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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 ...
An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. [2] The osculating circle provides a way to understand the local behavior of a curve and is commonly used in differential geometry and calculus.
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]
Note that changing F into –F would not change the curve defined by F(x, y) = 0, but it would change the sign of the numerator if the absolute value were omitted in the preceding formula. A point of the curve where F x = F y = 0 is a singular point, which means that the curve is not differentiable at this point, and thus that the curvature is ...
Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equation. Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.
In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse , the Pythagorean ...
Let x, y and z be three points in R n; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line.Let Π ⊆ R n be the Euclidean plane spanned by x, y and z and let C ⊆ Π be the unique Euclidean circle in Π that passes through x, y and z (the circumcircle of x, y and z).
Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the sum of the distances from the point to the intersections of a circle through the point with the curve, divided by the nth power of the diameter d. Laguerre showed that this number is independent of the diameter (Laguerre 1905).