The Museum of HP Calculators
The Museum of HP Calculators
This program is Copyright © 2004 by Jean-Marc Baillard and is used here by permission.
This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.
-These programs deal with the most usual geometric transformations Tf,
essentially projections, symmetries, rotations ( and a few similarities
).
-In the first 4 paragraphs, we compute the coordinates of M' = Tf(M)
assuming the coordinates of M ( and Tf itself ) are known:
1°) Orthogonal Projection and Symmetry ( Reflexion )
a) 2-dimensional spaces: Orthogonal
Projection and Symmetry / Line
b) 3-dimensional spaces
1- Orthogonal Projection and
Symmetry / Plane
2- Orthogonal Projection /
Line
c) n-dimensional spaces
1- Orthogonal Projection and
Symmetry / Hyperplane
2- Orthogonal Projection and
Symmetry / Line
2°) Projection and Symmetry
a) 2-dimensional spaces: Projection
and Symmetry / Line // Vector
b) 3-dimensional spaces
1- Projection and Symmetry
/ Plane // Vector
2- Projection / Line // Vectorial
Plane
c) n-dimensional spaces
1- Projection and Symmetry
/ Hyperplane // Vector
2- Projection and Symmetry
/ Line // Vectorial Hyperplane
3°) Rotations
a) 2-dimensional spaces
b) 3-dimensional spaces
c) n-dimensional spaces
4°) Similarity ( Only 3 examples )
a) 2-dimensional spaces
b) 3-dimensional spaces
c) n-dimensional spaces
-The other programs deal with the link between Matrices and Transformations.
5°) Transformation >>>> Matrix
a) 2-dimensional spaces
b) 3-dimensional spaces
c) n-dimensional spaces
6°) Product ( Composition ) of 2 Transformations
7°) Reciprocal of a Transformation
8°) A Few Tests ( up to 17x18 Matrices )
a) Projection ? Symmetry ?
b) Isometry ?
c) Similarity ?
9°) Matrix >>>> Transformation
a) 2-dimensional spaces
b) 3-dimensional spaces
c) n-dimensional spaces ( One example
only )
Remarks: -Some of these programs use
synthetic registers M N O P Q a
( do not confuse O and Q in the listings )
-Don't stop a running program if registers P and Q are used ( it could
alter their contents - without any "crash" however )
-When register "a" is used, the program must not be called as more than
a first level subroutine ( however, an alternative is suggested in this
case )
-If you're not familiar to synthetic programming, these 6 registers may
be replaced by any unused standard registers,
( for example: R39
R40 R41 R42 R43 R44 which are unused in all the
following examples )
-The basis is assumed to be orthonormal.
-In the "rotation" programs, the basis must be direct orthonormal,
and the sign of the rotation angle is determined by the right-hand
rule.
1°) Orthogonal Projection and Symmetry
a-1) 2-Dimensional Space ( Program#1 )
M(x;y)
|
|
------------|M'(x';y')-----------------
( L ) : y = mx + p
|
|
M"(x";y")
-Given m , p and M(x,y) , the following program computes
M'(x',y') and M"(x",y") where ( MM' ) is perpendicular
to ( L ) and M' is the midpoint of [MM"]
Data Registers: /
Flags: /
Subroutines: /
01 LBL "OPS2A"
02 STO M
03 RDN
04 STO N
05 RDN
06 ST- T
07 RDN
08 ST* Z
09 RDN
10 +
11 R^
12 X^2
13 SIGN
14 ST+ L
15 X<> L
16 /
17 STO Z
18 R^
19 *
20 +
21 ENTER^
22 ST+ Y
23 X<> N
24 -
25 R^
26 ST+ X
27 RCL M
28 -
29 RCL N
30 R^
31 CLA
32 END
( 54 bytes / SIZE 000 )
| STACK | INPUTS | OUTPUTS |
| T | m | y" |
| Z | p | x" |
| Y | y | y' |
| X | x | x' |
| L | / | x |
Example: Line (L) is defined by
y = 3x - 7 and M(1;2) ; find M' and M"
3 ENTER^
-7 ENTER^
2 ENTER^
1 XEQ "OPS2A" >>>> RDN 2.8
RDN 1.4 RDN 4.6 RDN 0.8
whence M'(2.8;1.4) and M"(4.6;0.8)
Note: If (L) // (Oy) , the program
doesn't work but you can use "OPS2" below
In this case however, the formulas are trivial.
a-2) 2-Dimensional
Space ( Program#2 )
M(x;y)
|
|
------------|M'(x';y')-----------------
( L ) : ax + by = c
|
|
M"(x";y")
Data Registers: • R00 =
c • R01 = a
• R02 = b ( these 3 registers are to be initialized before executing
"PS2" )
Flags: F01 CF01 = projection
SF01 = reflexion
Subroutines: /
01 LBL "OPS2"
02 STO Z
03 RCL 01
04 *
05 X<>Y
06 RCL 02
07 X<>Y
08 ST* Y
09 RDN
10 +
11 RCL 00
12 X<>Y
13 -
14 FS? 01
15 ST+ X
16 RCL 01
17 X^2
18 SIGN
19 CLX
20 RCL 02
21 ST* X
22 ST+ L
23 X<> L
24 /
25 RCL 02
26 X<>Y
27 *
28 ST+ Z
29 X<> L
30 RCL 01
31 *
32 X<>Y
33 +
34 END
( 52 bytes / SIZE 003 )
| STACK | INPUTS | OUTPUTS |
| Y | y | y' or y" |
| X | x | x' or x" |
| L | / | x |
(x';y') if Flag F01 is clear
(x";y") if Flag F01 is set
Example: (L): 3x + 4y = 9 M(1;2) Find M' and M"
9 STO 00 3 STO 01 4 STO 02
CF 01 2 ENTER^ 1 XEQ
"OPS2" >>>> 0.76 X<>Y
1.68 whence M'(0.76;1.68)
SF 01 2 ENTER^ 1
R/S >>>> 0.52 X<>Y
1.36 whence M"(0.52;1.36)
b) 3-Dimensional Space
b-1) Orthogonal Projection ( and Symmetry ) onto ( with respect to ) a Plane.
M(x;y;z)
|
|
/ ----------|----------------------------- /
/
| M'(x';y';z')
/
/
/
/-P----------------------------------------/
The plane (P) is defined by its equation ax + by + cz = d
|
( MM' ) is perpendicular to ( P ) and M' is the midpoint of [MM"]
( M' belongs to (P) )
|
M"(x";y";z")
Data Registers: • R00 =
d • R01 = a
• R02 = b • R03 = c ( these registers are
to be initialized before executing "OPS3" )
Flags: F01 CF01 for the projection
SF01 for the reflexion
Subroutines: /
01 LBL "OPS3"
02 STO M
03 RCL 01
04 *
05 X<>Y
06 STO N
07 RCL 02
08 *
09 +
10 X<>Y
11 STO O
12 RCL 03
13 *
14 +
15 RCL 00
16 X<>Y
17 -
18 FS? 01
19 ST+ X
20 RCL 01
21 X^2
22 RCL 02
23 X^2
24 RCL 03
25 X^2
26 +
27 +
28 /
29 RCL 03
30 RCL 02
31 RCL 01
32 R^
33 ST* Z
34 ST* T
35 *
36 RCL O
37 ST+ T
38 X<> N
39 ST+ Z
40 X<> M
41 +
42 CLA
43 END
( 64 bytes / SIZE 004 )
| STACK | INPUTS | OUTPUTS |
| Z | z | z' or z" |
| Y | y | y' or y" |
| X | x | x' or x" |
| L | / | x |
(x';y';z') if Flag F01 is clear
(x";y";z") if Flag F01 is set
Example: (P): 3x + 4y + 5z = 1 M(2;6;7) Find M' and M"
1 STO 00 3 STO 01 4 STO 02 5 STO 03
CF 01
7 ENTER^
6 ENTER^
2 XEQ "OPS3" >>>> -1.84
RDN 0.88 RDN 0.6
whence M'(-1.84;0.88;0.6)
SF 01
7 ENTER^
6 ENTER^
2 R/S
>>>> -5.68 RDN -4.24
RDN -5.8 whence M"(-5.68;-4.24;-5.8)
b-2)
Orthogonal Projection onto a Line.
-Line (L) is now defined by one of its points, say A(xA;yA;zA) and one vector U(a;b:c)
M(x;y;z)
|
A
|
----|------->U -------|------------
(L)
(MM') perpendicular to (L)
M'(x';y';z')
Data Registers: R00
unused
( registers R01 to R06 are to be initialized before executing "OPL3" )
• R01 = xA • R04 = a
• R02 = yA • R05 = b
• R03 = zA • R06 = c
Flags: /
Subroutines: /
01 LBL "OPL3"
02 RCL 01
03 -
04 RCL 04
05 *
06 X<>Y
07 RCL 02
08 -
09 RCL 05
10 *
11 +
12 X<>Y
13 RCL 03
14 -
15 RCL 06
16 *
17 +
18 RCL 04
19 X^2
20 RCL 05
21 X^2
22 RCL 06
23 X^2
24 +
25 +
26 /
27 RCL 03
28 RCL 06
29 RCL Z
30 *
31 +
32 RCL 05
33 R^
34 *
35 RCL 02
36 +
37 RCL 04
38 R^
39 *
40 RCL 01
41 +
42 END
( 52 bytes / SIZE 007 )
| STACK | INPUTS | OUTPUTS |
| Z | z | z' |
| Y | y | y' |
| X | x | x' |
Example: A(2;3;7) U(3;4;5) Find M' if M(9;4;1)
2 STO 01 3 STO
04
3 STO 02 4 STO
05
7 STO 03 5 STO
06
1 ENTER^
4 ENTER^
9 XEQ "OPL3" >>>> 1.7
RDN 2.6 RDN 6.5
Thus, M'(1.7;2.6;6.5)
Note: The axial symmetry is equivalent to
a rotation by 180° around the (L) axis ( cf 3°) b) below
)
c) n-Dimensional Space
c-1) Orthogonal Projection ( and Symmetry ) onto ( with respect to ) an Hyperplane.
M(x1;...;xn)
|
|
/ ----------|------------------------------/
/
| M'(x'1;...;x'n)
/
/
/
/-H ---------------------------------------/
The hyperplane (H) is defined by its equation a1x1
+ ......... + anxn = b
|
( MM' ) is perpendicular to ( P ) and M' is the midpoint of [MM"]
( M' belongs to (P) )
|
M"(x"1;...;x"n)
Data Registers: • R00 = b
( Registers R00 thru Rnn and Rdd thru Rff are to be initialized before
executing "OPSN" )
• R01 = a1 • R02 = a2
.................. • Rnn = an
• Rbb = x1 • Rbb+1 = x2
............ • Ree = xn
( ee - bb = ee' - bb' = n-1 )
and when the program stops: Rbb' = x'1
; Rbb'+1 = x'2 ;............;
Ree' = x'n ( These 2 blocks
of registers can't overlap )
Flags: F01
CF 01 means projection
SF 01 means reflexion.
