Hyperbolic Tetrahedron for the HP-41

Overview
 

 1°)  Volume of Hyperbolic Orthotetrahedra
 2°)  General Case

  a)  Volume of a Hyperbolic Tetrahedron
  b)  Dihedral Angles >>> Edge Lengths
  c)  Edge Lengths >>> Dihedral Angles
 
 

1°)  Volume of Hyperbolic Orthotetrahedra
 

-An hyperbolic Orthotetraedron may be defined by 3 dihedral angles A , B , C   ( the 3 other dihedral angles = 90° )

    provided  A , B , C < 90°  ;  A+B , B+C >= 90°  ;  (Sin A) (Sin C) < Cos B

-The volume is calculated by the formula:

   V = (1/4) [ L(A+d) - L(A-d) + L(C+d) - L(C-d) - L(90°-B+d) + L(90°-B+d) + 2.L(90°-d) ]

   where  d = Arc Tan [ sqrt( cos² B - sin² A  sin² C ) / cos A / cos C ]    and L = Lobachevsky function
 

Data Registers:  R00: temp  R01 = V ,  R02 = A , R03 = B , R04 = C , R05 = d
Flags: /
Subroutine:   "LOB"  ( cf "Lobachevsky Function for the HP-41" )
 
 

 
    ( 115 bytes / SIZE 006 )
 
 

   Where  A , B , C   are expressed in degrees                     ---Execution time = 46s---

Example1:     A = C = 36°  ,   B = 60°

   36  ENTER^
   60  ENTER^
   36  XEQ "VHOT"  >>>>   0.09332553955

-The Seifert-Weber space is obtained by 120 such orthotetrahedra, so multiplying the result by 120 yields.

    V = 11.19906475   ( the last decimal should be a 4 )

Example2:     A = 41°  ,   B = 67°  ,  C = 28°

  41  ENTER^
  67  ENTER^
  28     R/S      >>>>   Volume = 0.03420659993

Note:

-This routine does not check that  A , B , C < 90°  &  A+B , B+C >= 90°
 

2°)  General Case
 

     a)  Volume of a Hyperbolic Tetrahedron
 
 

                           *
                         * *  *
               A    *       *     *    C
                  *          B  *         *
             *                        *            *                                       Here,  A B C D E ( invisible ) F are the 6 dihedral angles ( not the edges lengths )
       *                                  *                   *
                            F     *           *        * D
                                                      *
 

-In the general case, the volume of a hyperbolic tetrahedron is more complicated ( see reference [2] ).
-It is the sum of 16 Lobachevsky functions.

-We assume that the 6 dihedral angles A , B , C , D , E , F are known and expressed in degrees.
-They must be stored in registers R01 to R06.
 

Data Registers:              R00: temp                             ( Registers R01 thru R06 are to be initialized before executing "HTHV" )

                                      •  R01 = A      •  R04 = D         R07 = Volume        R08 to R10: temp
                                      •  R02 = B      •  R05 = E
                                      •  R03 = C      •  R06 = F
Flags: /
Subroutine:  "LOB"  ( cf "Lobachevsky Function for the HP-41" )
 
 

 
     ( 242 bytes / SIZE 011 )
 
 

   Provided  R01 = A , R02 = B , R03 = C , R04 = D , R05 = E , R06 = F , all expressed in degrees

Example1:     A = 71° , B = 59° , C = 61° , D = 37° , E = 84° , F = 87°

   71  STO 01     37  STO 04
   59  STO 02     84  STO 05
   61  STO 03     87  STO 06

   XEQ "HTHV"  >>>>  V = 0.308555268                                               ---Execution time = 115s---

Example2:     A = 72° , B = 60° , C = 60° , D = 36° , E = 90° , F = 90°

   72  STO 01                    36  STO 04
   60  STO 02  STO 03      90  STO 05  STO 06

       R/S     >>>>  V = 0.1866510793

 multiplying by 60 gives again the volume of the Seifert-Weber space:   11.19906476   ( the last decimal should be a 4 )
 

     b)  Dihedral Angles >>> Edge Lengths
 

-Assuming the 6 dihedral angles ( A , B , C , D , E , F )  are known,
 "DA-EL" computes the 6 corresponding edges lengths  ( a , b , c , d , e , f )  and store them into R11 to R16.

