Descartes' theorem on the "total defect" of a polyhedron states the if the polyhedron is homeomorphic to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4π radians). The polyhedron need not be convex. A generalization says the number of circles in the total def ...

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Defect geometry, Defect geometry - Examples, Defect geometry - Descartes' theorem, Defect geometry - A potential error

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Descartes' theorem, Descartes' theorem - History, Descartes' theorem - Definition of curvature, Descartes' theorem - Descartes' theorem, Descartes' theorem - Special cases, Descartes' theorem - Complex Descartes theorem

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Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the third century BC, Apollonius of Perga devoted an entire book to the topic. Unfortunately the book, which was called On Tangencies, is not among his surviving work. René Descartes touched on the problem briefly in 1643, in a letter to Princess Elizabeth of Bohemia (as such things might go in those times). He came up with essentially the same solution as given in equation (1) bel ...

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Descartes' theorem, Descartes' theorem - History, Descartes' theorem - Definition of curvature, Descartes' theorem - Descartes' theorem, Descartes' theorem - Special cases, Descartes' theorem - Complex Descartes theorem

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A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. The points can only be those that are part of a conic section; within the set of a plane bisecting a cone. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually however, the circumference means the length of the circle, and the interior ...

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Circle - Mathematical definitions Circle - Properties
Circle - Chord properties Circle - Tangent properties Circle - Inscribed angle theorem Circle - Secant tangent and chord properties

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Apollonius of Perga [Pergaeus] (c. 262 BC–c. 190 BC) was a Greek geometer and astronomer, of the Alexandrian school, noted for his writings on conic sections. It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which we know them. The hypothesis of eccentric orbits, or equivalently, deferent and epicycles, to explain the apparent motion of the planets and ...

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Apollonius of Perga - De Rationis Sectione Apollonius of Perga - De Spatii Sectione Apollonius of Perga - De Sectione Determinata Apollonius of Perga - De Tactionibus Apollonius of Perga - De Inclinationibus Apollonius of Perga - De Locis Planis

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De Locis Planis is a collection of propositions relating to loci which are either straight lines or circles. Pappus gives somewhat full particulars of the propositions, and restorations were attempted by P. Fermat (Oeuvres, i., 1891, pp. 3-51), F. Schooten (Leiden, 1656) and, most successfully of all, by R. Simson (Glasgow, 1749). Other works of Apollonius are referred to by ancient writers, viz. Περι του πυριου, On the Burning-Glass, where the focal properties of the parabola probably fo ...

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Apollonius of Perga, Apollonius of Perga - De Rationis Sectione, Apollonius of Perga - De Spatii Sectione, Apollonius of Perga - De Sectione Determinata, Apollonius of Perga - De Tactionibus, Apollonius of Perga - De Inclinationibus, Apollonius of Perga - De Locis Planis

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The Ford circle associated with the fraction p/q is denoted by C[p/q] or C[p, q]. There is a Ford circle associated with every rational number. In addition, the line y = 1 is counted as a Ford circle - it can be thought of as the Ford circle associated with infinity, which is the case p = 1, q = 0. Two different Ford circles are either disjoint or tangent to one another. No two interiors of Ford circles intersect - even though there is a Ford circle tangent to the x ...

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Ford circle, Ford circle - History, Ford circle - Properties

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In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that If the circle is centred at the origin (0, 0), then this formula can be simplified to x2 + y2 = r2. The circle centred at the origin with radius 1 is called the unit circle. Expressed in parametric equations, (x, y) can be written as x = a + r cos(t)

In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that If the circle is centred at the origin (0, 0), then this formula can be simplified to: The circle centred at the origin with radius 1 is called the unit circle. Expressed in parametric equations, (x, ySee also:

Circle, Circle - Mathematical definitions, Circle - Properties, Circle - Chord properties, Circle - Tangent properties, Circle - Inscribed angle theorem, Circle - Secant tangent and chord properties

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In geometry, the circumcircle of a given two-dimensional geometric shape is a circle which contains the shape completely within it. For a triangle, it is the unique circle containing all three vertices. The center of this circumcircle is known as the shape's circumcenter. Note that although the circumcircle of an acute triangle is indeed the smallest circle containing this triangle, this is not true of obtuse triangles. Circumcircle - Cyclic polygons. At least three ver ...

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Circumcircle - Cyclic polygons Circumcircle - Circumcircles of triangles
Circumcircle - Circumcircle equation
Circumcircle - Circumcircle of circles

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De Spatii Sectione discussed the similar problem which requires the rectangle contained by the two intercepts to be equal to a given rectangle. An Arabic version of the De Rationis Sectione was found towards the end of the 17th century in the Bodleian library by Edward Bernard, who began a translation of it; Halley finished it and published it along with a restoration of the De Spatii Sectione in 1706. ...

