Hypercycle (geometry)
Encyclopedia
In hyperbolic geometry
, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line.
Given a straight line L and a point P not on L,
we 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 orthogonal segments from each point to L are called radii.
Their common length is called distance.
In the Poincaré disk
and half-plane
models of the hyperbolic plane, hypercycles are represented by lines and circles arcs that intersect the boundary circle/line at non-right angles. The representation of the axis intersects the boundary circle/line in the same points, but at right angles.
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 circles in Euclidean geometry
:
Let AB be the chord and M its middle point.
By symmetry the line R through M perpendicular to AB must be orthogonal to the axis L.
Therefore R is a radius.
Also by symmetry, R will bisect the arc AB.
Let us assume that a hypercycle C has two different axes
and 
.
Using the previous property twice with different chords we can determine two distinct radii
and 
. 
and 
will then have to be perpendicular to both 
and 
, giving us a rectangle. This is a contradiction because the rectangle is an impossible figure in hyperbolic geometry
.
If they have equal distance, we just need to bring the axes to coincide by a rigid motion and also all the radii will coincide; since the distance is the same, also the points of the two hypercycles will coincide.
Vice versa, if they are congruent the distance must be the same by the previous property.
Let the line K cut the hypercycle C in two points A and B. As before, we can construct the radius R of C through the middle point M of AB. Note that K is ultraparallel to the axis L because the have the common perpendicular R. Also, two ultraparallel lines have minimum distance at the common perpendicular and monotonically increasing distances as we go away from the perpendicular.
This means that the points of K inside AB will have distance from L smaller than the common distance of A and B from L, while the points of K outside AB will have greater distance. In conclusion, no other point of K can be on C.
Let
and 
be hypercycles intersecting in three points A, B, and C.
If
is the line orthogonal to AB through its middle point, we know that it is a radius of both 
and 
.
Similarly we construct
, the radius through the middle point of BC.
and 
are simultaneously orthogonal to the axes 
and 
of 
and 
, respectively.
We already proved that then
and 
must coincide (otherwise we have a rectangle).
Then
and 
have the same axis and at least one common point, therefore they have the same distance and they coincide.
If the points A, B, and C of an hypercycle are collinear then the chords AB and BC are on the same line K.
Let
and 
be the radii through the middle points of AB and BC.
We know that the axis L of the hypercycle is the common perpendicular of
and 
.
But K is that common perpendicular
.
Then the distance must be 0 and the hypercycle degenerates into a line.
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line.
Given a straight line L and a point P not on L,
we 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 orthogonal segments from each point to L are called radii.
Their common length is called distance.
In the Poincaré disk
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the straight lines of the hyperbolic geometry are segments of circles contained in the disk...
and half-plane
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry....
models of the hyperbolic plane, hypercycles are represented by lines and circles arcs that intersect the boundary circle/line at non-right angles. The representation of the axis intersects the boundary circle/line in the same points, but at right angles.
The hypercycles through a given point that share a tangent through that point converge towards a horocycle
Horocycle
In hyperbolic geometry, a horocycle is a curve whose normals all converge asymptotically. It is the two-dimensional example of a horosphere ....
as their distances go towards infinity.
Hypercycles in hyperbolic geometry have some properties similar to those of circles in Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
:
- A line perpendicular to a chord of a hypercycle at its midpoint is a radius and it bisects the arc subtended by the chord.
Let AB be the chord and M its middle point.
By symmetry the line R through M perpendicular to AB must be orthogonal to the axis L.
Therefore R is a radius.
Also by symmetry, R will bisect the arc AB.
- The axis and distance of an hypercycle is uniquely determined.
Let us assume that a hypercycle C has two different axes
and .Using the previous property twice with different chords we can determine two distinct radii
and . and will then have to be perpendicular to both and , giving us a rectangle. This is a contradiction because the rectangle is an impossible figure in hyperbolic geometryHyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
.
- Two hypercycles have equal distances IffIFFIFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
they are congruent.
If they have equal distance, we just need to bring the axes to coincide by a rigid motion and also all the radii will coincide; since the distance is the same, also the points of the two hypercycles will coincide.
Vice versa, if they are congruent the distance must be the same by the previous property.
- A straight line cuts a hypercycle in at most two points.
Let the line K cut the hypercycle C in two points A and B. As before, we can construct the radius R of C through the middle point M of AB. Note that K is ultraparallel to the axis L because the have the common perpendicular R. Also, two ultraparallel lines have minimum distance at the common perpendicular and monotonically increasing distances as we go away from the perpendicular.
This means that the points of K inside AB will have distance from L smaller than the common distance of A and B from L, while the points of K outside AB will have greater distance. In conclusion, no other point of K can be on C.
- Two hypercycles intersect in at most two points.
Let
and be hypercycles intersecting in three points A, B, and C.If
is the line orthogonal to AB through its middle point, we know that it is a radius of both and .Similarly we construct
, the radius through the middle point of BC. and are simultaneously orthogonal to the axes and of and , respectively.We already proved that then
and must coincide (otherwise we have a rectangle).Then
and have the same axis and at least one common point, therefore they have the same distance and they coincide.- No three points of a hypercycle are collinear.
If the points A, B, and C of an hypercycle are collinear then the chords AB and BC are on the same line K.
Let
and be the radii through the middle points of AB and BC.We know that the axis L of the hypercycle is the common perpendicular of
and .But K is that common perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
.
Then the distance must be 0 and the hypercycle degenerates into a line.