Euclidean geometry In
Euclidean geometry, the
triangle postulate states that the sum of the angles of a triangle is two
right angles. This postulate is equivalent to the
parallel postulate. In the presence of the other axioms of Euclidean geometry, the following statements are equivalent: •
Triangle postulate: The sum of the angles of a triangle is two right angles. •
Playfair's axiom: Given a straight line and a point not on the line, exactly one straight line may be drawn through the point parallel to the given line. •
Proclus' axiom: If a line intersects one of two parallel lines, it must intersect the other also. •
Equidistance postulate: Parallel lines are everywhere equidistant (i.e. the
distance from each point on one line to the other line is always the same.) •
Triangle area property: The
area of a triangle can be as large as we please. •
Three points property: Three points either lie on a line or lie on a
circle. •
Pythagoras' theorem: In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The spherical excess and the
area A of the triangle determine each other via the relation (called '''Girard's theorem'''):E = \frac{A}{r^2}where r is the
radius of the sphere, equal to r = \frac{1}{\sqrt{\kappa}} where \kappa > 0 is the constant curvature. The spherical excess can also be calculated from the three side lengths, the lengths of two sides and their angle, or the length of one side and the two adjacent angles (see
spherical trigonometry). In the limit where the three side lengths tend to 0, the spherical excess also tends to 0: the spherical geometry locally resembles the Euclidean one. More generally, the Euclidean law is recovered as a limit when the area tends to 0 (which does not imply that the side lengths do so). A spherical triangle is determined up to
isometry by E , one side length and one adjacent angle. More precisely, according to
Lexell's theorem, given a spherical segment [A, B] as a fixed side and a number 0^\circ , the set of points C such that the triangle ABC has spherical excess E is a circle through the antipodes A', B' of A and B. Hence, the
level sets of E form a
foliation of the sphere with two singularities A', B' , and the
gradient vector of E is orthogonal to this foliation.
Hyperbolic geometry Hyperbolic geometry breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane), the equidistance postulate (the points on one side of, and equidistant from, a given line do not form a line), and Pythagoras' theorem. A circle cannot have arbitrarily small
curvature, so the three points property also fails. Contrarily to the spherical case, the sum of the angles of a
hyperbolic triangle is less than 180°, and can be arbitrarily close to 0°. Thus one has an
angular defectD = 180^\circ - \text{sum of angles}. As in the spherical case, the angular defect D and the area A determine each other: one hasD = \frac{A}{r^2} where r = \frac{1}{\sqrt{-\kappa}} and \kappa is the constant
curvature. This relation was first proven by
Johann Heinrich Lambert. One sees that all triangles have area bounded by 180^\circ \times r^2 . As in the spherical case, D can be calculated using the three side lengths, the lengths of two sides and their angle, or the length of one side and the two adjacent angles (see
hyperbolic trigonometry). Once again, the Euclidean law is recovered as a limit when the side lengths (or, more generally, the area) tend to 0 . Letting the lengths all tend to infinity, however, causes D to tend to 180°, i.e. the three angles tend to 0°. One can regard this limit as the case of
ideal triangles, joining three
points at infinity by three bi-infinite
geodesics. Their area is the limit value A = 180^\circ \times {r^2}. Lexell's theorem also has a hyperbolic counterpart: instead of circles, the level sets become pairs of curves called
hypercycles, and the foliation is non-singular. == Taxicab geometry ==