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Apollonius of Perga, Apollonius of Perga - De Rationis Sectione, Apollonius of Perga - De Spatii Sectione, Apollonius of Perga - De Sectione Determinata, Apollonius of Perga - De Tactionibus, Apollonius of Perga - De Inclinationibus, Apollonius of Perga - De Locis Planis

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Circle - Chord properties. Chords equidistant from the centre of a circle are equal. Equal chords are equidistant from the centre. A line from the centre, perpendicular to a chord, bisects the chord. The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord. The perpendicular bisector of a chord passes t ...

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Circle, Circle - Mathematical definitions, Circle - Properties, Circle - Chord properties, Circle - Tangent properties, Circle - Inscribed angle theorem, Circle - Secant tangent and chord properties

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The circumcircle of a triangle is the unique circle on which all its three vertices lie. (This is not the same as the first definition for "thin" triangles where only two points would lie on the first definition's circumcircle.) The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint.) This is because the circumcenter is equidistant from any pair of the triangle's points, and all points on the perpendicular b ...

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Circumcircle, Circumcircle - Cyclic polygons, Circumcircle - Circumcircles of triangles, Circumcircle - Circumcircle equation, Circumcircle - Circumcircle of circles

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An Apollonian gasket can be constructed as follows. Start with three circles C1, C2 and C3, each one of which is tangent to the other two (in the general construction, these three circles can be any size, as long as they have common tangents). Apollonius discovered that there are two other non-intersecting circles, C4 and C5, which have the property that they are tangent to all three of the original circles - these are called Apollonian circles (see Descartes' theorem). Adding the two Apollonian circle ...

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Apollonian gasket, Apollonian gasket - Construction, Apollonian gasket - Variations, Apollonian gasket - Symmetries, Apollonian gasket - Links with hyperbolic geometry

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If two of the original generating circles have the same radius and the third circle has a radius that is two-thirds of this, then the Apollonian gasket has two lines of reflective symmetry; one line is the line joining the centres of the equal circles; the other is their mutual tangent, which passes through the centre of the third circle. These lines are perpendicular to one another, so the Apollonian ...

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Apollonian gasket, Apollonian gasket - Construction, Apollonian gasket - Variations, Apollonian gasket - Symmetries, Apollonian gasket - Links with hyperbolic geometry

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An Apollonian gasket can also be constructed by replacing one of the generating circles by a straight line, which can be regarded as a circle passing through the point at infinity. Alternatively, two of the generating circles may be replaced by parallel straight lines, which can be regarded as being tangent to one another at infinity. In this construction, the circles that are tangent to one of the t ...

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Apollonian gasket, Apollonian gasket - Construction, Apollonian gasket - Variations, Apollonian gasket - Symmetries, Apollonian gasket - Links with hyperbolic geometry

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In order to determine a circle completely, not only its radius (or curvature), but also its center must be known. The relevant equation is expressed most clearly if the coordinates (x, y) are interpreted as a complex number z = x + iy. The equation then looks similar to Descartes' theorem and is therefore called the complex Descartes theorem. Given four circles with curvatures ki and centers zi (for i = 1…4), the following equal ...

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Descartes' theorem, Descartes' theorem - History, Descartes' theorem - Definition of curvature, Descartes' theorem - Descartes' theorem, Descartes' theorem - Special cases, Descartes' theorem - Complex Descartes theorem

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Descartes' theorem is most easily stated in terms of the circles' curvature. The curvature of a circle is defined as k = ±1/r, where r is its radius. The larger a circle, the smaller is the magnitude of its curvature, and vice versa. The plus sign in k = ±1/r applies to a circle that is externally tangent to the other circles, like the three black circles in the image. For an internally tangent circle like the big red circle, that < ...

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Descartes' theorem, Descartes' theorem - History, Descartes' theorem - Definition of curvature, Descartes' theorem - Descartes' theorem, Descartes' theorem - Special cases, Descartes' theorem - Complex Descartes theorem

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If one of the three circles is replaced by a straight line, k3 (say) is zero and drops out of equation (1). Equation (2) then becomes much simpler: Descartes' theorem does not apply when two or all three circles are replaced by lines. Nor does the theorem apply when more than one circle is internally tangent, e.g. in the case of three nested circles all touching in one point. ...

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Descartes' theorem, Descartes' theorem - History, Descartes' theorem - Definition of curvature, Descartes' theorem - Descartes' theorem, Descartes' theorem - Special cases, Descartes' theorem - Complex Descartes theorem

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