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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).
The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8r 2 − 4p 2, where r is the circle radius, and p is the distance from the centre point to the point of intersection.
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 combinations thereof. [1] [2] [3]
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.
In Euclidean space, there is a unique circle passing through any given three non-collinear points P 1, P 2, P 3. Using Cartesian coordinates to represent these points as spatial vectors, it is possible to use the dot product and cross product to calculate the radius and center of the circle. Let
The distance from a point to a plane in three-dimensional Euclidean space [7] The distance between two lines in three-dimensional Euclidean space [8] The distance from a point to a curve can be used to define its parallel curve, another curve all of whose points have the same distance to the given curve. [9]
The recursion terminates when P is empty, and a solution can be found from the points in R: for 0 or 1 points the solution is trivial, for 2 points the minimal circle has its center at the midpoint between the two points, and for 3 points the circle is the circumcircle of the triangle described by the points. (In three dimensions, 4 points ...
d is the distance between the two points along a great circle of the sphere (see spherical distance), r is the radius of the sphere. The haversine formula allows the haversine of θ to be computed directly from the latitude (represented by φ) and longitude (represented by λ) of the two points: