Most geometries on the plane R2 are non-Euclidean, but first, what is Euclidean geometry? Euclid's The Elements is one of the most famous mathematical texts of all time. Although written around 300 B.C., its content (often in a simplified form) is still taught in every middle school and high school around the world. This is truly a testament to the fact that knowledge is cumulative. What is truly remarkable about Euclid's Elements is the fact that the entire edifice that is now known as Euclidean geometry was built from only five postulates.
1. One may draw a straight line from any point to any other.
2. One may produce a finite straight line continuously in a straight line.
3. One may produce a circle with any center and distance.
4. All right angles are equal to each other.
5. Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
There was much controversy over whether the fifth postulate could be derived from the other four, though now it is known to be independent. One way to prove it is independent is to construct models that satisfy the other four postulates and not the fifth. This is easily accomplished once one is willing to generalize the notion of arc length.
One could perform the same procedure used to calculate arc length in the last section, but with a different definition of ds, and one which contradicts the Pythagorean theorem. Clearly, if we're contradicting the Pythagorean theorem, we are moving outside of the bounds of Euclidean geometry!
The most general second degree polynomial in dx and dy that one could consider is ds2 = A(x,y)dx2 + B(x,y)dxdy + C(x,y)dy2. (5.15)
Euclidean geometry arises from Equation 5.15 through the very special choice A = C = 1, B = 0. Let us now consider a different choice, called the Poincare half-plane, that arises from Equation 5.15 by setting B again to zero, and A = C = y2. This has the counterintuitive property that points on the x-axis (that is, with y = 0) cause the formula for length to give ds = «>, which is not very helpful. Let's restrict the plane to y > 0, which means we're restricting our attention to the upper half-plane. Then, at least, Equation 5.15 with A = C = y2 is well-defined for all points we're considering.
Let y denote a path in the upper half-plane joining two points P, Q. Define the hyperbolic length of this path as follows: approximate the path with line segments as in Figure 5.3. Then, compute the hyperbolic length of each segment by dividing its ordinary length by y 1, where y1 is the y-coordinate of the leftmost point of the segment. Do this with N segments of equal Euclidean length, let N approach infinity, and observe what value the sum of the lengths of the segments seems to be approaching. Call this value LH [y], where the L is for length, and the H for hyperbolic.
Given points P, Q, consider all paths originating from P and ending at Q. It is of interest to know which path y has the smallest value of Lh[y]; such paths are called geodesies. Let's now consider whether or not Euclid's five postulates, reproduced above, hold in this new kind of geometry, assuming that we replace all occurrences of the word "line" with the word "geodesic." For many choices of the functions A, B, C in Equation 5.15, one obtains a plane geometry that will satisfy the first four of Euclid's axioms by designating the geodesies to be lines. By the definition of "geodesic," this automatically satisfies the postulate that any two points must have a line containing them.
Thinking of lines as geodesies, rather than as perfect straight Euclidean lines, is a very natural thing to do. After all, the shortest path along the surface of the Earth is not a straight line; it is rather like traveling along the equator, which is a circle that has the same radius as the radius of the Earth. In other words, in a general curved space, we will define the word line so that the well-worn and time-honored adage "A straight line is the shortest distance between two points" is still true!
With our generalized definition of a line, it is then straightforward to generalize all of the definitions of Euclidean geometry: a line segment is part of a line; a triangle is the region bounded by three line segments, and so on. Angles are defined as usual: When two smooth curves intersect, measure the angle between their tangent lines at the point of intersection.
For the Poincare half-plane model, one can explicitly find all of the geodesics (whereas in many curved spaces such simple explicit solutions for the geodesics are not possible). They come in two types: vertical lines, and Euclidean-circles centered on the x-axis (though remember that the x-axis is not part of the space). Figure 5.4 shows several of the second type of geodesic, which combine to form triangles.
It's easy to see by visual estimation that the angles in AABC from Figure 5.4 do not add up to n radians (or 180°), as they would in Euclidean geometry. As it turns out, somewhat magically, the difference between n and the sum of the angles will equal the area of the triangle, where area is interpreted using the non-Euclidean metric:
For more detail about the Poincare half-plane, and for proofs of all these statements, see Stahl (1993) and Spivak (1980). Once you've understood integration, however, why not try to
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Figure 5.4 Several geodesics in the Poincare half-plane model of hyperbolic geometry combine to form a hyperbolic triangle.
prove some of these properties of the Poincare half-plane model for yourself?
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