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The circle of thirds is related to the Circle of fifths. The circle of fifths is composed of the twelve Major keys in the order (C, G, D, A, E, B, F#/G♭, D♭, A♭, E♭, B♭, F) going clockwise. The circle of fifths can also be drawn as a circle of the associated minor keys in the order (Am, Em, Bm, F#m, C#m, G#m, D#m, B♭m, Fm, Cm, Gm, Dm).
Chord diagrams are conventionally visualized by arranging the objects in their order around a circle, and drawing the pairs of the matching as chords of the circle. The number of different chord diagrams that may be given for a set of 2 n {\displaystyle 2n} cyclically ordered objects is the double factorial ( 2 n − 1 ) ! ! {\displaystyle (2n ...
A circle with five chords and the corresponding circle graph. In graph theory, a circle graph is the intersection graph of a chord diagram.That is, it is an undirected graph whose vertices can be associated with a finite system of chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
A circle with 1st-order contact (tangent) A circle with 2nd-order contact (osculating) A circle with 3rd-order contact at a vertex of a curve. For each point S(t) on a smooth plane curve S, there is exactly one osculating circle, whose radius is the reciprocal of κ(t), the curvature of S at t.
As a circle has a clockwise order and a counterclockwise order, any set with a cyclic order has two senses. A bijection of the set that preserves the order is called an ordered correspondence . If the sense is maintained as before, it is a direct correspondence , otherwise it is called an opposite correspondence . [ 10 ]
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Examples of osculating curves of different orders include: The tangent line to a curve C at a point p, the osculating curve from the family of straight lines. The tangent line shares its first derivative with C and therefore has first-order contact with C. [1] [2] [4] The osculating circle to C at p, the osculating curve from the family of circles.
The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph ; more generally, intersection graphs of interior-disjoint geometric objects ...