Hyperbolic Tetrahedron

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" )

01 LBL "VHOT"
02 DEG
03 STO 04
04 SIN
05 RCL Z
06 STO 02
07 SIN
08 *
09 X^2
10 CHS
11 X<>Y
12 STO 03
13 COS
14 X^2

15 +
16 SQRT
17 RCL 02
18 COS
19 RCL 04
20 COS
21 *
22 /
23 ATAN
24 STO 05
25 RCL 02
26 +
27 XEQ "LOB"
28 STO 01

29 RCL 02
30 RCL 05
31 -
32 XEQ "LOB"
33 ST- 01
34 RCL 04
35 RCL 05
36 +
37 XEQ "LOB"
38 ST+ 01
39 RCL 04
40 RCL 05
41 -
42 XEQ "LOB"

43 ST- 01
44 90
45 RCL 03
46 -
47 RCL 05
48 +
49 XEQ "LOB"
50 ST- 01
51 90
52 RCL 03
53 -
54 RCL 05
55 -
56 XEQ "LOB"

57 ST+ 01
58 90
59 RCL 05
60 -
61 XEQ "LOB"
62 ST+ X
63 RCL 01
64 +
65 4
66 /
67 STO 01
68 END

( 115 bytes / SIZE 006 )

STACK	INPUTS	OUTPUTS
Z	A	/
Y	B	/
X	C	Volume

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" )

01 LBL "HTHV"
02 DEG
03 RCL 01
04 RCL 04
05 +
06 STO 07
07 1
08 P-R
09 RCL 02
10 RCL 05
11 +
12 ST+ 07
13 1
14 P-R
15 X<>Y
16 ST+ T
17 RDN
18 +
19 RCL 03
20 RCL 06
21 +
22 ST+ 07
23 1
24 P-R
25 X<>Y
26 ST+ T
27 RDN
28 +
29 RCL 04
30 RCL 05
31 +
32 RCL 06

33 +
34 1
35 P-R
36 X<>Y
37 ST+ T
38 RDN
39 +
40 RCL 02
41 RCL 03
42 +
43 RCL 04
44 +
45 1
46 P-R
47 X<>Y
48 ST+ T
49 RDN
50 +
51 RCL 01
52 RCL 03
53 +
54 RCL 05
55 +
56 1
57 P-R
58 X<>Y
59 ST+ T
60 RDN
61 +
62 RCL 01
63 RCL 02
64 +

65 RCL 06
66 +
67 1
68 P-R
69 X<>Y
70 ST+ T
71 RDN
72 +
73 RCL 07
74 1
75 P-R
76 X<>Y
77 ST+ T
78 RDN
79 +
80 CHS
81 X<>Y
82 R-P
83 X^2
84 STO 07
85 X<>Y
86 STO 08
87 RCL 01
88 SIN
89 RCL 04
90 SIN
91 *
92 RCL 02
93 SIN
94 RCL 05
95 SIN
96 *

97 +
98 RCL 03
99 SIN
100 RCL 06
101 SIN
102 *
103 +
104 ST+ X
105 RCL 07
106 RCL Y
107 X^2
108 -
109 SQRT
110 R-P
111 X<>Y
112 ENTER^
113 CHS
114 RCL 08
115 ST- Z
116 -
117 PI
118 R-D
119 +
120 STO 09
121 X<>Y
122 STO 08
123 CLST
124 STO 07
125 XEQ 01
126 RCL 01
127 RCL 02
128 RCL 04

129 RCL 05
130 XEQ 01
131 RCL 01
132 RCL 03
133 RCL 04
134 RCL 06
135 XEQ 01
136 RCL 02
137 RCL 03
138 RCL 05
139 RCL 06
140 XEQ 01
141 RCL 08
142 X<> 09
143 STO 08
144 GTO 00
145 LBL 01
146 +
147 +
148 +
149 STO 10
150 RCL 08
151 +
152 2
153 /
154 XEQ "LOB"
155 ST+ 07
156 RCL 09
157 RCL 10
158 +
159 2
160 /

161 XEQ "LOB"
162 ST- 07
163 RTN
164 LBL 00
165 PI
166 R-D
167 RCL 01
168 RCL 02
169 RCL 03
170 XEQ 01
171 PI
172 R-D
173 RCL 01
174 RCL 05
175 RCL 06
176 XEQ 01
177 PI
178 R-D
179 RCL 02
180 RCL 04
181 RCL 06
182 XEQ 01
183 PI
184 R-D
185 RCL 03
186 RCL 04
187 RCL 05
188 XEQ 01
189 2
190 ST/ 07
191 RCL 07
192 END

( 242 bytes / SIZE 011 )

STACK	INPUT	OUTPUT
X	/	Volume

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" )

01 LBL "DA-EL"
02 XEQ 01
03 STO 11
04 RCL 01
05 X<> 02
06 STO 01
07 RCL 04
08 X<> 05
09 STO 04
10 XEQ 01
11 STO 12
12 RCL 01
13 X<> 03
14 STO 01
15 RCL 04
16 X<> 06
17 STO 04
18 XEQ 01
19 STO 13
20 RCL 01
21 X<> 05
22 STO 01
23 RCL 02
24 X<> 04
25 STO 02
26 XEQ 01

27 STO 14
28 RCL 01
29 X<> 06
30 STO 01
31 RCL 03
32 X<> 04
33 STO 03
34 XEQ 01
35 STO 15
36 RCL 01
37 X<> 02
38 STO 01
39 RCL 04
40 X<> 05
41 STO 04
42 XEQ 01
43 STO 16
44 RCL 01
45 X<> 06
46 X<> 04
47 X<> 03
48 STO 01
49 RCL 02
50 X<> 05
51 STO 02
52 GTO 02

53 LBL 01
54 RCL 01
55 1
56 P-R
57 STO 00
58 X<>Y
59 X^2
60 STO 07
61 RCL 05
62 COS
63 STO 08
64 ST* Z
65 X^2
66 -
67 RCL 06
68 COS
69 STO 09
70 ST* Z
71 X^2
72 -
73 X<>Y
74 ST+ X
75 -
76 RCL 00
77 RCL 07
78 RCL 02

79 COS
80 STO 07
81 ST* Z
82 X^2
83 -
84 RCL 03
85 COS
86 STO 10
87 ST* Z
88 X^2
89 -
90 X<>Y
91 ST+ X
92 -
93 *
94 RCL 07
95 RCL 08
96 *
97 RCL 09
98 ST* 07
99 RCL 10
100 *
101 +
102 RCL 00
103 RCL 04
104 COS

105 STO 09
106 *
107 -
108 RCL 00
109 *
110 RCL 09
111 +
112 RCL 07
113 +
114 RCL 08
115 RCL 10
116 *
117 +
118 X<>Y
119 SQRT
120 /
121 ACOSH
122 RTN
123 LBL 02
124 RCL 14
125 RCL 13
126 RCL 12
127 RCL 11
128 END

( 174 bytes / SIZE 017 )

STACK	INPUTS	OUTPUTS
T	/	d
Z	/	c
Y	/	b
X	/	a

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 /

01 LBL "EL-DA"
02 XEQ 01
03 STO 11
04 RCL 01
05 X<> 02
06 STO 01
07 RCL 04
08 X<> 05
09 STO 04
10 XEQ 01
11 STO 12
12 RCL 01
13 X<> 03
14 STO 01
15 RCL 04
16 X<> 06
17 STO 04
18 XEQ 01
19 STO 13
20 RCL 01
21 X<> 05
22 STO 01
23 RCL 02
24 X<> 04
25 STO 02
26 XEQ 01

53 LBL 01
54 RCL 01
55 COSH
56 STO 00
57 LASTX
58 SINH
59 X^2
60 STO 07
61 RCL 02
62 COSH
63 STO 08
64 ST* Z
65 X^2
66 +
67 RCL 06
68 COSH
69 STO 09
70 ST* Z
71 X^2
72 +
73 X<>Y
74 ST+ X
75 -
76 RCL 00
77 RCL 07
78 RCL 03

79 COSH
80 STO 07
81 ST* Z
82 X^2
83 +
84 RCL 05
85 COSH
86 STO 10
87 ST* Z
88 X^2
89 +
90 X<>Y
91 ST+ X
92 -
93 *
94 RCL 07
95 RCL 09
96 *
97 RCL 08
98 ST* 07
99 RCL 10
100 *
101 +
102 RCL 00
103 RCL 04
104 COSH

105 STO 08
106 *
107 -
108 RCL 00
109 *
110 RCL 07
111 -
112 RCL 08
113 +
114 RCL 09
115 RCL 10
116 *
117 -
118 X<>Y
119 SQRT
120 /
121 ACOS
122 RTN
123 LBL 02
124 RCL 14
125 RCL 13
126 RCL 12
127 RCL 11
128 END

( 180 bytes / SIZE 017 )

STACK	INPUTS	OUTPUTS
T	/	d
Z	/	c
Y	/	b
X	/	a

---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"

hp41programs

Hyperbolic Tetrahedron for the HP-41