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This then gives the circumference as. Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of " -sphere," with geometers referring to the number of coordinates in the underlying space and topologists referring to the dimension of the surface itself Coxeter , p. As a result, geometers call the circumference of the usual circle the 2-sphere, while topologists refer to it as the 1-sphere and denote it. The circle is a conic section obtained by the intersection of a cone with a plane perpendicular to the cone 's symmetry axis. It is also a Lissajous curve. A circle is the degenerate case of an ellipse with equal semimajor and semiminor axes i.

The interior of a circle is called a disk. The generalization of a circle to three dimensions is called a sphere , and to dimensions for a hypersphere. The region of intersection of two circles is called a lens.

Area and circumference of circles

The region of intersection of three symmetrically placed circles as in a Venn diagram , in the special case of the center of each being located at the intersection of the other two, is called a Reuleaux triangle. In Cartesian coordinates , the equation of a circle of radius centered on is. In pedal coordinates with the pedal point at the center, the equation is. The circle having as a diameter is given by. The parametric equations for a circle of radius can be given by.

The plots above show a sequence of normal and tangent vectors for the circle. In polar coordinates , the equation of the circle has a particularly simple form. The equation of a circle passing through the three points for , 2, 3 the circumcircle of the triangle determined by the points is. The center and radius of this circle can be identified by assigning coefficients of a quadratic curve. Completing the square gives. Four or more points which lie on a circle are said to be concyclic. Three points are trivially concyclic since three noncollinear points determine a circle.

In trilinear coordinates , every circle has an equation of the form. The center of a circle given by equation 35 is given by. In exact trilinear coordinates , the equation of the circle passing through three noncollinear points with exact trilinear coordinates , , and is. An equation for the trilinear circle of radius with center is given by Kimberling , p. Beyer, W. Casey, J.

Coolidge, J. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, Courant, R. What Is Mathematics? Oxford, England: Oxford University Press, pp. Coxeter, H. Regular Polytopes, 3rd ed. New York: Dover, Washington, DC: Math. Dunham, W. New York: Wiley, pp. Eppstein, D. Hilbert, D. Geometry and the Imagination. The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass.

Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ratio other than 1 of distances to two fixed foci, A and B. That circle is sometimes said to be drawn about two points. The proof is in two parts. First, one must prove that, given two foci A and B and a ratio of distances, any point P satisfying the ratio of distances must fall on a particular circle. Let C be another point, also satisfying the ratio and lying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angle APB , since the segments are similar:.

Since the interior and exterior angles sum to degrees, the angle CPD is exactly 90 degrees, i. Second, see  : p. A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A , B , and C are as above, then the circle of Apollonius for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:.

Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [ A , B ; C , P ] is on the unit circle in the complex plane. If C is the midpoint of the segment AB , then the collection of points P satisfying the Apollonius condition. Thus, if A , B , and C are given distinct points in the plane, then the locus of points P satisfying the above equation is called a "generalised circle. In this sense a line is a generalised circle of infinite radius.

In every triangle a unique circle, called the incircle , can be inscribed such that it is tangent to each of the three sides of the triangle. About every triangle a unique circle, called the circumcircle , can be circumscribed such that it goes through each of the triangle's three vertices.

A tangential polygon , such as a tangential quadrilateral , is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon. A cyclic polygon is any convex polygon about which a circle can be circumscribed , passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.

The circle can be viewed as a limiting case of each of various other figures:. Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In p -norm , distance is determined by. Thus, a circle's circumference is 8 r.