Subroutines: /
01 LBL "OPSN"
02 STO N
03 X<>Y
04 STO M
05 INT
06 CHS
07 LASTX
08 FRC
09 E3
10 *
11 +
12 1
13 +
14 STO O
15 CLX
16 STO P
17 LASTX
18 RCL 00
19 LBL 01
20 RCL IND Y
21 X^2
22 ST+ P
23 X<> L
24 RCL IND M
25 *
26 -
27 ISG Y
28 CLX
29 ISG M
30 GTO 01
31 FS? 01
32 ST+ X
33 RCL P
34 /
35 RCL O
36 ST- M
37 SIGN
38 LBL 02
39 RCL IND X
40 RCL Z
41 *
42 RCL IND M
43 +
44 STO IND N
45 CLX
46 SIGN
47 ST+ N
48 +
49 ISG M
50 GTO 02
51 RCL M
52 RCL N
53 ENTER^
54 DSE X
55 E3
56 /
57 +
58 RCL O
59 ST- Z
60 -
61 CLA
62 END
( 101 bytes )
| STACK | INPUTS | OUTPUTS |
| Y | bbb.eee(M) | bbb.eee(M) |
| X | bbb' | bbb.eee(M') |
| L | / | n |
Example: (H): x + 2y + 3z + 6t = 10 Find the coordinates of M' and M" if M(4;7;1;9)
10 STO 00
1 STO 01 2 STO 02
3 STO 03 6 STO 04
and if we choose registers R05 thru R08 for M ,
4 STO 05 7 STO 06 1 STO 07 9
STO 08 ( control number = 5.008 )
-If we want to get M' in registers R09 thru R12 and M" in R13 thru
R16
CF 01
5.008 ENTER^
9 XEQ
"OPSN" yields ( in 5 seconds ) 9.012
RCL 09 >>>> 2.7
RCL 10 >>>> 4.4
RCL 11 >>>> -2.9
RCL 12 >>>> 1.2 whence
M'(2.7;4.4;-2.9;1.2)
-Likewise, SF 01 5.008 ENTER^
13 R/S
gives 13.016 and we recall registers
R13 thru R16 to get M"(1.4;1.8;-6.8;-6.6)
c-2)
Orthogonal Projection ( and Symmetry ) onto ( with respect to ) a Line.
-Line (L) is determined by one of its points A(a1;a2;.....;an) and one vector U(u1;u2;....;un)
M(x1;...;xn)
|
|
A
|
----|------->U -------|-M'(x'1;...;x'n)-----------------------------
(L)
(MM") perpendicular to (L) ; MM' = M'M"
|
|
|
M"(x"1;...;x"n)
Data Registers: R00
unused
( Registers R01 thru R2n and Rbb thru Ree are to be initialized before
executing "OPSLN" )
• R01 = a1 • R02 = a2
.................. • Rnn = an
• Rn+1 = u1 • Rn+2 = u2 ................
• R2n = un
• Rbb = x1 • Rbb+1 = x2
............ • Ree = xn
and when the program stops: Rbb' = x'1
; Rbb'+1 = x'2 ;............;
Ree' = x'n ( These last
2 blocks of registers can't overlap )
Flags: F01
CF 01 = projection
SF 01 = reflexion.
Subroutines: /
01 LBL "OPSLN"
02 STO N
03 X<>Y
04 STO M
05 INT
06 CHS
07 LASTX
08 FRC
09 E3
10 *
11 +
12 1
13 +
14 STO O
15 LASTX
16 +
17 STO P
18 LASTX
19 ENTER^
20 CLX
21 STO Q
22 LBL 01
23 RCL IND M
24 RCL IND Z
25 -
26 RCL IND P
27 X^2
28 ST+ Q
29 X<> L
30 *
31 +
32 ISG Y
33 CLX
34 ISG P
35 CLX
36 ISG M
37 GTO 01
38 RCL Q
39 /
40 RCL O
41 ST- P
42 PI
43 INT
44 10^X
45 /
46 ISG X
47 LBL 02
48 RCL IND P
49 RCL Z
50 *
51 RCL IND Y
52 +
53 STO IND N
54 CLX
55 SIGN
56 ST+ N
57 ST+ P
58 RDN
59 ISG X
60 GTO 02
61 FC? 01
62 GTO 04
63 RCL O
64 ST- M
65 ST- N
66 LBL 03
67 RCL IND N
68 ST+ X
69 RCL IND M
70 -
71 STO IND N
72 ISG N
73 CLX
74 ISG M
75 GTO 03
76 LBL 04
77 RCL M
78 RCL N
79 ENTER^
80 DSE X
81 E3
82 /
83 +
84 RCL O
85 ST- Z
86 -
87 CLA
88 END
( 143 bytes )
| STACK | INPUTS | OUTPUTS |
| Y | bbb.eee(M) | bbb.eee(M) |
| X | bbb' | bbb.eee(M') |
| L | / | n |
Example: A(2;4;7;8) U(1;2;3;6) M(4;7;1;9) Find M' and M"
2 STO 01 1 STO 05
and, for instance, 4 STO 09
4 STO 02 2 STO 06
7 STO 10
7 STO 03 3 STO 07
1 STO 11
8 STO 04 6 STO 08
9 STO 12 ( control number
= 9.012 )
CF 01 9.012 ENTER^
13 XEQ "OPSLN" >>>> 13.016
( in 5 seconds ) and recalling registers R13 to R16, we get:
M'(1.92;3.84;6.76;7.52)
SF 01 9.012 ENTER^
13 R/S
>>>> 13.016 ( in 7 seconds ) -------------------------------------------
M"(-0.16;0.68;12.52;6.04)
2°) Projection and Symmetry
a) 2-Dimensional
Space
-The direction of projection and symmetry is now determined by a vector V(a';b')
M(x;y) V
\
\
\
\
------------\ M'(x';y')-----------------
( L ) : ax + by = c
(MM') // V and MM' = M'M"
\
\
M"(x";y")
Data Registers: • R00 =
c • R01 = a
• R02 = b
• R03 = a' • R04 = b' ( these 5 registers
are to be initialized before executing "PS2" )
Flags: F01 CF01 = projection
SF01 = symmetry
Subroutines: /
01 LBL "PS2"
02 STO M
03 RCL 01
04 *
05 X<>Y
06 STO N
07 RCL 02
08 *
09 +
10 RCL 00
11 X<>Y
12 -
13 FS? 01
14 ST+ X
15 RCL 01
16 RCL 03
17 *
18 RCL 02
19 RCL 04
20 *
21 +
22 /
23 RCL 04
24 RCL 03
25 RCL Z
26 ST* Z
27 *
28 RCL N
29 ST+ Z
30 X<> M
31 +
32 CLA
33 END
( 50 bytes / SIZE 005 )
| STACK | INPUTS | OUTPUTS |
| Y | y | y' or y" |
| X | x | x' or x" |
| L | / | x |
(x';y') if Flag F01 is clear
(x";y") if Flag F01 is set
Example: (L): 2x + 3y = 7 ; V(4;-3) ; M(1;2) Find M' and M"
7 STO 00 2 STO 01
3 STO 02
4 STO 03 -3 STO 04
CF 01
2 ENTER^
1 XEQ "PS2" >>>> 5 X<>Y
-1 whence M'(5;-1)
SF 01
2 ENTER^
1 R/S
>>>> 9 X<>Y -4 whence M"(9;-4)
b) 3-Dimensional Space
b-1) Projection ( and Symmetry ) onto ( with respect to ) a Plane.
-The direction of the projection is determined by a vector V(a';b';c')
M(x;y;z) V
\
\
\
\
/ --------\------------------------------- /
/
\ M'(x';y';z')
/
/
/
/-P----------------------------------------/
The plane (P) is defined by its equation ax + by + cz = d
\
( MM' ) // V and MM' = M'M"
( M' belongs to (P) )
\
M"(x";y";z")
Data Registers: • R00 =
d • R01 = a
• R02 = b • R03 = c
• R04 = a' • R05 = b' • R06 = c'
( these 7 registers are to be initialized before executing "PS3" )
Flags: F01 CF01 for the projection
SF01 for the symmetry
Subroutines: /
01 LBL "PS3"
02 STO M
03 RCL 01
04 *
05 X<>Y
06 STO N
07 RCL 02
08 *
09 +
10 X<>Y
11 STO O
12 RCL 03
13 *
14 +
15 RCL 00
16 X<>Y
17 -
18 FS? 01
19 ST+ X
20 RCL 01
21 RCL 04
22 *
23 RCL 02
24 RCL 05
25 *
26 +
27 RCL 03
28 RCL 06
29 *
30 +
31 /
32 RCL 06
33 RCL 05
34 RCL 04
35 R^
36 ST* T
37 ST* Z
38 *
39 RCL O
40 ST+ T
41 X<> N
42 ST+ Z
43 X<> M
44 +
45 CLA
46 END
( 66 bytes / SIZE 007 )
| STACK | INPUTS | OUTPUTS |
| Z | z | z' or z" |
| Y | y | y' or y" |
| X | x | x' or x" |
| L | / | x |
(x';y';z') if Flag F01 is clear
(x";y";z") if Flag F01 is set
Example: (P): 2x + 3y + 7z = 4 ; V(6;1;-2) M(1;4;5) Find M' and M"
4 STO 00 2 STO 01
3 STO 02 7 STO 03
6 STO 04 1 STO 05 -2 STO 06
CF 01
5 ENTER^
4 ENTER^
1 XEQ "PS3" >>>> -269
RDN -41 RDN 95 whence M'(-269;-41;95)
SF 01
5 ENTER^
4 ENTER^
1 R/S
>>>> -539 RDN -86 RDN
185 whence M"(-539;-86;185)
b-2) Projection ( and Symmetry
) onto ( with respect to ) a Line.