Formulae:

  Cosh a = c₃₄ / sqrt ( c₃₃.c₄₄ )   with

    c₃₃ = 1 - 2 Cos A Cos E Cos F - Cos² A - Cos² E - Cos² F
    c₄₄ = 1 - 2 Cos A Cos B Cos C - Cos² A - Cos² B - Cos² C

    c₃₄ = Cos D + Cos A Cos B Cos E + Cos A Cos C Cos F + Cos B Cos F - Cos² A Cos D + Cos C Cos E

-Similar formulae might be used for the other edges but the order of the dihedral angles is changed in R01 to R06
  in order to use always the formulas above.
 

Data Registers:              R00 & R07 to R10: temp                        ( Registers R01 thru R06 are to be initialized before executing "DA-EL" )

                                      •  R01 = A      •  R04 = D                          R11 = a         R14 = d
                                      •  R02 = B      •  R05 = E         give            R12 = b        R15 = e
                                      •  R03 = C      •  R06 = F                           R13 = c        R16 = f
Flags: /
Subroutines:  /

-Line 121  ACOSH  is an M-code routine ( inverse hyperbolic cosine )
 which may be replaced by  XEQ "ACH"   ( cf "Hyperbolic functions for the HP-41" )
 
 

 
   ( 174 bytes / SIZE 017 )
 
 

 
Example:      A = 71° , B = 59° , C = 61° , D = 37° , E = 84° , F = 87°

   71  STO 01     37  STO 04
   59  STO 02     84  STO 05
   61  STO 03     87  STO 06

   XEQ "DA-EL"  >>>>   a = 1.203427992  = R11                            ---Execution time = 36s---
                            RDN    b = 2.439545899 = R12
                            RDN    c = 2.642741752 = R13
                            RDN    d = 3.020541806 = R14

    and   R15 = e = 2.114152483  ,   R16 = f = 1.938907707

Notes:

-Registers R01 to R06 are modified during the calculations, but their original content is restore at the end ( lines 44 to 51 )
-This program works in all angular modes.
 

     c)  Edge Lengths >>> Dihedral Angles
 

-Assuming now that the 6 edge lengths  ( a , b , c , d , e , f )  are known,
 "EL-DA" computes the 6 corresponding dihedral angles ( A , B , C , D , E , F )  and store them into R11 to R16.

Formulae:

  Cos A = c'₁₂ / sqrt ( c'₁₁.c'₂₂ )   with

    c'₁₁ = -1 - 2 Cosh a Cosh b Cosh f + Cosh² a + Cos² b + Cos² f
    c'₂₂ = -1 - 2 Cosh a Cosh c Cosh e + Cosh² A + Cosh² c + Cosh² e

    c'₁₂ = Cosh d + Cosh a Cosh c Cosh f + Cosh a Cosh b Cosh e - Cosh b Cosh c - Cosh² a Cosh d - Cosh e Cosh f

-Similar formulae might be used for the other angles but the order of the edge lengths is changed in R01 to R06
  in order to use always the formulas above.
 

Data Registers:              R00 & R07 to R10: temp                       ( Registers R01 thru R06 are to be initialized before executing "EL-DA" )

                                      •  R01 = a      •  R04 = d                           R11 = A         R14 = D
                                      •  R02 = b      •  R05 = e         give            R12 = B         R15 = E
                                      •  R03 = c      •  R06 = f                            R13 = C         R16 = F
Flags: /
Subroutines:  /

-2 M-Code routines SINH & COSH are used but you can also:

  -add  LBL 00  E^X  1/X  ST+ L  CLX  2  ST/ L  X<> L  RTN   after line 122
  -replace all the  COSH  by  XEQ 00
  -and replace lines 57-58 by  RCL 01  E^X-1  LASTX  CHS  E^X-1  -  2  /
 
 
 

 
   ( 180 bytes / SIZE 017 )
 
 

         ---Execution time = 39s---

Example:

-With the 6 edges lengths found in the previous paragraph, we find the original dihedral angles:

    A = 71° , B = 59° , C = 61° , D = 37° , E = 84° , F = 87°

 with small roundoff-errors in the last decimals.

Notes:

-Registers R01 to R06 are modified during the calculations, but their original content is restore at the end ( lines 44 to 51 )
-This program works in all angular modes.
 

References:

[1]  John G. Ratcliffe - "Foundations of Hyperbolic Manifolds" -  which may be dowloaded freely  here
[2]  D. Derevnin & A. Mednykh - "Volume of Hyperbolic Tetrahedron"
[3]  J. Murakami & A. Ushijima - " A Volume Formula for Hyperbolic Tetrahedra in Terms of Edge Lengths"