C.a.R. > Applications > Non-Euclidean Geometry > Other Geometries
For Version 4.9, I have revised the non-Euclidean geometries. The hyperbolic Poincaré geometry had long been a part of the C.a.R. distribution. Since Eric Hakenholz did some great work to introduce and realize the spherical geometry, I thought I had to include some version of it into the normal distribution of C.a.R. His realization is very nice and includes a 3D view to visualize what is going on. You find it here in this magazine in several articles. Below you see a picture of one of Eric's applets.
However, I wanted to do something more simple, omitting his 3D view, i.e., restrict myself to the right half of the picture above. Moreover, I wanted to build my own macro set for this geometry, so I could gain control over what exactly is happening. Usually, it turns out that C.a.R. has to be modified to make such a large project run as smoothly as possible. That is the advantage I have over the normal user!
Indeed, it turned out that there is a bug in Java causing a speed loss for exactly the things that are needed in this geometry. For Version 4.9, I used a different approach to restricted intersections and arcs. This speeds up many constructions, but especially the non-Euclidean geometries.
I want to explain here, how the geometries are realized in C.a.R. Since my French is not very good, I do not know well, what Eric has said about elliptic geometry in this magazine. I am sure I repeat some of it here. I am asking for your patience.
Spherical geometry on one hemisphere
In spherical geometry, the lines are the great circles. This is natural, since the shortest path from one point to another runs along a great circle.
Obviously, we run into troubles, since great circles intersect in two antipodes. The usual workaround is to identify those antipodes. This is a bit abstract, and it is easier to restrict this geometry to the southern hemisphere excluding the equator. We will assume this restriction in the sequel. If we do this, we get the usual behavior of lines requested by the incidence axiom: There is exactly one line through every two points.
Note that there are no parallel lines, since each two great circles intersect in one point of our geometry. If great circles pass our southern hemisphere they intersect in two antipodes, one of them in the southern hemisphere. This can easily be visualized, once we realize that great circles pass antipodes on the equator.
Distances between points are measured on the ball as usual, as well as angles. We must do this, or we will not get a useful realization of the spherical geometry. Thus circles are the usual circles on a sphere. For the moment, we allow circles of any radius. Note however, that circles with radius larger than half the perimeter of the ball are empty. Circles with radius equal to a quarter of the perimeter are great circles, i.e. lines.
As in Euclidean or Poincaré geometry, we can order the points on a line, keeping in mind that we only look at the part of the great circle in the southern hemisphere. Then it is clear, which point is between which points on one line. Thus we can define line segments. We also have exactly two sides of lines. Points on different sides cannot be connected with a line segment. Moreover, line segments are again the shortest connection between two points in our geometry. A funny fact is that lines are special circles in this geometry. For each line there is a point with the same distance to all points of the line.
However, contrary to the other geometries, we cannot extend a line segment by any given amount. Our lines are of finite size. This is important for the proofs of many results in ordinary geometry. We cannot even lay a segment congruent to another segment on each side of a point on a line. In fact, we can do it only on one side. On the other side, the segment may overshoot the southern hemisphere. Note however, that we can construct a line at any angle on any side of a given line through any vertex on the line.
Clearly, we get all axioms and laws of congruence for a triangle. E.g., two triangles with two congruent sides, and a congruent included angle are congruent. In fact, we get an even stronger result: Two triangles with three congruent angles are congruent. To see this, remember or look up the result that the sum of angles minus π is equal to the area of the triangle on the unit ball. Note, that the sum of angles is always larger than a straight angle, measured as π. This is the reason for the name "elliptic", meaning "with excess".
It is interesting to see what changes when we identify antipodes. Then two points are different, if and only if they are not antipodes of each other.
First of all, line segments are parts of great circles of length less than half the perimeter. The antipodal line segment identifies with the line segment, of course. Each two different points can be connected with a line segment, as usual.
However, lines do no longer have two sides. I.e., since we identify antipodes, each two points not on a great circle can be connected with a line segment not intersecting the given great circle.
Of course, a circle around a point is identified with the circle of same size around its antipode. As long as the circle has a radius less than a quarter of the perimeter, it is different from a line. Moreover, we can easily define an inside and an outside of the circle. Points on different sides cannot be connected with each other by a line segment. If the radius of the circle approaches a quarter of the perimeter, the circle becomes a great circle and looses its outside.
We use the projection from the north pole to a tangent plane at the south pole. This is called a stereographic projection of the ball. It is a fact, that this projection preserves angles. It also maps circles to circles, but not the midpoint to the midpoint. The equator is mapped to a circle around the south pole, the southern hemisphere to its inside. Let us call the image of the equator the basic circle, and its inside, the image of the southern hemisphere, the basic point set.
Here is, what happens, if we project a triangle on the circle contained in the southern hemisphere, and its circumcircle. The basic circle is shown in green. AB and BC are shown as line segments, AC is shown as a line, i.e. a projected great circle.
The great circles, which are our elliptic lines, become circles passing through opposite points on the basic circle. See the "line" AC in the picture above. An exception are great circles through the south pole, which become straight lines. The circles on the ball become ordinary circles, but the midpoints are usually not the ordinary midpoints. An exception are circles around the south pole. In the above picture U is the projection of the center of the circle, but not the center of the projected circle.
