Universal hyperbolic geometry is a new
approach to the subject. Its main features are
- works over a general field, not of characteristic
zero. In particular there is a complete theory over
the rational numbers, and also there are complete
theories over a finite prime field
- it extends the familiar Beltrami Klein model to
the outside of the unit disk
- connects more naturally with Einstein's special
theory of relativity in the geometrical context of
- allows faster and more accurate calculations
- allows simpler and more effective constructions
- many new theorems
The main reference paper
is the paper
`Universal Hyperbolic Geometry I: Trigonometry'
available on the ArXiV.
worksheets and Java applets
Below you may download a number of Geometer's Sketchpad
worksheets which illustrate various constructions and
theorems from Universal Hyperbolic Geometry. To run
these files you will need a copy of the commercially
available program `The
Geometer's Sketchpad'. More GSP worksheets will
be added, so be sure to come back periodically and have
Some of the GSP worksheets have Java applets associated
to them that you can run in your browser without having
Geometer's Sketchpad. These do not has as much functionality
as the GSP worksheets, but still allow you to manipulate
points and lines and see how configurations change.
MORE JAVA APPLETS WILL BE ADDED...
These worksheets give an affine picture of the projective
plane associated to the quadratic form x^2+y^2-z^2 in
3 dimensional space, via the plane z=1. This is essentially
the Beltrami Klein view, in which hyperbolic
points are represented by ordinary points, and hyperbolic
lines are represented by ordinary lines. Points lying
on the null circle are null points, and lines tangent
to the null circle are null lines. Note that the constructions
work equally well inside or outside the null circle.
Each worksheet shows the null circle, in blue,
together with control points that allow you to change
the center and size of the null circle. All worksheets
come with a number of built-in programs that
perform various functions, such as creating the pole
(dual) of a line, or the polar (dual) of a point, or
measuring a quadrance or spread. Some of the objects
in a worksheet may be manually manipulated into different
positions, these are usually points, and are usually
in red, and labelled.
Each worksheet contains a short description of what
it explains. Some of the worksheets also contain various
measurements, usually of quadrance between
points, of spreads between lines, or of quadreas
or quadreals of triangles. These measurements
are dynamic, in that they typically change as the contral
points controlling the illustrations are moved. Some
of them don't change, because of a theorem that states
they have some fixed value.
The basic setup showing the null circle, and containing
all the programs.
Applet for Point_line_duality Sows the duality between
a (hyperbolic) point and a (hyperbolic) line.
Applet for Polar_construction Shows how the polar
of a point is constructed using projective constructions
involving the null circle.
Shows a side, namely a set of two points, together with
the corresponding perpendicular point and perpendicular
line, and the opposite points and opposite lines.
Shows a triangle and its dual trilateral, whose points
are the duals of the lines of the triangle, and whose
lines are the duals of the point of the triangle.
Applet for Reflection_in_a_point Shows how to reflect
a point in another point, which is the same as the reflection
in the dual line.
Calculates the three quadrances and three spreads of
a triangle. Valid inside and outside the unit disk.
Illustrates Pythagoras' theorem for a right triangle.
This is a deformation of the Euclidean Pythagoras' theorem,
with an additional quadratic term.
Illustrates the Triple quad formula satisfied by the
three quadrances formed by three collinear points. This
is a deformation of the Euclidean Triple quad formula,
with an additional cubic term. It also agrees with the
Euclidean Triple spread formula.
Illustrates the Triple spread formula satisfied by the
three spreads formed by three concurrent lines. This
is the same as the Euclidean Triple spread formula.
Illustrates Thales' theorem, which has the same form
as in the Euclidean case. Note that only this particular
ratio is independent of the choice of right triangle.
midlines.gsp Shows how to construct the two midpoints
of a side, which are perpendicular, as well as the two
midlines of the side, known in Euclidean geometry as
Gives a construction for the orthocenter and ortholine
of a triangle. These are dual.
A hyperbolic triangle has four circumcenters, and dually
MORE GSP WORKSHEETS WILL BE ADDED...
Built-in GSP programs
Here is a list of the various in-built GSP programs
available on the above worksheets.
Polar Constructs the dual line of
a point, which is the usual polar of that point with
respect to the null circle.
Pole Constructs the dual point of
a line, which is the usual pole of that line with respective
to the null circle.
SideEnds Constructs the perpendicular
point and perpendicular line of a side, along with the
opposite points and opposite lines.
TriangleDual Constructs the dual of
a triangle, namely the trilateral whose lines are the
dual of the points of the triangle, and whose points
are the dual of the lines of the triangle.
Midpoints_of_sides Constructs the
two midpoints of a side, which lie on the common line
and make equal quadrances with the two points.
Midlines_of_sides Constructs the two
midlines of a side. These are called perpendicular bisectors
in Euclidean geometry. They are dual to the midpoints
of the side.
the four circumcenters of a triangle, namely the common
intersections of the midlines, as well as the four circumlines,
which are the common joins of the midpoints. The circumcenters
and circumlines are dual.
Bilines_of_vertex Constructs the two
bilines of a vertex, which pass through the common point
and make equal spreads with the two lines.
Bipoints_of_vertex Constructs the
two bipoints of a side. These are dual to the two bilines
of the vertex.
Incenters_inlines Constructs the four
incenters of a triangle, namely the common intersections
of the bilines, as well as the four inlines, which are
the common joins of the bipoints. The incenters and
inlines are dual.
Raise_perpendicular Constructs a line
through a first point perpendicular to the line joining
the first point to the second point.
Drop_perpendicular Constructs a line
perpendicular to the line formed by two points through
a third point.
a circle with a given center and through a given point.
Quadrance Measures the quadrance between
Spread Measures the spread between
Quadrances_spreads Measures the three
quadrances and the three spreads of a triangle.
Quadrea Measures the quadrea of a
Quadreal Measures the quadreal of
Circle_center_tangent Construct the
circle with a given center and a given tangent.
Circumcircles Contructs the four circumcircles
through three given points. These generally look like
ellipses or hyperbolas in the BJ model, differing considerably
depending on whether the center is inside or outside
the null circle.
Reflection_point Consturcts a center
of reflection and then reflects a point about that center.
Altitudes_Orthocenter Constructs the
altitudes and the orthocenter of a triangle.