Angle of parallelism

<h2 id="construction">Construction</h2>
<a href="/facts/J%C3%A1nos_Bolyai/JeXBranr">János Bolyai</a> discovered a construction which gives the asymptotic parallel s to a line r passing through a point A not on r.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> Drop a perpendicular from A onto B on r. Choose any point C on r different from B. Erect a perpendicular t to r at C. Drop a perpendicular from A onto D on t. Then length DA is longer than CB, but shorter than CA. Draw a circle around C with radius equal to DA. It will intersect the segment AB at a point E. Then the angle BEC is independent of the length BC, depending only on AB; it is the angle of parallelism. Construct s through A at angle BEC from AB.

sin
        ⁡
        B
        E
        C
        =
        
          
            
              sinh
              ⁡
              
                B
                C
              
            
            
              sinh
              ⁡
              
                C
                E
              
            
          
        
        =
        
          
            
              sinh
              ⁡
              
                B
                C
              
            
            
              sinh
              ⁡
              
                D
                A
              
            
          
        
        =
        
          
            
              sinh
              ⁡
              
                B
                C
              
            
            
              sin
              ⁡
              
                A
                C
                D
              
              sinh
              ⁡
              
                C
                A
              
            
          
        
        =
        
          
            
              sinh
              ⁡
              
                B
                C
              
            
            
              cos
              ⁡
              
                A
                C
                B
              
              sinh
              ⁡
              
                C
                A
              
            
          
        
        =
        
          
            
              sinh
              ⁡
              
                B
                C
              
              tanh
              ⁡
              
                C
                A
              
            
            
              tanh
              ⁡
              
                C
                B
              
              sinh
              ⁡
              
                C
                A
              
            
          
        
        =
        
          
            
              cosh
              ⁡
              
                B
                C
              
            
            
              cosh
              ⁡
              
                C
                A
              
            
          
        
        =
        
          
            
              cosh
              ⁡
              
                B
                C
              
            
            
              cosh
              ⁡
              
                C
                B
              
              cosh
              ⁡
              
                A
                B
              
            
          
        
        =
        
          
            1
            
              cosh
              ⁡
              
                A
                B
              
            
          
        
        
        .
      
    
    {\displaystyle \sin BEC={\frac {\sinh {BC}}{\sinh {CE}}}={\frac {\sinh {BC}}{\sinh {DA}}}={\frac {\sinh {BC}}{\sin {ACD}\sinh {CA}}}={\frac {\sinh {BC}}{\cos {ACB}\sinh {CA}}}={\frac {\sinh {BC}\tanh {CA}}{\tanh {CB}\sinh {CA}}}={\frac {\cosh {BC}}{\cosh {CA}}}={\frac {\cosh {BC}}{\cosh {CB}\cosh {AB}}}={\frac {1}{\cosh {AB}}}\,.}

See <a href="/facts/Hyperbolic_triangle/hTQldn7s">Trigonometry of right triangles</a> for the formulas used here.

<h2 id="history">History</h2>
The angle of parallelism was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien" by <a href="/facts/Nikolai_Lobachevsky/0fe8z6jY">Nikolai Lobachevsky</a>.
This publication became widely known in English after the Texas professor <a href="/facts/G._B._Halsted/40yRg39F">G. B. Halsted</a> produced a translation in 1891. (Geometrical Researches on the Theory of Parallels)
The following passages define this pivotal concept in hyperbolic geometry:

The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism) which we will here designate by Π(p) for AD = p.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>: 13 <a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
<h2 id="demonstration">Demonstration</h2>
 
