Spherical Coordinates for the HP-41

Overview
 

 1°)  3-Dimensional Space

  a)  For the Physicists & Astronomers  ( Focal & M-Code Routines )
  b)  For the Mathematicians

 2°)  n-Dimensional Space

  a)  For the Physicists & Astronomers
  b)  For the Mathematicians
 

1°)  3-Dimensional Space
 

     a)  For the Physicists & Astronomers
 

-We use the following formulae:

       x = r cos b cos L
       y = r cos b sin L
       z = r sin b

   where    x , y , z = rectangular coordinates,    r  = ( x² + y² + z² )^1/2  ,   L  = longitude ,  b = latitude

-However, the results can be obtained more easily by the R-P and P-R functions:
 T-register is saved and no data register is used !
-These programs are also listed in "Transformation of Coordinates"
 
 

01  LBL "R-S"  02  R-P  03  X<>Y  04  RDN  05  R-P  06  R^  07  X<>Y  08  RTN  09  LBL "S-R"  10  X<>Y  11  RDN  12  P-R  13  R^  14  X<>Y  15  P-R  16  END

 
( 30 bytes / SIZE 000 )
 
 

STACK	INPUTS1	OUTPUTS1	INPUTS2	OUTPUTS2
T	T	T	T	T
Z	z	b	b	z
Y	y	L	L	y
X	x	r	r	x

 
Example:     x = 2 ; y = 4 ; z = 7         ( in DEG mode )

     7  ENTER^
     4  ENTER^
     2  XEQ "R-S"     r  =  8.306623863
                  RDN     L  = 63°43494882
                  RDN     b =  57°42644430

-And if you execute "S-R" ( or simply press R/S )  with these numbers in X , Y , Z you'll get  x = 2 , y = 4 , z = 7 again

- "R-S" and "S-R" work in all angular modes
 

  >>>  M-Code Routine
 

-These routines perform the same calculations
-There is no check for alpha data and you'll sometimes get -0 instead of 0 ...
 

093   "S"
02D   "-"
012    "R"
104   CLRF 8
02B   JNC+05
092   "R"
02D   "-"
013   "S"
108   SETF 8
2A0  SETDEC
0B8  C=Y
10E  A=C ALL
10C  ?FSET 8
02B   JNC+05
078   C=Z
0A8  Y=C
0AE  A<>C ALL
068   Z=C
0F8   C=X
128   L=C
070   N=C ALL
351   ?NCXQ
084   21D4
10C  ?FSET 8
1D5   ?NCXQ
079   1E75
10C   ?FSET 8
125   ?NCXQ
074   1D49
0E8   X=C
078   C=Z
0A8   Y=C
0AE   A<>C ALL
0F0   C<>N ALL
068   Z=C
0F8   C=X
070   N=C ALL
351   ?NCXQ
084   21D4
10C  ?FSET 8
1D5   ?NCXQ
079   1E75
10C   ?FSET 8
125   ?NCXQ
074   1D49
0E8   X=C
10C   ?FSET 8
023   JNC+04
0F0   C<>N ALL
0A8   Y=C
3E0   RTN
078   C=Z
0A8  Y=C
0F0   C<>N ALL
068   Z=C
3E0   RTN

( 56 words )
 
 

STACK	INPUTS1	OUTPUTS1	INPUTS2	OUTPUTS2
T	T	T	T	T
Z	z	b	b	z
Y	y	L	L	y
X	x	r	r	x
L	/	x	/	r

 
-The same examples return the same results.
-Here, the original content of register X is saved in L-register.

-It's only slightly faster than the focal programs.
-Even in M-Code, the trigonometric functions remain slow.
 

     b)  For the Mathematicians
 

-Slightly different formulas are employed:

       x = r cos µ
       y = r sin µ cos (phi)
       z = r sin µ sin (phi)

   where    x , y , z = rectangular coordinates,    r  = ( x² + y² + z² )^1/2  as before
 
 
 

01  LBL "R-S2"  02  RDN  03  R-P  04  R^  05  R-P  06  RTN  07  LBL "S-R2"  08  P-R  09  RDN  10  P-R  11  R^  12  END

 
( 28 bytes / SIZE 000 )
 
 

STACK	INPUTS1	OUTPUTS1	INPUTS2	OUTPUTS2
T	T	T	T	T
Z	z	phi	phi	z
Y	y	µ	µ	y
X	x	r	r	x

 
Example:     x = 2 ; y = 4 ; z = 7         ( in DEG mode )

     7  ENTER^
     4  ENTER^
     2  XEQ "R-S2"     r  =  8.306623863
                   RDN      L  = 76°06790845
                   RDN      b =  60°25511870

-And if you execute "S-R2" ( or simply press R/S )  with these numbers in X , Y , Z you'll get  x = 2 , y = 4 , z = 7 again

- "R-S2" and "S-R2" also work in all angular modes.
 

