In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis).
Given a straight line L and a point P not on L, one can construct a hypercycle by taking all points Q on the same side of L as P, with perpendicular distance to L equal to that of P. The line L is called the axis, center, or base line of the hypercycle. The lines perpendicular to L, which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between L and the hypercycle are called the radii. Their common length is called the distance or radius of the hypercycle. [1]
The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.
Hypercycles in hyperbolic geometry have some properties similar to those of lines in Euclidean geometry:
Hypercycles in hyperbolic geometry have some properties similar to those of circles in Euclidean geometry:
In the hyperbolic plane of constant curvature −1, the length of an arc of a hypercycle can be calculated from the radius r and the distance between the points where the normals intersect with the axis d using the formula l = d cosh r. [2]
In the Poincaré disk model of the hyperbolic plane, hypercycles are represented by lines and circle arcs that intersect the boundary circle at non-right angles. The representation of the axis intersects the boundary circle in the same points, but at right angles.
In the Poincaré half-plane model of the hyperbolic plane, hypercycles are represented by lines and circle arcs that intersect the boundary line at non-right angles. The representation of the axis intersects the boundary line in the same points, but at right angles.
The congruence classes of Steiner parabolas in the hyperbolic plane are in one-to-one correspondence with the hypercycles in a given half-plane H of a given axis. In an incidence geometry, the Steiner conic at a point P produced by a collineation T is the locus of intersections LT(L) for all lines L through P. This is the analogue of Steiner's definition of a conic in the projective plane over a field. The congruence classes of Steiner conics in the hyperbolic plane are determined by the distance between P and T(P) and the angle of rotation induced by T about T(P). Each Steiner parabola is the locus of points whose distance from a focus F is equal to the distance to a hypercycle directrix that is not a line. Assuming a common axis for the hypercycles, the location of F is determined by as follows. Fixing , the classes of parabolas are in one-to-one correspondence with ∈ (0,π/2). In the conformal disk model, each point P is a complex number with |P| Let the common axis be the real line and assume the hypercycles are in the half-plane H with
Im(P). Then the vertex of each parabola will be in H, and the parabola is symmetric about the line through the vertex perpendicular to the axis. If the hypercycle is at distance from the axis, with , then F = ((1-tan)/(1+tan)). In particular, F = 0 when π/4. In this case, the focus is on the axis; equivalently, inversion in the corresponding hypercycle leaves H invariant. This is the harmonic case, that is, the representation of the parabola in any inversive model of the hyperbolic plane is a harmonic, genus 1 curve.
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.
In elementary geometry, two geometric objects are perpendicular if their intersection forms right angles at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines, between a line and a plane, and between two planes.
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word line may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it or by a single letter.
In hyperbolic geometry, the angle of parallelism , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism.
In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel.
A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal.
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.
In hyperbolic geometry, a horocycle, sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere.
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point, which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement.
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.
The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field.
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed. The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. As in Euclidean geometry, where ancient Greek mathematicians used a compass and idealized ruler for constructions of lengths, angles, and other geometric figures, constructions can also be made in hyperbolic geometry.