In the <a href="/facts/Poincar%C3%A9_half-plane_model/4mg3Mj2b">Poincaré half-plane model</a> of the hyperbolic plane (see <a href="/facts/Hyperbolic_motion/3XveHh2l">Hyperbolic motions</a>), one can establish the relation of Φ to a with <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>. Let Q be the semicircle with diameter on the x-axis that passes through the points (1,0) and (0,y), where y > 1. Since Q is tangent to the unit semicircle centered at the origin, the two semicircles represent parallel hyperbolic lines. The y-axis crosses both semicircles, making a right angle with the unit semicircle and a variable angle Φ with Q. The angle at the center of Q subtended by the radius to (0, y) is also Φ because the two angles have sides that are perpendicular, left side to left side, and right side to right side. The semicircle Q has its center at (x, 0), x < 0, so its radius is 1 − x. Thus, the radius squared of Q is

x
          
            2
          
        
        +
        
          y
          
            2
          
        
        =
        (
        1
        −
        x
        
          )
          
            2
          
        
        ,
      
    
    {\displaystyle x^{2}+y^{2}=(1-x)^{2},}

hence

x
        =
        
          
            
              1
              2
            
          
        
        (
        1
        −
        
          y
          
            2
          
        
        )
        .
      
    
    {\displaystyle x={\tfrac {1}{2}}(1-y^{2}).}

The <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> of the <a href="/facts/Poincar%C3%A9_half-plane_model/4mg3Mj2b">Poincaré half-plane model</a> of hyperbolic geometry parametrizes distance on the ray {(0, y) : y > 0 } with <a href="/facts/Logarithmic_measure/VFXU56H9">logarithmic measure</a>. Let the hyperbolic distance from (0, y) to (0, 1) be a, so: log y − log 1 = a, so y = ea where <a href="/facts/E_(mathematical_constant)/coEtSBk8">e</a> is the base of the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a>. Then 
the relation between Φ and a can be deduced from the triangle {(x, 0), (0, 0), (0, y)}, for example:

tan
        ⁡
        ϕ
        =
        
          
            y
            
              −
              x
            
          
        
        =
        
          
            
              2
              y
            
            
              
                y
                
                  2
                
              
              −
              1
            
          
        
        =
        
          
            
              2
              
                e
                
                  a
                
              
            
            
              
                e
                
                  2
                  a
                
              
              −
              1
            
          
        
        =
        
          
            1
            
              sinh
              ⁡
              a
            
          
        
        .
      
    
    {\displaystyle \tan \phi ={\frac {y}{-x}}={\frac {2y}{y^{2}-1}}={\frac {2e^{a}}{e^{2a}-1}}={\frac {1}{\sinh a}}.}

<ul><li><a href="/facts/Marvin_Greenberg/CCNK2m0M">Marvin J. Greenberg</a> (1974) Euclidean and Non-Euclidean Geometries, pp. 211–3, <a href="/facts/W.H._Freeman_%26_Company/sJ3zg32G">W.H. Freeman & Company</a>.</li>
<li><a href="/facts/Robin_Hartshorne/mCBLyIW6">Robin Hartshorne</a> (1997) Companion to Euclid pp. 319, 325, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0821807978.</li>
<li><a href="/facts/Jeremy_Gray/Vbo8J5mk">Jeremy Gray</a> (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, <a href="/facts/Clarendon_Press/CYyTzzWG">Clarendon Press</a>, Oxford (See pages 113 to 118).</li>
<li><a href="/facts/B%C3%A9la_Ker%C3%A9kj%C3%A1rt%C3%B3/ImD3BbEM">Béla Kerékjártó</a> (1966) Les Fondements de la Géométry, Tome Deux, §97.6 Angle de parallélisme de la géométry hyperbolique, pp. 411,2, Akademiai Kiado, Budapest.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Non-Euclidean Geometry" by Roberto Bonola, page 104, Dover Publications. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Nikolai Lobachevsky (1840) G. B. Halsted translator (1891) Geometrical Researches on the Theory of Parallels <a href="/wiki/Nikolai_Lobachevsky" target="_blank">/wiki/Nikolai_Lobachevsky</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Bonola, Roberto (1955). Non-Euclidean geometry : a critical and historical study of its developments (Unabridged and unaltered republ. of the 1. English translation 1912. ed.). New York, NY: Dover. ISBN 0-486-60027-0. <a href="0-486-60027-0" target="_blank">0-486-60027-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Angle of parallelism open-in-new

Angle of parallelism