-Line (L) is defined by one of its points A(xA;yA;zA)
and a vector U(a;b;c)
-The direction of projection is determined by a ( vectorial ) plane
(P): a'x + b'y + c'z = 0
A(xA;yA;zA)
\
\ U
\
M(x;y;z)-----------M'(x';y';z')---------M"(x";y";z")
\
/---------------------\--------------------/
/
\
/
/
/
/-P----------------------------------------/
( MM' ) // (P) and MM' = M'M"
\
\ (L)
Data Registers: R00 unused
• R01 = xA • R02 = yA • R03 = zA
• R04 = a • R05 = b • R06 = c
• R07 = a' • R08 = b' • R09 = c'
( these 9 registers are to be initialized before executing "PS3" )
Flags: F01 CF01 = projection
SF01 = symmetry
Subroutines: /
01 LBL "PS3L"
02 STO M
03 RCL 01
04 -
05 RCL 07
06 *
07 X<>Y
08 STO N
09 RCL 02
10 -
11 RCL 08
12 *
13 +
14 X<>Y
15 STO O
16 RCL 03
17 -
18 RCL 09
19 *
20 +
21 RCL 04
22 RCL 07
23 *
24 RCL 05
25 RCL 08
26 *
27 +
28 RCL 06
29 RCL 09
30 *
31 +
32 /
33 RCL 03
34 RCL 06
35 RCL Z
36 *
37 +
38 RCL 05
39 R^
40 *
41 RCL 02
42 +
43 RCL 04
44 R^
45 *
46 RCL 01
47 +
48 FC? 01
49 GTO 01
50 2
51 ST* T
52 ST* Z
53 *
54 RCL O
55 ST- T
56 X<> N
57 ST- Z
58 X<> M
59 -
60 LBL 01
61 CLA
62 END
( 84 bytes / SIZE 010 )
| STACK | INPUTS | OUTPUTS |
| Z | z | z' or z" |
| Y | y | y' or y" |
| X | x | x' or x" |
(x';y';z') if Flag F01 is clear
(x";y";z") if Flag F01 is set
Example: A(4;6;-1) U(2;3;7) (P): 6x + y - 2z = 0 ; M(1;4;2) Find M' and M"
4 STO 01 6 STO 02
-1 STO 03
2 STO 04 3 STO 05
7 STO 06
6 STO 07 1 STO 08
-2 STO 09
CF 01
2 ENTER^
4 ENTER^
1 XEQ "PS3L" >>>> -48
RDN -72 RDN -183 whence
M'(-48;-72;-183)
SF 01
2 ENTER^
4 ENTER^
1 R/S
>>>> -97 RDN -148 RDN
-368 whence M"(-97;-148;-368)
c) n-Dimensional Space
c-1) Projection ( and Symmetry ) onto ( with respect to ) an Hyperplane.
-The direction of projection is defined by a vector V(v1;v2;....;vn)
M(x1;...;xn) V
\ \
\ \
/ ---------\-------------------------------/
/
\ M'(x'1;...;x'n)
/
/
/
/-H ---------------------------------------/
The hyperplane (H) is defined by its equation a1x1
+ ......... + anxn = b
\
( MM' ) // V and M' is the midpoint of [MM"]
( M' belongs to (P) )
\
M"(x"1;...;x"n)
Data Registers: • R00 = b
( Registers R00 thru R2n and Rdd thru Rff are to be initialized before
executing "PSN" )
• R01 = a1 • R02 = a2
.................. • Rnn = an
• Rn+1 = v1 • Rn+2 = v2 ................
• R2n = vn
• Rbb = x1 • Rbb+1 = x2
............ • Ree = xn
( ee - bb = ee' - bb' = n-1 )
and when the program stops: Rbb' = x'1
; Rbb'+1 = x'2 ;............;
Ree' = x'n ( These 2 blocks
of registers can't overlap )
Flags: F01
CF 01 = projection
SF 01 = symmetry.
Subroutines: /
01 LBL "PSN"
02 STO N
03 X<>Y
04 STO M
05 INT
06 CHS
07 LASTX
08 FRC
09 E3
10 *
11 +
12 1
13 +
14 STO O
15 ST+ O
16 0
17 LBL 01
18 RCL IND Y
19 RCL IND O
20 *
21 +
22 DSE O
23 DSE Y
24 GTO 01
25 STO P
( synthetic )
26 CLX
27 SIGN
28 RCL 00
29 LBL 02
30 RCL IND Y
31 RCL IND M
32 *
33 -
34 ISG Y
35 CLX
36 ISG M
37 GTO 02
38 FS? 01
39 ST+ X
40 RCL P
41 /
42 RCL O
43 ST- M
44 ISG X
45 LBL 03
46 RCL IND X
47 RCL Z
48 *
49 RCL IND M
50 +
51 STO IND N
52 CLX
53 SIGN
54 ST+ N
55 +
56 ISG M
57 GTO 03
58 RCL M
59 RCL N
60 ENTER^
61 DSE X
62 E3
63 /
64 +
65 RCL O
66 ST- Z
67 -
68 CLA
69 END
( 112 bytes )
| STACK | INPUTS | OUTPUTS |
| Y | bbb.eee(M) | bbb.eee(M) |
| X | bbb' | bbb.eee(M') |
| L | / | n |
Example: (H): x + 2y + 3z + 6t = 10 and V(2;4;7;-5) Find the coordinates of M' and M" if M(4;7;1;9)
10 STO 00 1 STO 01
2 STO 05 and, for
instance: 4 STO 09
2 STO 02
4 STO 06
7 STO 10
3 STO 03
7 STO 07
1 STO 11
6 STO 04
-5 STO 08
9 STO 12 ( control number
= 9.012 )
CF 01 9.012 ENTER^
14 XEQ "PSN" >>>> 14.017
and we recall registers R14 thru R17 to get M'(-126;-253;-454;334)
SF 01 9.012 ENTER^
14 R/S
>>>> 14.017 --------------------------------------------
M"(-256;-513;-909;659)
c-2)
Projection ( and Symmetry ) onto ( with respect to ) a Line.
-Line (L) passes through A(a1;a2;.....;an)
and is parallel to a vector U(u1;u2;....;un)
-The direction of projection is given by a ( vectorial ) hyperplane
(H): a'1x1 + ......... + a'nxn
= 0
A(a1;a2;.....;an)
\
\
\
M(x1;...;xn)-----------M'(x'1;...;x'n)---------M"(x"1;...;x"n)
\
/---------------------\--------------------/
/
\
/
/
/
/-H----------------------------------------/
( MM' ) // (H) and MM' = M'M"
\
\ (L)
Data Registers: R00
unused
( Registers R01 thru R3n and Rbb thru Ree are to be initialized before
executing "PSLN" )
• R01 = a1 • R02 = a2
.................. • Rnn = an
• Rn+1 = u1 • Rn+2 = u2 ................
• R2n = un
• R2n+1 = a'1 • R2n+2 = a'2 .............
• R3n = a'n
• Rbb = x1 • Rbb+1 = x2
............ • Ree = xn
and when the program stops: Rbb' = x'1
; Rbb'+1 = x'2 ;............;
Ree' = x'n ( These last
2 blocks of registers can't overlap )
Flags: F01
CF 01 = projection
SF 01 = symmetry.
Subroutines: /
01 LBL "PSLN"
02 STO N
03 X<>Y
04 STO M
05 INT
06 CHS
07 LASTX
08 FRC
09 E3
10 *
11 +
12 1
13 +
14 ENTER^
15 STO O
16 ST+ X
17 ST+ Y
18 .1
19 %
20 ST+ Z
21 CLX
22 STO Q
23 RDN
24 LBL 01
25 RCL IND Y
26 RCL IND Y
27 *
28 ST+ Q
29 CLX
30 SIGN
31 -
32 DSE Y
33 GTO 01
34 ST+ X
35 STO P
36 SIGN
37 ST+ P
38 ENTER^
39 CLX
40 LBL 02
41 RCL IND M
42 RCL IND Z
43 -
44 RCL IND P
45 *
46 +
47 ISG Y
48 CLX
49 ISG P
50 CLX
51 ISG M
52 GTO 02
53 RCL Q
54 /
55 RCL O
56 ST- P
57 ST- P
58 PI
59 INT
60 10^X
61 /
62 ISG X
63 LBL 03
64 RCL IND P
65 RCL Z
66 *
67 RCL IND Y
68 +
69 STO IND N
70 CLX
71 SIGN
72 ST+ N
73 ST+ P
74 RDN
75 ISG X
76 GTO 03
77 FC? 01
78 GTO 05
79 RCL O
80 ST- M
81 ST- N
82 LBL 04
83 RCL IND N
84 ST+ X
85 RCL IND M
86 -
87 STO IND N
88 ISG N
89 CLX
90 ISG M
91 GTO 04
92 LBL 05
93 RCL M
94 RCL N
95 ENTER^
96 DSE X
97 E3
98 /
99 +
100 RCL O
101 ST- Z
102 -
103 CLA
104 END
( 168 bytes )
| STACK | INPUTS | OUTPUTS |
| Y | bbb.eee(M) | bbb.eee(M) |
| X | bbb' | bbb.eee(M') |
| L | / | n |
Example: A(2;8;1;3) ; U(2;4;7;-5) and (H): x + 2y + 3z + 6t = 0 Find the coordinates of M' and M" if M(4;7;1;9)
2 STO 01
2 STO 05 1
STO 09 and, for instance:
4 STO 13
8 STO
02 4
STO 06 2 STO 10
7 STO 14
1 STO
03 7
STO 07 3 STO 11
1 STO 15
3 STO
04 -5
STO 08 6 STO 12
9 STO 16 ( control number
= 13.016 )
CF 01 13.016 ENTER^
17 XEQ "PSLN" >>>> 17.020
and we recall registers R17 thru R20 to get M'(74;152;253;-177)
SF 01 13.016 ENTER^
17 R/S
>>>> 17.020 --------------------------------------------
M"(144;297;505;-363)
3°) Rotation
a) 2-Dimensional Space
-Given a point A(xA;yA) and
an angle µ , "ROT2" calculates M'(x';y') if we know M(x;y)
-This program works in all angular modes.
M'(x';y')
/
/
/
/ µ
A /-------------- M(x;y)
AM = AM' and ( AM;AM' ) = µ
Data Registers: • R00 =
µ • R01 = xA
• R02 = yA ( these registers are to
be initialized before executing "ROT2" )
Flags: /
Subroutines: /
01 LBL "ROT2"
02 RCL 01
03 -
04 RCL 00
05 RCL 02
06 R^
07 -
08 P-R
09 RCL 00
10 R^
11 P-R
12 X<> Z
13 -
14 RCL 02
15 +
16 X<> Z
17 +
18 RCL 01
19 +
20 END
( 31 bytes / SIZE 003 )
| STACK | INPUTS | OUTPUTS |
| Y | y | y' |
| X | x | x' |
Example: A(6;7) µ = 21° Find M' if M(4;3)
21 STO 00 6 STO 01 7 STO 02 Set the HP-41 in DEG mode
3 ENTER^
4 XEQ "ROT2" >>>> 5.566310945
X<>Y 2.548942395 whence
M'( 5.566310945 ; 2.548942395 )
b) 3-Dimensional Space
-Given M(x;y;z) this program computes M'(x';y';z') = r(M)
where r is the rotation by an angle µ around an
axis (L) passing through a point A(xA;yA;zA)
and parallel to a vector U(a;b;c)
U
|
| A
|
|
| M'(x';y';z')
The vectors NM and NM' are both orthogonal to
U
and lengths NM and NM' are equal.
