Non-Euclidean Geometry - Poincaré Model
Joel Althoff
In 1901, in one of his popular philosophical writings, Henri Poincaré described an imaginary universe occupying the interior of a disc in the Euclidean plane. This disc has come to be know as the Poincaré disc.
In Poincaré's model of hyperbolic geometry, points are represented by the points interior to a Euclidean circle. Lines are represented a little differently though.
There are two ways lines are represented in the Poincaré model. The first is by all open chords that pass through the center of the circle. Hence, all open diameters. The second way to represent lines are by open arcs of circles orthogonal to the Euclidean space. To get a visual explanation, see the figure below.
In the above figure, line l is an open diameter to the circle O. Line m is an open arc orthogonal to the circle O. Meaning, if one were to extend the radius of O and the radius of the circle on which m lies both to one of the points where m intersects O, the two radii would be orthogonal.
Sounds simple enough, right? Next we will talk about how we would go about constructing a line if we are given two points.
As stated earlier, there are two ways to represent lines, called p-lines in the Poincaré model. Similarly, there are more than one way to construct p-lines. This web page will concentrate on the two methods used in the java applets.
Method 1
In this method, let us assume that the two points given happen to fall on an open diameter to the circle. The construction is easy, the p-line is simply the open diameter incident with both points (and the center of the circle).
Method 2
In this method, let us assume that the two points given (P and Q) are not incident with the same open diameter. The construction is a little more complex. The first step I took was to find the inverse of one of the points. One can do this a number of ways. The method we will use here is as follows:
- Draw the ray, call it R, from the center of circle O, through the point in which we are trying to find the inverse for, call it P.
- Construct a perpendicular to R through P.
- This perpendicular intersects O at two points. Call them T and U.
- Next, draw the radii of O to T and U respectively.
- Construct a perpendicular to the radius incident with T, call it t. This will be tangent to O.
- Similarly, construct a perpendicular to the radius incident with U, call it u. This will also be tangent to O.
- The inverse of P, call it P' is the intersection of t and u.
A diagram of the above steps is featured below.
Now that we have P', we have almost all we need. The circle containing the p-line incident with Q and P is also incident with P'. Three noncollinear points form a circle. So, all that is left is for us to form a circle using points P,Q, and P'.
To draw the circle, we perform the following steps:
- Connect P and Q with a line segment, call it pq.
- Connect P and P' with a line segment, call it pp'.
- Construct the perpendicular bisector to each line.
- Where the two perpendicular bisectors intersect is the center of the circle incident with P,P',and Q.
- Draw a circle, call it C with center at the intersection of the two perpendicular bisectors and a radius equal the distance between that center and P.
- The part of C inside O is the p-line incident with Q and P.
A diagram of the above steps is featured below.
Pictured below is the applet that demonstrates the two types of p-lines. The applet is interactive. Feel free to move around the points inside the disc. To access the applet, click HERE or on the picture below.
Pictured below is the applet that demonstrates the construction described in the above mentioned method 2. You can access the applet by clicking HERE or on the picture below.
Most of the information I obtained about p-lines came from Marvin Greenberg's Euclidean and Non-Euclidean Geometries (3rd ed.).
The Java Applets contained in these pages utilize the JavaSketchPad, which is a product from Key Curriculum Press®