2°)  N-Dimensional Space
 

     a)  For the Physicists & Astronomers
 

-The formulas are generalized as follows:

      x_n  =  r sin µ_n-1
     x_n-1 =  r cos µ_n-1 sin µ_n-2
     x_n-2 =  r cos µ_n-1 cos µ_n-2 sin µ_n-3
     ..........................................................

      x₁ =  r cos µ_n-1 cos µ_n-2 cos µ_n-3  ..............  cos µ₁
 

Data Registers:               R00 = unused                 ( Registers R01 thru Rnn are to be initialized before executing "RN-SN" or "SN-RN" )

                                      •  R01 = x₁                                    •  R01 = r
                                      •  R02 = x₂                                    •  R02 = µ₁
                                      .................                  or               .................

                                      •  Rnn = x_n                                     •  Rnn = µ_n-1

  >>>  When the program stops, the new coordinates have replaced the inputs in R01 thru Rnn

Flags: /
Subroutines:  /
 
 

01  LBL "RN-SN"  02   E3  03  /  04  ISG X  05  ENTER^  06  ISG X  07  RCL 01

08  LBL 01  09  RCL IND Y    10  X<>Y  11  R-P  12  X<>Y  13  STO IND Z  14  RDN

15  ISG Z  16  ISG Y  17  GTO 01  18  STO 01  19  RTN  20  LBL "SN-RN"  21  ENTER^

22  DSE Y  23  RCL 01  24  LBL 02  25  RCL IND Y    26  X<>Y  27  P-R  28  X<>Y

29  STO IND Z    30  RDN  31  DSE Y  32  DSE Z  33  GTO 02  34  STO 01  35  END

 
    ( 67 bytes / SIZE nnn+1 )
 
 

STACK	INPUT1	OUTPUT1	INPUT2	OUTPUT2
X	n > 1	r	n > 1	x1

 
Example:     In a 4-dimensional space,   x = 2 ; y = 4 ; z = 7 ; t = 10        ( in DEG mode )

    2   STO 01
    4   STO 02
    7   STO 03
   10  STO 04

    4   XEQ "RN-SN"   >>>>   13 = R01 = r    and

         R02 = µ₁ = 63°43494882
         R03 = µ₂ = 57°42644430
         R04 = µ₃ = 50°28486277

-Then   4  XEQ "SN-RN"  stores the 4 initial numbers  2  4  7  10  in  R01 to R04  again

-These routines work in all angular mode.
-For n = 10 , execution time is about 8 seconds.
 

     b)  For the Mathematicians
 

-The formulas are generalized differently:

      x₁  =  r cos µ₁
      x₂  =  r sin µ₁ cos µ₂
      x₃  =  r sin µ₁ sin µ₂ cos µ₃
     ..........................................................

      x_n =  r sin µ₁ sin µ₂ sin µ₃  ..............  sin µ_n-1
 

Data Registers:               R00 = unused                 ( Registers R01 thru Rnn are to be initialized before executing "RN-SN2" or "SN-RN2" )

                                      •  R01 = x₁                                    •  R01 = r
                                      •  R02 = x₂                                    •  R02 = µ₁
                                      .................                  or               .................

                                      •  Rnn = x_n                                     •  Rnn = µ_n-1

  >>>  When the program stops, the new coordinates have replaced the inputs in R01 thru Rnn

Flags: /
Subroutines:  /
 
 

01  LBL "RN-SN2"  02  STO Y  03  RCL IND X  04  DSE Y  05  LBL 01  06  RCL IND Y  07  R-P

08  RDN  09  STO IND Z       10  X<> T  11  DSE Z  12  DSE Y  13  GTO 01  14  STO 01

15  RTN  16  LBL "SN-RN2"  17   E3  18  /  19  ISG X  20  ENTER^  21  ISG X

22  RCL 01  23  LBL 02  24  RCL IND Y      25  X<>Y  26  P-R  27  STO IND T  28  RDN

29  ISG Z  30  ISG Y  31  GTO 02  32  STO IND Z       33  END

 
     ( 71 bytes / SIZE nnn+1 )
 
 

STACK	INPUT1	OUTPUT1	INPUT2	OUTPUT2
X	n > 1	r	n > 1	x1

 
Example:     In a 4-dimensional space,   x = 2 ; y = 4 ; z = 7 ; t = 10        ( in DEG mode )

    2   STO 01
    4   STO 02
    7   STO 03
   10  STO 04

    4   XEQ "RN-SN2"   >>>>   13 = R01 = r    and

         R02 = µ₁ = 81°15011690
         R03 = µ₂ = 71°85640421
         R04 = µ₃ = 55°00797980

-Then   4  XEQ "SN-RN2"  stores the 4 initial numbers  2  4  7  10  in  R01 to R04  again

-These routines work in all angular mode.
-For n = 10 , execution time is about 8 seconds.
-The example above actually returns the polar form of the quaternion  q = 2 + 4 i + 7 j + 10 k