| /
The angle ( NM ; NM' ) = µ
| / µ
|/-------------- M(x;y;z)
N
Data Registers: • R00 = µ
( registers R00 to R06 are to be initialized before executing "ROT3" )
• R01 = xA • R04 = a
• R02 = yA • R05 = b
R07 to R12: temp
• R03 = zA • R06 = c
Flags: /
Subroutine: "OPL3" ( cf 1°) b-2)
Orthogonal projection onto a line )
01 LBL "ROT3"
02 STO 07
03 RDN
04 STO 08
05 X<>Y
06 STO 09
07 X<>Y
08 R^
09 XEQ "OPL3"
10 STO 10
11 ST- 07
12 RDN
13 STO 11
14 ST- 08
15 X<>Y
16 STO 12
17 ST- 09
18 RCL 07
19 RCL 00
20 COS
21 *
22 ST+ 10
23 RCL 08
24 LASTX
25 *
26 ST+ 11
27 RCL 09
28 LASTX
29 *
30 ST+ 12
31 RCL 00
32 SIN
33 RCL 04
34 X^2
35 RCL 05
36 X^2
37 RCL 06
38 X^2
39 +
40 +
41 SQRT
42 /
43 RCL 05
44 RCL 09
45 *
46 RCL 06
47 RCL 08
48 *
49 -
50 *
51 ST+ 10
52 CLX
53 RCL 06
54 RCL 07
55 *
56 RCL 04
57 RCL 09
58 *
59 -
60 *
61 ST+ 11
62 CLX
63 RCL 04
64 RCL 08
65 *
66 RCL 05
67 RCL 07
68 *
69 -
70 *
71 RCL 12
72 +
73 RCL 11
74 RCL 10
75 END
( 97 bytes / SIZE 013 )
| STACK | INPUTS | OUTPUTS |
| Z | z | z' |
| Y | y | y' |
| X | x | x' |
Example: µ = 24° A(2;3;7) U(4;6;9) Find M' = r(M) if M(1;4;8)
24 STO 00 2 STO 01
4 STO 04
3 STO 02 6 STO 05
7 STO 03 9 STO 06
Set the HP-41 in DEG mode
8 ENTER^
4 ENTER^
1 XEQ "ROT3" >>>> 1.009250426
RDN 3.497956694 RDN 8.330584237
whence M'( 1.009250426 ; 3.497956694
; 8.330584237 )
c) n-Dimensional Space
-Though the word "rotation" has a more general meaning in n-dimensional
spaces, "ROTN" computes M' = r(M)
where r is the rotation by an angle µ in the plane
(P) defined by a point A(a1;...;an) and 2 non-colinear
vectors U(u1;...;un) , V(v1;...;vn)
|
|
| M'(x'1;...;x'n)
The vectors NM and NM' are both parallel to the
plane (A;U;V) and lengths NM and NM' are equal.
| /
The angle ( NM ; NM' ) = µ
| / µ
|/-------------- M(x1;...;xn)
N
|
|
| M'1
| /
| / µ
M1 , M'1 are the orthogonal projections of
M , M' onto (P).
A |/-------------- M1
The vectors U , V , AM1 , AM'1
are in the plane (P)
/ \
The angle ( AM1 ; AM'1 ) = µ
/ \ V
/ U
-The vertical dotted line actually symbolizes the (n-2) dimensional
subspace (S) passing through A and orthogonal to (P).
-r may also be called a rotation by an angle µ "around" the subspace
(S).
Data Registers: • R00 = µ
( Registers R00 thru R3n and Rbb thru Ree are to be initialized before
executing "ROTN" )
• R01 = a1 • R02 = a2
.................. • Rnn = an
• Rn+1 = u1 • Rn+2 = u2 ................
• R2n = un
• R2n+1 = v1 • R2n+2 = v2 ..............
• R3n = vn
• Rbb = x1 • Rbb+1 = x2
............ • Ree = xn
and when the program stops: Rbb' = x'1
; Rbb'+1 = x'2 ;............;
Ree' = x'n ( These last
2 blocks of registers can't overlap )
Flags: /
Subroutines: /
01 LBL "ROTN"
02 X<>Y
03 STO M
04 INT
05 CHS
06 LASTX
07 FRC
08 E3
09 *
10 +
11 STO T
12 +
13 E3
14 /
15 +
16 STO N
17 SIGN
18 +
19 ENTER^
20 ST+ X
21 STO O
22 .1
23 %
24 +
25 +
26 0
27 STO P
( synthetic )
28 LBL 01
29 RCL IND O
30 X^2
31 ST+ P
32 X<> L
33 RCL IND Z
34 *
35 -
36 DSE O
37 DSE Y
38 GTO 01
39 RCL P
40 ST/ Y
41 CLX
42 STO Q
43 SIGN
44 RCL O
45 +
46 STO O
47 LASTX
48 +
49 LBL 02
50 RCL IND O
51 RCL Z
52 *
53 RCL IND Y
54 +
55 STO IND N
56 X^2
57 ST+ Q
58 SIGN
59 ST+ O
60 +
61 ISG N
62 GTO 02
63 DSE X
64 PI
65 INT
66 /
67 ST- N
68 ST- O
69 RCL N
70 LASTX
71 SIGN
72 X<> Q
73 SQRT
74 LBL 03
75 ST/ IND Y
76 ISG Y
77 GTO 03
78 CLST
79 LBL 04
80 RCL IND M
81 RCL IND Q
82 -
83 RCL IND O
84 X<>Y
85 *
86 ST+ Y
87 X<> L
88 RCL IND N
89 *
90 ST+ Z
91 CLX
92 SIGN
93 ST+ M
94 ST+ O
95 ST+ Q
96 RDN
97 ISG N
98 GTO 04
99 RCL P
100 SQRT
101 /
102 RCL Q
103 DSE X
104 ST- M
105 ST- N
106 ST- O
107 CLX
108 RCL 00
109 X<>Y
110 P-R
111 LASTX
112 -
113 RCL 00
114 R^
115 P-R
116 ST- L
117 X<> L
118 ST- T
119 RDN
120 -
121 RCL P
122 SQRT
123 /
124 LBL 05
125 RCL IND O
126 RCL Y
127 *
128 RCL IND N
129 R^
130 ST* Y
131 RDN
132 +
133 RCL IND M
134 +
135 STO IND N
136 CLX
137 SIGN
138 ST+ M
139 ST+ O
140 RDN
141 ISG N
142 GTO 05
143 RCL M
144 RCL N
145 RCL O
146 DSE X
147 2
148 /
149 ST- Z
150 -
151 CLA
152 END
( 235 bytes )
| STACK | INPUTS | OUTPUTS |
| Y | bbb.eee(M) | bbb.eee(M) |
| X | bbb' | bbb.eee(M') |
| L | / | n |
Example: µ = 24° A(1;2;4;7;6) U(3;1;4;2;5) V(2;7;3;1;4) Find M' = r(M) if M(4;6;1;2;3)
24 STO 00 1 STO 01
3 STO 06 2
STO 11 and, for instance:
4 STO 16
2 STO 02
1 STO 07 7
STO 12
6 STO 17
4 STO 03
4 STO 08 3
STO 13
1 STO 18
7 STO 04
2 STO 09 1
STO 14
2 STO 19
6 STO 05
5 STO 10 4
STO 15
3 STO 20 ( control
number = 16.020 )
16.020 ENTER^
21 XEQ "ROTN"
>>>> ( in 15 seconds ) 21.025 and we recall
registers R21 to R25 to get:
M'( 3.515304126 ; 4.098798832 ; 0.262179052
; 1.768429201 ; 2.009053978 )
4°) Examples of a more complex Transformation:
Similarity
-A similarity preserves the ratios of distances:
Tf is a similarity iff , for all points M , N
M'N' = k.MN where M' = Tf(M) ; N' = Tf(N) and k is a constant
= ratio of magnification.
-In the 3 following examples, Tf is the product of an homothecy h (
homothetic center = the origin O ; M1 = h(M)
iff OM1 = k.OM ( vectorial identity
) )
and an isometry Is: Tf = hoIs
a) 2-Dimensional
Space
-Here, k = 3 ( line 09 )
Is = Rot
where R is the reflexion with respect to line (L): ax + by = c
and t is the translation defined by vector U(a';b')
( a translation consists of a constant offset )
Data Registers: • R00 =
c • R01 = a
• R02 = b
• R03 = a' • R04 = b' ( these 5 registers
are to be initialized before executing "PS2" )
Flag: F01
Subroutine: "OPS2"
01 LBL "SIM2"
02 SF 01
03 XEQ "OPS2"
04 RCL 04
05 ST+ Z
06 CLX
07 RCL 03
08 +
09 3
the ratio of magnification may of course be stored into any unused data
register.
10 ST* Z
11 *
12 END
( 29 bytes / SIZE 005 )
| STACK | INPUTS | OUTPUTS |
| Y | y | y' |
| X | x | x' |
Example: (L): 3x + 4y = 9 U(4;-3) Calculate M' if M(6;1)
9 STO 00 3 STO 01 4 STO 02 4 STO 03 -3 STO 04
1 ENTER^
6 XEQ "SIM2" >>>> 20.64
RDN -18.48 whence M'(20.64;-18.48)
b) 3-Dimensional Space
-Now, Tf = hoIs with k = 2 ( line 11 ) , Is = rot
where r is the rotation by an angle µ around an axis
(L) passing through a point A(xA;yA;zA)
and parallel to a vector U(a;b;c)
and t is the translation defined by the same vector U
Data Registers: • R00 = µ
( registers R00 to R06 are to be initialized before executing "ROT3" )
• R01 = xA • R04 = a
• R02 = yA • R05 = b
R07 to R12: temp
• R03 = zA • R06 = c
Flags: /
Subroutine: "ROT3"
01 LBL "SIM3"
02 XEQ "ROT3"
03 RCL 06
04 ST+ T
05 CLX
06 RCL 05
07 ST+ Z
08 CLX
09 RCL 04
10 +
11 2
the ratio of magnification may be stored into any unused data register.
12 ST* T
13 ST* Z
14 *
15 END
( 33 bytes / SIZE 013 )
| STACK | INPUTS | OUTPUTS |
| Z | z | z' |
| Y | y | y' |
| X | x | x' |
Example: µ = 49° A(2,3,4) U(2,4,8) Compute M' if M(1,4,7)
49 STO 00 2 STO 01
2 STO 04
3 STO 02 4 STO 05
4 STO 03 8 STO 06
7 ENTER^
4 ENTER^
1 XEQ "SIM3" >>>> 7.772478150
RDN 13.85810556 RDN 30.62782768
whence M'( 7.772478150 ; 13.85810556 ; 30.62782768
)
c) n-Dimensional Space
-The following program calculates M'(x'1;...;x'n) = Tf ( M(x1;...;xn) ) where Tf = hotoR is the product ( composition ) of 3 transformations:
h is an homothecy ( ratio = k ; homothetic
center = the origin O )
t is a translation defined by a vector V(v1;v2;....;vn)
R is the reflection with respect to an hyperplane
(H): a1x1 + ......... + anxn
= b
Data Registers: • R00 = b
( Registers R00 thru R2n+1 and Rbb thru Ree are to be
initialized before executing "SIMN" )
• R01 = a1 • R02 = a2
.................. • Rnn = an
• Rn+1 = v1 • Rn+2 = v2 ................
