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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 ...
The dimension of a complete bipartite graph,, for , can be drawn as in the figure to the right, by placing m vertices on a circle whose radius is less than a unit, and the other two vertices one each side of the plane of the circle, at a suitable distance from it.
The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle. Given the length y of a chord and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle that will fit around the two lines: = +.
Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon. Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon. The fifth vertex is the rightmost intersection of the horizontal line with the original circle.
For many years the proof of the four-vertex theorem remained difficult, but a simple and conceptual proof was given by Osserman (1985), based on the idea of the minimum enclosing circle. [10] This is a circle that contains the given curve and has the smallest possible radius. If the curve includes an arc of the circle, it has infinitely many ...
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 ...
Following Archimedes' argument in The Measurement of a Circle (c. 260 BCE), compare the area enclosed by a circle to a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius. If the area of the circle is not equal to that of the triangle, then it must be either greater or less.
Both vertices and angles at the vertices of a triangle are denoted by the same upper case letters A, B, and C. Sides are denoted by lower-case letters: a, b, and c. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (see arc length).