If " C " is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector " N ".
12.
:: If the osculating circle has radius 1, the parabola is given by 2 " " x " 2; but this does not give a satisfactory solution.
13.
If the derivative of curvature ?'( " t " ) is zero, then the osculating circle will have 3rd-order contact and the curve is said to have a vertex.
14.
For each point " S " ( " t " ) on a locus of the centers of all the osculating circles ( also called " centers of curvature " ) is the evolute of the curve.
15.
In more familiar terms, if " & lambda; " is the contact lift of a curve " & gamma; " in the plane, then the preferred cycle at each point is the osculating circle.
16.
The radius of curvature of the path is ? as found from the rate of rotation of the tangent to the curve with respect to arc length, and is the radius of the osculating circle at position " s ".
17.
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.
18.
For general motion of a particle ( as opposed to simple circular motion ), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous osculating circle of its motion, not to a fixed center of polar coordinates.
19.
The sign of the radius of curvature indicates the direction in which the osculating circle moves if it is parameterized in the same direction as the curve at the point of contact : it is positive if the circle moves in a counterclockwise sense, and negative otherwise.
20.
When describing general motion, the actual forces acting on a particle are often referred to the instantaneous osculating circle tangent to the path of motion, and this circle in the general case is not centered at a fixed location, and so the decomposition into centrifugal and Coriolis components is constantly changing.
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