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Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path ...
The following can be used to find the versine of a given constant radius curve: [2] The Hallade method is to use the chord to continuously measure the versine in an overlapping pattern along the curve. The versine values for the perfect circular curve would have the same number. [3]
The vectors T and N at two points on a plane curve, a translated version of the second frame (dotted), and δT the change in T. Here δs is the distance between the points. In the limit dT / ds will be in the direction N. The curvature describes the rate of rotation of the frame.
The circle S and the curve C have the common tangent 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.
Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. For a rectifiable curve these approximations don't get arbitrarily large (so the curve has a finite length).
When they exist, the osculating circles to parallel curves at corresponding points are concentric. [13] As for parallel lines, a normal line to a curve is also normal to its parallels. When parallel curves are constructed they will have cusps when the distance from the curve matches the radius of curvature.
A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results: = + +. This will exactly fit a simple curve to three points.