• R2n = vn • R2n+1 = k
• Rbb = x1 • Rbb+1 = x2
............ • Ree = xn
and when the program stops: Rbb' = x'1
; Rbb'+1 = x'2 ;............;
Ree' = x'n ( These last
2 blocks of registers can't overlap )
Flag: F01
Subroutine: "OPSN"
01 LBL "SIMN"
02 SF 01
03 XEQ "OPSN"
04 LASTX
05 ISG X
06 CLX
07 STO M
08 LASTX
09 +
10 RDN
11 RCL IND T
12 STO T
13 LBL 01
14 CLX
15 RCL IND M
16 ST+ IND Y
17 R^
18 ST* IND Z
19 RDN
20 ISG M
21 CLX
22 ISG Y
23 GTO 01
24 X<> L
25 -
26 CLA
27 END
( 53 bytes )
| STACK | INPUTS | OUTPUTS |
| Y | bbb.eee(M) | bbb.eee(M) |
| X | bbb' | bbb.eee(M') |
| L | / | n |
Example: (H): x + 2y + 3z + 6t = 10 V(2;5;2;-3) k = 4 Find the coordinates of M' if M(4;7;1;9)
10 STO 00
1 STO 01 2 STO 02
3 STO 03 6 STO 04
2 STO 05 5 STO 06
2 STO 07 -3 STO 08 4 STO 09
For instance, 4 STO 10 7 STO
11 1 STO 12 9 STO 13
( control number = 10.013 )
and if we want to get M' in registers R14 to R17:
10.013 ENTER^
14 XEQ
"SIMN" >>>> 14.017 ( in 6 seconds
) and recalling registers R14 to R17 we find:
M'( 13.6 ; 27.2 ; -19.2 ; -38.4 )
Note: If registers R2n+1 to R3n are already
used by one of the transformations ( for instance: "ROTN" or "PSN" )
you may wish
to store k into register R3n+1 instead of R2n+1.
In this case, simply add
LASTX + after line 09
-The last 3 programs are only 3 examples among numerous other geometric
transformations ...
5°) Transformation >>>> Matrix
a) 2-Dimensional Space
M'(x';y') may be calculated from M(x;y) by means of linear formulas:
x' = ax + by + c
y' = a'x + b'y + c'
-The following program computes and stores these 6 coefficients in registers R05 thru R10
Data Registers: Those used by the transformation itself which are to be initialized before executing "TM2"
and when the program stops: R05 = a ; R07 = b ; R09 =
c
R06 = a' ; R08 = b' ; R10 = c' ( R11: temp
)
Flags: ? ( for example: CF 01 for a projection
, SF 01 for a reflexion ... )
Subroutines: The transformation itself
01 LBL "TM2"
02 ASTO 11
03 CLST
04 SIGN
05 XEQ IND 11
06 STO 05
07 X<>Y
08 STO 06
09 CLST
10 SIGN
11 X<>Y
12 XEQ IND 11
13 STO 07
14 X<>Y
15 STO 08
16 CLST
17 XEQ IND 11
18 STO 09
19 ST- 05
20 ST- 07
21 X<>Y
22 STO 10
23 ST- 06
24 ST- 08
25 5.01002
26 END
( 48 bytes / SIZE 012 )
Example: Find the matrix of the projection onto line (L) 2x + 3y = 7 in the direction of the vector U(4;-3)
1°) We store the elements of the transformation into the proper
registers: 7 STO 00 2 STO 01
3 STO 02 4 STO 03 -3 STO 04
CF 01 ( projection )
2°) We place the name of the corresponding program in the alpha
register: [alpha] "PS2" [alpha]
3°) XEQ "TM2" >>>> 5.01002 = the control
number of the matrix, and we get
[ [
9 12 -28 ]
in registers R05 R07
R09 respectively.
Thus: x' = 9x + 12y -28 defines this projection.
[ -6 -8 21
] ]
R06 R08 R10
y' = -6x -8y + 21
Note: Do not use "OPS2A" which requires different
inputs.
b) 3-Dimensional
Space
M'(x';y';z') may be calculated from M(x;y;z) by means of linear formulas:
x' = ax + by + cz + d
y' = a'x + b'y + c'z + d'
z' = a"x + b"y + c"z + d"
-The following program computes and stores these 12 coefficients in registers R13 thru R24
Data Registers: Those used by the transformation itself which are to be initialized before executing "TM3"
and when the program stops: R13 = a ; R16 = b ; R19 =
c ; R22 = d
R14 = a' ; R17 = b' ; R20 = c' ; R23 = d'
R15 = a" ; R18 = b" ; R21 = c" ; R24 = d" ( R25:
temp )
Flags: ? ( for example: CF 01 for a projection
, SF 01 for a reflexion ... )
Subroutines: The transformation itself
01 LBL "TM3"
02 ASTO 25
03 CLST
04 SIGN
05 XEQ IND 25
06 STO 13
07 RDN
08 STO 14
09 X<>Y
10 STO 15
11 CLST
12 SIGN
13 X<>Y
14 XEQ IND 25
15 STO 16
16 RDN
17 STO 17
18 X<>Y
19 STO 18
20 CLST
21 SIGN
22 X<> Z
23 XEQ IND 25
24 STO 19
25 RDN
26 STO 20
27 X<>Y
28 STO 21
29 CLST
30 XEQ IND 25
31 STO 22
32 ST- 13
33 ST- 16
34 ST- 19
35 RDN
36 STO 23
37 ST- 14
38 ST- 17
39 ST- 20
40 X<>Y
41 STO 24
42 ST- 15
43 ST- 18
44 ST- 21
45 13.02403
46 END
( 85 bytes / SIZE 026 )
Example: Find the matrix defining the symmetry with respect to the plane (P) 2x + 3y + 7z = 4 in the direction of the vector U(-2;4;-1)
1°) 4 STO 00 2 STO 01
3 STO 02 7 STO 03 -2 STO 04
4 STO 05 -1 STO 06 SF 01
( symmetry )
2°) [alpha] "PS3" [alpha]
3°) XEQ "TM3" >>>> 13.02403 = control
number of the matrix and we get:
[ [
9 12 28 -16 ]
R13 R16 R19 R22
x' = 9x + 12y + 28z -16
[ -16 -23 -56 32 ]
in registers R14
R17 R20 R23 respectively,
and y' = -16x -23y -56z + 32
defines this symmetry.
[ 4 6 15
-8 ] ]
R15 R18 R21 R24
z' = 4x + 6y + 15z -8
c) n-Dimensional Space
-M'(x'1,....,x'n) may be computed from M(x1,....,xn) by the formulas:
x'1 = a11.x1 + a12.x2
+ ......... + a1n.xn + b1
x'2 = a21.x1 + a22.x2
+ ......... + a2n.xn + b2
.................................................................
x'n = an1.x1 + an2.x2
+ ......... + ann.xn + bn
-"TMN" calculates and stores these n(n+1) coefficients into registers R4n+3 thru Rn2+5n+2 ( in column order )
Data Registers: R00 thru R3n are used
by the subroutine ( at least most of them , which are to be initialized
before executing "TMN" )
R3n+1 = Tf name R3n+2: temp ;
R3n+3 to R4n+2 contain successively ( 1,0,....,0 ) .......( 0,......,0,1)
( 0,........,0)
and when the program stops:
R4n+3 = a11 ................... Rn2+3n+3 =
a1n Rn2+4n+3 = b1
.........................................................................................
R5n+2 = an1 ................... Rn2+4n+2 =
ann Rn2+5n+2 = bn
Flags: ? ( perhaps SF 01 for
a reflexion , .... )
Subroutine: The transformation itself
( "OPSN" "OPSLN" "PSN" "PSLN" "ROTN" ... )
-If you don't have an HP-41CX,
replace line 51 by the 7 instructions: STO Z
0 LBL 04 STO IND T ISG T
GTO 04 RDN
and line 25 -------------------
STO Z 0 LBL 00 STO IND T
ISG T GTO 00 RDN
01 LBL "TMN"
02 STO Y
03 3
04 *
05 1
06 +
07 ASTO IND X
08 1
09 +
10 RCL Y
11 E3
12 /
13 ISG X
14 STO IND Y
15 X<>Y
16 .1
17 %
18 +
19 +
20 ST+ Y
21 X<>Y
22 INT
23 LBL 01
24 X<>Y
25 CLRGX
26 1
27 -
28 RCL IND X
29 X<>Y
30 ST+ Y
31 1
32 STO IND Z
33 +
34 ISG L
35 CLX
36 ST- L
37 R^
38 RCL IND L
39 RDN
40 XEQ IND T
41 LASTX
42 +
43 INT
44 RCL Y
45 1
46 -
47 RDN
48 ISG IND T
49 GTO 01
50 X<>Y
51 CLRGX
52 STO Z
53 2
54 -
55 RDN
56 RCL IND T
57 RDN
58 XEQ IND T
59 LBL 02
60 RCL X
61 LASTX
62 STO T
63 -
64 LBL 03
65 RCL IND Y
66 ST- IND Y
67 X<> L
68 -
69 DSE Z
70 GTO 03
71 X<>Y
72 ISG X
73 GTO 02
74 ISG Y
75 X<> L
76 E5
77 /
78 +
79 END
( 122 bytes / SIZE n2+5n+3 )
| STACK | INPUTS | OUTPUTS |
| X | n | bbb.eeenn |
where bbb = 4n+3 ; eee = n2+5n+2
; n = the dimension of the space = the number of rows of the matrix.
bbb.eeenn is the control number of the matrix.
Example: Tf is the symmetry with respect to (H): x + 2y + 3z + 6t = 10 parallel to the vector V(2;4;7;-5)
1°) 10 STO 00 1 STO 01
2 STO 02 3 STO 03 6
STO 04 2 STO 05 4 STO 06
7 STO 07 -5 STO 08
( these numbers are used by "PSN"
)
2°) We place the name of the subroutine in the "alpha" register
[alpha] PSN [alpha]
3°) We set flag F01 ( Tf is a symmetry ) SF 01
3°) 4 XEQ "TMN" >>>> 24.04304
( in 36 seconds ) and we get the matrix in registers R24 thru
R43:
-3 -8 -12
-24 40
R24 R28 R32 R36 R40
-8 -15 -24 -48
80
in registers R25
R29 R33 R37 R41
respectively.