Angles are just measured as before. E.g., the images of rectangular circles and lines remain rectangular. Likewise, all other intersection angles are preserved. It is well known that angles in a triangle on the sphere add up to more than 180°. They are measured as angles between tangents to the circles.
Distances are somewhat difficult to measure in the projection. But we only need to be able to transport a circle to some other place keeping its size. Then we can at least compare distances. I will show you later how to do this.
Since we can easily draw and handle circles and lines in C.a.R., we represent the elliptic geometry in this projection. Then we create macros replacing the usual construction tools.
For this article, I want to go into details on the macros of the elliptic geometry. Along the way, I also want to investigate why these constructions are necessarily as they are. I.e., I want to prove that the macros are correct. We use the following assumptions:
Lines are represented as circles passing through opposite points of the basic circle, with the exception of lines through the center of the basic circle, which are actually straight lines.
Circles are represented as circles, but the midpoint is different from the midpoint of the representing circle.
Angles are preserved, e.g., diameters of a circle are perpendicular to the circle itself.
First of all, we have to construct "lines" through two points. We know that our lines are circles passing through opposite points on the basic circle, which is the image of the equator. To construct such a line, look at the following picture.
The given points are A and B. The black circle is the solution of the problem. It passes through opposite points of the basic circle, and through A and B. From the secant theorem, we get r^2=MA*MC. This helps us, to find the point C and thus our black circle. C is the "inverse reflection" of A at the basic circle.
How do we handle circles? Have a look at the following picture.
We are searching for the black circle around the given point A through the given point B. We know that the projection preserves angles. So we construct the line through A and B shown in blue. This line is a diameter of the circle and must be perpendicular to it. Moreover the circle must be symmetric to the line MA, since the line MA is also a diameter of the circle and thus perpendicular to it. With this information, we can construct the midpoint A' and then the black circle.
Next, we want to create a circle with a give size. In C.a.R. this is called a "circle using three points".
Given are the distance AB, and the center C. The red circle is our target and has radius AB. The points A and C define a line, which is shown in blue. Let P be the center of this great circle on one side of the ball. We find P by intersecting perpendicular lines to AC through A and C. AB equals AQ, since it is on the circle around A through B. The other blue circle is the circle around P through Q. Thus PQ equals PQ'. Since PA equals PC it follows that CQ' equals AQ and in turn equals AB, as requested.
In fact, any circle around P through A and C would have done the trick. We took the great circle for convenience. In fact, this construction is like reflecting the circle on the middle perpendicular of AC.
Exercise: It remains to show, how to construct the dotted line perpendicular to AC through A. But it is an easy exercise to find a circle through A intersecting the basic circle in opposite points and going into a predefined direction.
We like to have much more macros. But we stumble into difficulties even for the middle perpendicular of a line segment AB. Trying the ordinary construction with circles around A through B and around B through A fails, if the segment is too long. Then those circled no longer intersect. Thus we take a different approach.
With the above exercise, we can construct four lines each 45° to our given segment AB. The intersections are both on the middle perpendicular. This construction is stable. To see that it is correct, we must use the congruence laws twice. First the triangle above AB and below are congruent. It follows that the triangle left of the red line and right to it are congruent. Thus the red line is the middle perpendicular.
The hyperbolic Poincaré geometry is similar, and yet very different. First of all, we can also view it as a stereographic projection of the southern hemisphere. This time, the lines are not great circles, but circles, passing the equator in perpendicular direction. Circles are again circles, but with different midpoint, and angles are preserved.
The same construction as above now looks like this.
As you see, the lines are now represented by circles perpendicular to the basic circle. Angles in a triangle add to less than 180°.
This time, we do not have self evident congruence laws for our geometry. In fact, even the distances back on the ball are measured in some complicated way. The easiest way to understand this is by complex function theory. We do not go into details, and just assume the usual geometric laws.
Moreover, this geometry is a full geometry with a plane of infinite size. That is the most important difference to the spherical geometry above. In fact, it can be shown that in any geometry obeying the usual laws, angles in a triangle add to less than 180° or to 180° exactly.
The details are very much like at the elliptic geometry.
E.g., let us construct the line through A and B. We are looking for a circle perpendicular to the basic circle through A and B. From the secant tangent theorem, we get r^2=MA*MC. Again, we can construct or compute a third point C on the black circle, and we are able to construct the black circle.
All the other constructions are very similar. The middle perpendicular is easier, since we can now intersect the circles around A through B and around B through A as usual. Thus, copying a circle to another place is easier too.
What we haven't shown is that these constructions do indeed yield a geometry. I.e., the main point is to show the congruence laws. In the case of the elliptic geometry this may be trivial, if we do it on the sphere. But then we are facing the problem to show that our stereographic projection does indeed preserve angles and map circles to circles. In the case of the hyperbolic geometry, it requires some additional work.
What we showed was, that assuming we know the three principles mentioned above, we can construct the geometries using compass and ruler constructions in the plane. This was my intention here.