be ultraparallel if they do not intersect and are not limiting parallel. The ultraparallel theorem states that every pair of (distinct) ultraparallel lines...

distance PB and is called the angle of parallelism. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic...

line l. Ultra parallel lines have single common perpendicular (ultraparallel theorem), and diverge on both sides of this common perpendicular. In spherical...

Sylow theorems Transcendence of e and π (as corollaries of Lindemann–Weierstrass) Tychonoff's theorem (to do) Ultrafilter lemma Ultraparallel theorem Urysohn's...

the angle bisector for ᗉ IAI'. Case 2c: IB' is ultraparallel to I'B. Using the ultraparallel theorem, construct the common perpendicular of IB' and I'B...

motions. Such a development enables one to methodically prove the ultraparallel theorem by successive motions. Quite often, it appears there are two or...

Models of the hyperbolic plane Pseudosphere Schwarz–Ahlfors–Pick theorem Ultraparallel theorem Notes "Distance formula for points in the Poincare half plane...

For some hyperbolic motions in the half-plane see the Ultraparallel theorem....

Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem. Later in the...

because in hyperbolic geometry the second definition holds only for ultraparallel lines. From the beginning, the postulate came under attack as being...

intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry, the lines converge toward each other and intersect...

theorem from his work Geometrical Investigations on the Theory of Parallels. Lobachevsky observes, using a combination of his 16th and 23rd theorems,...

do not meet ℓ {\displaystyle \ell } are called non-intersecting or ultraparallel lines. Since hyperbolic geometry and Euclidean geometry are both built...

straight lines into three distinct classes: incident, asymptotic and ultraparallel. Furthermore, he outlined the fundamental concept of angle of parallelism...