-14 -28 -41 -84
140
R26 R30 R34 R38 R42
10 20 30
61 -100
R27 R31 R35 R39 R43
x' = -3x - 8y - 12z - 24t + 40
-Thus, this symmetry may be defined by:
y' = -8x - 15y - 24z - 48t + 80
z' = -14x - 28y - 41z - 84t + 140
t' = 10x + 20y + 30z + 61t - 100
Note: "TMN" may of course be used with other
subroutines than those listed above ( "OPSN" "PSN" ...etc... )
but they must do the same modifications in the stack, i-e:
| STACK | INPUTS | OUTPUTS |
| Y | bbb.eee(M) | bbb.eee(M) |
| X | bbb' | bbb.eee(M') |
| L | / | n |
-However, it doesn't matter if they modify registers R3n+3 to R4n+2
-If you need, say 1, extra register: R3n+1,
( for instance to store a ratio of magnification ), simply replace
line 05 by 2
----------------- 2 -------------s: R3n+1 and R3n+2 , replace
line 05 by 3
-If you need n extra registers: R3n+1 to R4n ( for instance to
store the coordinates of a vector defining a translation ), replace line
03 by 4
------------2n -------------: R3n+1 to R5n, replace line 03 by
5 ... and so on ...
6°) Product ( Composition ) of 2 Transformations
-If 2 mappings Tf1 and Tf2 are defined
by their matrices, the following program computes the matrix of the product
Tf1oTf2
-The coefficients of the 1st matrix are to be stored into consecutive
registers in column order and likewise for the
2nd matrix.
-These 2 arrays are represented by control numbers of the form:
bbb.eeenn1 and bbb.eeenn2
where bbb is the first register , eee is the last register
of the block and nn = the number of rows
( all bbb > 0 ; eee - bbb + 1 = n(n+1) )
-You also have to specify the first register of the product:
Rbbb3
Data Registers: R00: temp
and the 3n(n+1) registers of the 3 matrices.
Flags: F10
Subroutines: /
-If you don't want to use register a, replace lines 56 and 40 with the 4 lines RCL M FRC ISG X INT and delete line 14
01 LBL "COMP"
02 STO 00
03 STO O
04 RDN
05 STO N
06 SF 10
07 ISG X
08 CF 10
09 X<>Y
10 STO M
11 FRC
12 ISG X
13 INT
14 STO a
15 .1
16 %
17 ST- M
18 X<>Y
19 STO Q
20 DSE 00
21 LBL 01
22 CLX
23 STO P
( synthetic )
24 RCL M
25 RCL N
26 LBL 02
27 RCL IND Y
28 RCL IND Y
29 *
30 ST+ P
31 CLX
32 SIGN
33 +
34 ISG Y
35 GTO 02
36 SIGN
37 ST+ 00
38 ST+ M
39 CLX
40 RCL a
41 X^2
42 RCL 00
43 RCL O
44 -
45 X<Y?
46 FS? 10
47 X<0?
48 GTO 03
49 RCL IND T
50 ST+ P
51 LBL 03
52 X<> P
53 STO IND 00
54 DSE Q
55 GTO 01
56 RCL a
57 STO Q
58 ST- M
59 ISG N
60 GTO 01
61 1
62 %
63 RCL 00
64 +
65 E3
66 /
67 RCL O
68 +
69 CF 10
70 CLA
71 END
( 120 bytes )
| STACK | INPUTS | OUTPUTS |
| Z | bbb.eeenn1 | / |
| Y | bbb.eeenn2 | / |
| X | bbb3 | bbb.eeenn3 |
Example1: Find the matrix of the composition
of s1 defined by x' = 0.8 x - 0.6
y + 0.4 and s2 defined
by x' = -0.8 x - 0.6 y - 1.2
y' = -0.6 x - 0.8 y + 1.2
y' = -0.6 x + 0.8 y - .4
-Let's store these 12 coefficients, for instance:
[ [ 0.8 -0.6 0.4 ]
into R01 R03 R05
and [ [ -0.8 -0.6
-1.2 ] into R07 R09
R11 respectively.
[ -0.6 -0.8 1.2 ] ]
R02 R04 R06
[ -0.6 0.8 -0.4
] ] R08 R10
R12
( control number = 1.00602 )
( control number = 7.01202 )
-Then, 1.00602 ENTER^
7.01202 ENTER^
13 XEQ "COMP" >>>> 13.01803
( in 9 seconds ) and we get the coefficients of s1os2
in registers R13 to R18
[ [ -0.28 -0.96 -0.32 ]
in registers R13 R15 R17
respectively
[ 0.96 -0.28 2.24 ]
]
R14 R16 R18
-Actually, s1 is the reflexion with respect
to (L): x + 3y = 2 and s2 is the reflexion
with respect to (L'): 3x + y = -2
-Therefore, s1os2 is the rotation r by
the angle 2µ around the point A(-1;1)
where µ = 53.1301024° = the angle between (L)
and (L') and A = the point of intersection of (L) and (L') as states
a well known theorem.
Example2: This program may also be used to compute the coordinates of M' = Tf (M)
-For example, if M(2;5) and r is the rotation above ( its coefficients in registers R13 thru R18 ) Let's store 2 and 5 into registers R21 R22 respectively and:
13.01802 ENTER^
21.02202 ENTER^
25
XEQ "COMP" >>>> 25.02602 and we find
in registers R25 and R26 M'( -5.68 ; 2.76
)
-Remark that the control number of M must be 21.02202 and not 21.022 ( bbb.eeenn and not bbb.eee )
-In general, this product is not commutative.
7°) Reciprocal of a Transformation
- Tf is defined by the formulas:
x'1 = a11.x1 + a12.x2
+ ......... + a1n.xn + b1
x'2 = a21.x1 + a22.x2
+ ......... + a2n.xn + b2
.................................................................
x'n = an1.x1 + an2.x2
+ ......... + ann.xn + bn
-"INVTF" computes a'i,j and b'i with
x1 = a'11.x'1 + a'12.x'2
+ ......... + a'1n.x'n + b'1
i-e the coefficients of Tf -1
x2 = a'21.x'1 + a'22.x'2
+ ......... + a'2n.x'n + b'2
.................................................................
xn = a'n1.x'1 + a'n2.x'2
+ ......... + a'nn.x'n + b'n
Data Registers:
R00: temp ( Registers R01 thru Rn2+n
are to be initialized before executing "INVTF" )
• R01 = a1,1 • Rn+1
= a1,2 .......................................
• Rn2-n+1 = a1,n • Rn2+1
= b1
• R02 = a2,1 • Rn+2
= a2,2 .......................................
• Rn2-n+2 = a2,n • Rn2+2
= b2
.....................................................................................................................................
• Rnn = an,1 • R2n
= an,2 ........................................
• Rn2 = an,n
• Rn2+n = bn
and when the program stops, R00 = det A and
R01 thru Rn2+n contain the coefficients of the reciprocal
( in column order )
( Rn2+n+1 to R2n2+n are used for
temporary data storage: in fact a'i,j are also stored in these
registers )
Flag: ( F01 )
Subroutine: "LS" ( cf "Linear and
non-linear systems for the HP-41" )
01 LBL "INVTF"
02 ENTER^
03 X^2
04 ENTER^
05 ST+ X
06 LASTX
07 ST+ Z
08 +
09 E3
10 /
11 +
12 ISG X
13 CLRGX
If you don't have an HP-41CX, replace line 13 by 0
LBL 00 STO IND Y ISG Y GTO 00
CLX RCL Z X^2 - ( 9 lines
)
14 X<>Y
15 1
16 +
17 E5
18 /
19 +
20 SIGN
21 LBL 01
22 STO IND L
23 ISG L
24 GTO 01
25 LASTX
26 FRC
27 E-5
28 -
29 1
30 +
31 E-7
32 XEQ "LS"
33 LASTX
34 FRC
35 ENTER^
36 ISG X
37 INT
38 STO Z
39 X^2
40 X<>Y
41 E3
42 *
43 INT
44 LBL 02
45 RCL IND X
46 STO IND Z
47 RDN
48 DSE X
49 DSE Y
50 GTO 02
51 1
52 CHS
53 LBL 03
54 ST* IND Y
55 DSE Y
56 DSE T
57 GTO 03
58 END
( 96 bytes / SIZE 2n2+n+1 )
| STACK | INPUTS | OUTPUTS |
| X | n | -1 |
x' = x + y + z + 4
Example: Tf is defined
by y' = x + 2y +3z + 1
Find the reciprocal transformation.
z' = x + 3y + 6z + 8
1 1 1 4
R01 R04 R07 R10
-Store 1 2
3 1 into registers
R02 R05 R08 R11
respectively
1 3 6 8
R03 R06 R09 R12
3 -3 1 -17
-Then 3 XEQ "INVTF" >>>>
( 28 seconds ) -1 and we get -3
5 -2 23 in the same registers
( and R00 = det A = 1 )
1 -2 1 -10
x = 3x' - 3y + z - 17
-Thus, y = -3x' + 5y' -2z' + 23
define the reciprocal.
z = x' - 2y' + z' - 10
Notes: -Meaningless results will be
obtained if the matrix A = [ ai,j ] has no inverse (
check that det A is significantly different from 0 in register R00
)
-We must
have n < 13.
8°) A Few Tests
a) Projection ? Symmetry ?
-Let Tf defined by the formulas:
x'1 = a11.x1 + a12.x2
+ ......... + a1n.xn + b1
x'2 = a21.x1 + a22.x2
+ ......... + a2n.xn + b2
.................................................................
x'n = an1.x1 + an2.x2
+ ......... + ann.xn + bn
-"PS?" tests if this transformation is a projection or a symmetry
Data Registers:
R00: temp ( Registers R01 thru Rn2+n
are to be initialized before executing "PS?" )
• R01 = a1,1 • Rn+1
= a1,2 .......................................
• Rn2-n+1 = a1,n • Rn2+1
= b1
• R02 = a2,1 • Rn+2
= a2,2 .......................................
• Rn2-n+2 = a2,n • Rn2+2
= b2
.....................................................................................................................................
• Rnn = an,1 • R2n
= an,2 ........................................
• Rn2 = an,n
• Rn2+n = bn
Flags: F00 and F03: If
flag F00 is still set when the program stops, Tf is a projection or a constant
mapping.
If flag F03 --------------------------------- , Tf is a symmetry.
