a geometric object deviates from being flat. ...
The osculating circle of a mathematically defined curve shares location, first derivative, and second derivative with the curve, just as a tangent line shares location and first derivative. Osculate literally means to kiss; the term is used because osculation is a more gentle form of contact than simple tangency. In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...
In mathematics, the derivative is one of the two central concepts of calculus. ...
In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ...
It has been suggested that this article or section be merged with Contact (mathematics). ...
In mathematics, the word tangent has two distinct, but etymologically-related meanings: one in geometry, and one in trigonometry. ...
The centers of the osculating circles form the evolute of the curve. Some points on the curve will form vertices where there is a higher degree of contact. In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. ...
A ellipse (red) and its evolute (blue), the dots are the vertices of the curve, each vertex corresponds to a cusp on the evolute. ...
In lay terms Imagine driving a car along a curved road on a vast flat plane. Suddenly lock the steering wheel in its present position at one point along the road. Thereafter, the car moves in a circle that "kisses" the road at the point of locking. The curvature 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. Curvature refers to a number of loosely related concepts in different areas of geometry. ...
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