Subroutines: /
-If you don't want to use register a, replace lines 78 , 55 and 20 with
the 5 lines RCL M FRC ISG X
CLX INT and delete line 06
01 LBL "PS?"
02 SF 00
03 SF 03
04 ENTER^
05 STO O
06 STO a
07 X^2
08 LASTX
09 E5
10 /
11 +
12 STO M
13 +
14 STO N
15 INT
16 STO 00
17 LBL 01
18 CLX
19 STO P ( synthetic
)
20 RCL a
21 X^2
22 RCL 00
23 X<=Y?
24 GTO 01
25 RCL IND 00
26 STO P ( synthetic
)
27 LBL 01
28 RCL M
29 RCL N
30 LBL 02
31 RCL IND Y
32 RCL IND Y
33 *
34 ST+ P
35 CLX
36 SIGN
37 -
38 DSE Y
39 GTO 02
40 X<> P
41 RCL IND 00
42 RCL Y
43 -
44 ABS
45 E-7
( or another "small" number )
46 X<=Y?
47 CF 00
48 X<> Z
49 RCL 00
50 1
51 ST- M
52 -
53 ENTER^
54 STO 00
55 RCL a
56 ST/ Z
57 MOD
58 X<>Y
59 INT
60 -
61 ABS
62 CHS
63 X=0?
64 SIGN
65 X<0?
66 CLX
67 -
68 ABS
69 E-7
( or another "small" number )
70 X<=Y?
71 CF 03
72 FC? 00
73 FS? 03
74 X<0?
75 GTO 03
76 DSE O
77 GTO 01
78 RCL a
79 STO O
80 ST+ M
81 DSE N
82 GTO 01
83 LBL 03
84 CLA
85 END
( 135 bytes / SIZE n2+n+1 )
| STACK | INPUTS | OUTPUTS |
| X | n | / |
Example1: A mapping is given by
x' = 9x + 12y + 28z -16
y' = -16x -23y -56z +32
z' = 4x + 6y +15z - 8
9 12 28 -16
R01 R04 R07 R10
-We store -16 -23
-56 32 into registers
R02 R05 R08 R11
respectively
4 6 15
-8
R03 R06 R09 R12
-Then 3 XEQ "PS?" >>>> 28 seconds later, flag F00 is clear and flag F03 is set: the transformation is a symmetry but not a projection.
Example2: If x' = 9x + 12y -28
we store 9
12 -28 into
R01 R03 R05 respectively
y' = -6x - 8y + 21
-6 -8 21
R02 R04 R06
and 2 XEQ "PS?" >>>> 13 seconds later, flag F00 is set and flag F03 is clear: the transformation is projection but not a symmetry.
-If Flags F00 and F03 are clear, the mapping is neither a projection
nor a symmetry.
-If both flags are set, the mapping = Identity.
-Actually, this program computes the elements of TfoTf:
If TfoTf = Id we have a symmetry
If TfoTf = Tf we have a projection or a constant.
-This program can test up to a 17x18 matrix.
b) Isometry ?
-Let Tf defined by the formulas:
x'1 = a11.x1 + a12.x2
+ ......... + a1n.xn + b1
x'2 = a21.x1 + a22.x2
+ ......... + a2n.xn + b2
.................................................................
x'n = an1.x1 + an2.x2
+ ......... + ann.xn + bn
-"PS?" tests if this transformation is an isometry ( i-e if it preserves the distances )
Data Registers:
R00: temp ( Registers R01 thru Rn2
are to be initialized before executing "ISO?" ) Registers Rn2+1
to Rn2+n are actually unused.
• R01 = a1,1 • Rn+1
= a1,2 .......................................
• Rn2-n+1 = a1,n • Rn2+1
= b1
• R02 = a2,1 • Rn+2
= a2,2 .......................................
• Rn2-n+2 = a2,n • Rn2+2
= b2
.....................................................................................................................................
• Rnn = an,1 • R2n
= an,2 ........................................
• Rn2 = an,n
• Rn2+n = bn
Flags: F02: If flag F02
is still set when the program stops, Tf is an isometry.
Subroutines: /
-If you don't want to use register a, replace lines 55 and 34 with the 5 lines RCL M FRC ISG X CLX INT and delete line 04
01 LBL "ISO?"
02 SF 02
03 STO O
04 STO a
05 X^2
06 STO 00
07 LASTX
08 E5
09 /
10 +
11 STO M
12 STO N
13 LBL 01
14 CLX
15 STO P
( synthetic )
16 RCL M
17 RCL N
18 LBL 02
19 RCL IND Y
20 RCL IND Y
21 *
22 ST+ P
23 DSE Y
24 RDN
25 DSE Y
26 GTO 02
27 X<> P
28 RCL 00
29 1
30 ST- M
31 -
32 ENTER^
33 STO 00
34 RCL a
35 ST/ Z
36 MOD
37 X<>Y
38 INT
39 -
40 ABS
41 CHS
42 X=0?
43 SIGN
44 X<0?
45 CLX
46 -
47 ABS
48 E-7
49 X<=Y?
50 CF 02
51 X<=Y?
52 GTO 03
53 DSE O
54 GTO 01
55 RCL a
56 STO O
57 ST+ M
58 SIGN
59 ST- N
60 RCL 00
61 X#0?
62 GTO 01
63 LBL 03
64 CLA
65 END
( 104 bytes / SIZE n2+1 )
| STACK | INPUTS | OUTPUTS |
| X | n | / |
Example: A mapping is given by
x' = 0.8.x - 0.6.y +1 we store
0.8 -0.6 1 into
registers R01 R03 R05 ( in fact,
registers R05 & R06 are unused )
y' = 0.6.x + 0.8.y +2
0.6 0.8 2
R02 R04 R06
-Then 2 XEQ "ISO?" >>>> 6 seconds later, Flag F02 is still set: we have an isometry ( actually a rotation )
-This program works up to n = 17.
c) Similarity
?
-Let Tf defined by the formulas:
x'1 = a11.x1 + a12.x2
+ ......... + a1n.xn + b1
x'2 = a21.x1 + a22.x2
+ ......... + a2n.xn + b2
.................................................................
x'n = an1.x1 + an2.x2
+ ......... + ann.xn + bn
-"SIM?" tests if this transformation is a similarity ( i-e if it preserves the ratios of distances )
Data Registers:
R00: temp ( Registers R01 thru Rn2
are to be initialized before executing "SIM?" ) Registers
Rn2+1 to Rn2+n are actually unused.
• R01 = a1,1 • Rn+1
= a1,2 .......................................
• Rn2-n+1 = a1,n • Rn2+1
= b1
• R02 = a2,1 • Rn+2
= a2,2 .......................................
• Rn2-n+2 = a2,n • Rn2+2
= b2
.....................................................................................................................................
• Rnn = an,1 • R2n
= an,2 ........................................
• Rn2 = an,n
• Rn2+n = bn
Flags: F02: If flag F02
is still set when the program stops, Tf is a similarity.
Subroutines: /
-If you don't want to use register a, replace lines 41 and 13 with the
5 lines RCL M FRC ISG X
CLX INT and delete line 04
01 LBL "SIM?"
02 SF 02
03 STO O
04 STO a
05 X^2
06 STO 00
07 LASTX
08 E5
09 /
10 +
11 STO M
12 STO N
13 E99
14 STO P ( synthetic )
15 CHS
16 STO Q
17 LBL 01
18 RCL M
19 RCL N
20 ENTER^
21 CLX
22 LBL 02
23 RCL IND Z
24 SIGN
25 CLX
26 RCL IND Z
27 ST* L
28 X<> L
29 DSE Z
30 +
31 DSE Z
32 GTO 02
33 +
34 RCL 00
35 ENTER^
36 SIGN
37 ST- M
38 -
39 ENTER^
40 STO 00
41 RCL a
42 ST/ Z
43 MOD
44 X<>Y
45 INT
46 X=Y?
47 GTO 03
48 R^
49 ABS
50 PI
( lines 50 to 53 computes a "small" number without any digit entry line:
this would disrupt status register P )
51 CHS
52 10^X
53 X^2
54 X<Y?
55 CF 02
56 X<Y?
57 GTO 05
58 GTO 04
59 LBL 03
60 X<> P
61 R^
62 X>Y?
63 X<>Y
64 STO P ( synthetic
)
65 X<> Q
66 R^
67 X<Y?
68 X<>Y
69 STO Q
70 LBL 04
71 DSE O
72 GTO 01
73 RCL a
74 STO O
75 ST+ M
76 SIGN
77 ST- N
78 RCL 00
79 X#0?
80 GTO 01
81 RCL Q
82 RCL P
83 RCL Y
84 RCL Y
85 -
86 5 E-7
( or another "small" number )
87 X<Y?
88 CF 02
89 R^
90 R^
91 +
92 2
93 /
94 SQRT
95 LBL 05
96 CLA
97 END
( 151 bytes / SIZE n2+1 )
| STACK | INPUTS | OUTPUTS |
| X | n | k |
where k = the ratio of magnification ( if Flag F02 is still set at the end )
Example: Tf is given
by x' = 7.68.x - 6.4.y - 0.24.z
+ 9.28
y' = -4.24.x - 4.8.y - 7.68.z - 3.04
z' = -4.8.x - 6.y + 6.4.z
+ 19.2
7.68 -6.4 -0.24
9.28
R01 R04 R07 R10
-We store -4.24
-4.8 -7.68 -3.04
into registers R02 R05
R08 R11 respectively
( Registers R10 to R12 are actually unused )
-4.8 -6
6.4 19.2
R03 R06 R09 R12
3 XEQ "SIM?" >>>> 10 ( execution
time = 19 seconds ), Flag F02 is still set: Tf is a similarity
and the ratio of magnification k = 10
-This program may also be used to test an isometry by comparing
k with 1 ( if flag F02 is set )
-Up to 17x18 matrices can be dealt with.
9°) Matrix >>>> Transformation
a) 2-Dimensional Space
-"MT2" gives the specifications of a transformation Tf defined by the following formulas
x' = ax + by + c
y' = a'x + b'y + c'
-This program finds the projections, symmetries and isometries.
Data Registers:
• R01 = a • R03 = b • R05 = c
( these 6 registers are to be initialized before executing "MT2" )
• R02 = a' • R04 = b' • R06 = c'
R00 and R07 to R19: temp
when the program stops, R00 thru R04 contain the required data
Flags: F00 F01 F02 F03 F10
Subroutines: PS? ISO?
COMP and LS ( cf "linear and non-linear systems for the
HP-41" )
Note: If you don't have an X-Functions Module,
you may, for instance,
replace
all the REGMOVEs by XEQ 14 and add the 21 following instructions
after line 154
RTN
LBL 14
INT
LASTX
FRC
E3
*
INT
ST- L
E3
ST* L
LBL 15
CLX
RCL IND Z
STO IND Y
ISG Y
CLX
ISG Z
CLX
DSE L
GTO 15
( This alternative is not necessarily the best! )
01 LBL "MT2"
02 14.001006
03 STO 12
04 7.000005
05 STO 13
06 1.014006
07 REGMOVE
08 2
09 XEQ "PS?"
10 FS? 00
11 FC? 03
12 GTO 00
13 "ID"
14 PROMPT
15 LBL 00
16 CF 01
17 2
18 XEQ "ISO?"
19 FS? 02
20 GTO 01
21 FC? 00
22 FS? 03
23 GTO 00
24 "TF NOT FOUND"
25 PROMPT
26 LBL 00
27 XEQ 03
28 XEQ 06
29 RCL 12
30 REGMOVE
31 CLX
32 FS? 03
33 SIGN
34 ST+ 01
35 ST+ 04
36 SIGN
37 CHS
38 XEQ 04
39 RCL 01
40 RCL 04
41 +
42 X=0?
43 1/X
44 RCL 03
45 CHS
46 STO 10
47 1
48 STO 11
49 RCL 04
50 X=0?
51 GTO 00
52 STO 10
53 CLX
54 STO 11
55 LBL 00
56 RCL 13
57 REGMOVE
58 "PROJ"
59 FS? 03
60 "SYM"
61 "~/L//V"
( append / L // V )
62 PROMPT
63 LBL 01
64 XEQ 03
65 RCL 01
66 RCL 04
67 +
68 GTO IND X
69 LBL 00
70 RCL 19
71 STO 02
72 RCL 18
73 STO 01
74 "TRANS"
75 PROMPT
76 LBL 01
77 14.01902
78 STO Y
79 6
80 XEQ "COMP"
81 RCL 12
82 REGMOVE
83 2
84 ST/ 10
85 ST/ 11
86 RCL 10
87 ST- 05
88 RCL 11
89 ST- 06
90 XEQ 03
91 XEQ 06
92 RCL 13
93 REGMOVE
94 "REFL/L"
95 FC? 03
96 "~*TRANS"
( append *TRANS )
97 PROMPT
98 LBL 02
99 RCL 06
100 STO 02
101 RCL 05
102 STO 01
103 RCL 15
104 RCL 14
105 R-P
106 RDN
107 STO 00
108 "ROT"
109 PROMPT
110 LBL 03
111 1
112 CHS
113 ST+ 01
114 ST+ 04
115 LBL 04
116 ST* 05
117 ST* 06
118 1.00602
119 E-7
120 XEQ "LS"
121 6
122 LBL 05
123 RCL IND X
124 ABS
125 E-7
( or another "small" number )
126 X<=Y?
127 GTO 00
128 CLX
129 STO IND Z
130 LBL 00
131 X<> Z
132 DSE X
133 GTO 05
134 RTN
135 LBL 06
136 RCL 05
137 STO 07
138 RCL 01
139 STO 08
140 RCL 03
141 STO 09
142 RCL 04
143 X=0?
144 RTN
145 RCL 06
146 STO 07
147 CLX
148 STO 08
149 SIGN
150 STO 09
151 END
( 314 bytes / SIZE 020 )
Example1: Tf is defined
by x' = 9x + 12y -28
we store 9 12 -28
into registers R01 R03
R05 respectively
y' = -6x - 8y + 21
-6 -8 21
R02 R04 R06
XEQ "MT2" >>>> ( in 35 seconds ) "PROJ/L//V" meaning we have a projection onto a line (L) in the direction of a vector V
with (L): 1 x + 1.5 y = 3.5 the coefficients
1 1.5 3.5 in registers
R01 R02 R00 respectively
and V( -4/3 ; 1 ) in registers R03 R04
respectively
Example2: x' = 17x + 24y - 56
we store 17 24 -56
into R01 R03 R05 respectively
y' = -12x - 17y + 42
-12 -17 42
R02 R04 R06
XEQ "MT2" >>>> ( 35 seconds ) "SYM/L//V" Tf is a symmetry with respect to line (L) in the direction of vector V
with (L): 1 x + 1.5 y = 3.5 the coefficients
1 1.5 3.5 in registers
R01 R02 R00 respectively
and V( -4/3 ; 1 ) in registers R03 R04
respectively
Example3: x' = 0.28x - 0.96y
+ 2.16 these 6 coefficients are stored into
R01 to R06 ( in column order )
y' = -0.96x -0.28y + 2.88
XEQ "MT2" >>>> ( 51 seconds ) "REFL/L" Tf is a reflexion with respect to line (L): 1 x + 4/3 y = 3 ( in registers R01 R02 R00 respectively )
Example4: x' = 0.28x - 0.96y
- 1.84 these 6 coefficients are stored into
R01 to R06 ( in column order )
y' = -0.96x -0.28y + 5.88
XEQ "MT2" >>>> ( 41 seconds ) "REFL/L*TRANS" Tf is the product of a reflexion / a line (L) and a translation defined by U parallel to (L)
(L): 1 x + 4/3 y = 3 ( in R01 R02 R00 ) and U( -4 ; 3 ) in registers R03 R04
Example5: x' = 0.8x - 0.6y
+ 2.2 we store these 6 numbers in R01 to
R06
y' = 0.6x + 0.8y - 0.6
XEQ "MT2" >>>> ( 24 seconds ) "ROT" Tf is a rotation by an angle µ around a point A µ is in R00 and the coordinates of A in R01 and R02
Here, µ = 36.86989765 and A( 2 ; 3 )
Example6: x' = x + 3
we store 1 0 3
into R01 thru R06
y' = y + 7
0 1 7
XEQ "MT2" >>>> ( 17 seconds ) "TRANS" Tf is a translation defined by a vector U Here U(3;7) is registers R01 and R02
-Likewise, with x' = x
and 1 0 0
in R01 to R06 XEQ "MT2" displays "ID"
in 14 seconds but these transformations are trivial!
y' = y
0 1 0
Example7: x' = 2x + 3y + 1
in this case, the HP-41 displays "TF NOT FOUND" after 7 seconds:
Tf is neither a symmetry nor a projection nor an isometry!
y' = 3x + 4y + 7
Remarks: -Examples 3 to 6 illustrate all
the possible isometries in a 2-dimensional Euclidean space.
-A central symmetry is equivalent to a rotation by 180°
-The HP-41 displays "DATA ERROR" ( line 43 ) if Tf is constant (
a = b = a' = b' = 0 ).
-You may also replace line 42-43 with X#0? GTO 00 RCL
19 STO 02 RCL 18 STO 01 "CST" PROMPT
LBL 00
-Every similarity
can be expressed as the product of an homothecy ( arbitrary center, hereafter
we choose the origin O ) and an isometry.
Therefore, we can use "SIM?" first and then "MT2" if a similarity
has been found.
For instance: x' = 2.8x - 9.6y
- 18.4 1°) Store these
6 numbers into R01 to R06
y' = -9.6x - 2.8y + 58.8 2°)
2 XEQ "SIM?" >>>> Flag F02 is still set at the end and
we have k = 10 ( ratio of magnification )
3°) Divide the contents of registers R01 to R06 by k (
ST/ 01 ST/ 02 ST/ 03 ST/ 04 ST/ 05 ST/ 06
)
4°) XEQ "MT2" >>>> we find the results of example
4 above
-Thus this transformation is the product of 3 transformations:
Tf = hosot
where h is an homothecy ( homothetic center = the origin
O ; ratio of magnification = 10 )
s is the reflexion with respect to (L): x + 4/3 y = 3
t is the translation defined by the vector U(-4;3)
-It's not difficult to modify "MT2" to take all similarities into account,
but it costs several bytes:
-Change line 18 by XEQ "SIM?" , add ST/ 01 ST/ 02
ST/ 03 ST/ 04 ST/ 05 ST/ 06
ST/ 14 ST/ 15 ST/ 16 ST/ 17 ST/ 18 ST/ 19
STO 20 2 XEQ "PS?" after line 63 ,
and modify the text lines if k is different from 1.
b) 3-Dimensional
Space
-"MT3" determines the characteristics of a transformation Tf defined by the formulas
x' = ax + by + cz + d
y' = a'x + b'y + c'z + d'
z' = a"x + b"y + c"z + d"
-This program identifies the projections, symmetries and isometries.
Data Registers:
• R01 = a • R04 = b • R07 = c
• R10 = d
( these 12 registers are to be initialized before executing "MT3" )
• R02 = a' • R05 = b' • R08 = c'
• R11 = d'
• R03 = a" • R06 = b" • R09 = c" •
R12 = d"
R00 and R13 to R37: temp
when the program stops, R00 thru R09 contain the solution.
Flags: F00 F01 F02 F03 F06 F07 F10
Subroutines: PS? ISO?
COMP and LS ( cf "linear and non-linear systems for the
HP-41" )
Note: If you don't have an X-Functions Module,
you may, for instance,
replace
all the REGMOVEs by XEQ 14 and add the 21 following instructions
after line 358
RTN
LBL 14
INT
LASTX
FRC
E3
*
INT
ST- L
E3
ST* L
LBL 15
CLX
RCL IND Z
STO IND Y
ISG Y
CLX
ISG Z
CLX
DSE L
GTO 15
-The "append" character is replaced by ~ in this listing.
01 LBL "MT3"
02 23.001012
03 STO 35
04 13.00001
05 STO 36
06 23.03403
07 STO 37
08 1.023012
09 REGMOVE
10 3
11 XEQ "PS?"
12 FS? 00
13 FC? 03
14 GTO 00
15 "ID"
16 PROMPT
17 LBL 00
18 CF 01
19 3
20 XEQ "ISO?"
21 FS? 02
22 GTO 01
23 FC? 00
24 FS? 03
25 GTO 00
26 "TF NOT FOUND"
27 PROMPT
28 LBL 00
29 XEQ 07
30 SF 07
31 2
32 RCL 01
33 RCL 05
34 RCL 09
35 +
36 +
37 X>Y?
38 ACOS
If Tf is a constant mapping ( a = a' = a" = b = b' = b" = c = c' = c" =
0 ), the HP-41 displays "DATA ERROR" ( line 38 )
39 X=Y?
Lines 37-38 may be replaced by
40 CF 07
X<=Y? GTO 00 RCL 34 STO 03 RCL 33 STO
02 RCL 32 STO 01 "CST" PROMPT LBL 00
41 X=Y?
but is it really worthwhile?
42 XEQ 05
43 FC? 07
44 XEQ 06
45 FS? 07
46 XEQ 04
47 RCL 35
48 REGMOVE
49 CLX
50 FS? 03
51 SIGN
52 ST+ 01
53 ST+ 05
54 ST+ 09
55 SIGN
56 CHS
57 XEQ 08
58 FS? 07
59 XEQ 06
60 FS? 07
61 GTO 00
62 RCL 07
63 STO 22
64 RCL 04
65 STO 21
66 RCL 01
67 STO 20
68 X#0?
69 GTO 00
70 SIGN
71 STO 21
72 RCL 08
73 STO 22
74 RCL 05
75 X#0?
76 GTO 00
77 STO 21
78 SIGN
79 STO 22
80 LBL 00
81 RCL 36
82 REGMOVE
83 "PROJ"
84 FS? 03
85 "SYM"
86 FS? 07
87 "~/P//V"
88 FC?C 07
89 "~/L//P"
90 PROMPT
91 LBL 01
92 XEQ 07
93 RCL 01
94 RCL 05
95 RCL 09
96 +
97 +
98 GTO IND X
99 LBL 00
100 RCL 34
101 STO 03
102 RCL 33
103 STO 02
104 RCL 32
105 STO 01
106 "TRANS"
107 PROMPT
108 LBL 01
109 RCL 37
110 STO Y
111 8
112 XEQ "COMP"
113 RCL 35
114 REGMOVE
115 2
116 ST/ 17
117 ST/.18
118 ST/ 19
119 RCL 17
120 ST- 10
121 RCL 18
122 ST- 11
123 RCL 19
124 ST- 12
125 XEQ 07
126 XEQ 04
127 RCL 36
128 REGMOVE
129 "REFL/P"
130 FC? 03
131 "~*TRANS"
132 PROMPT
133 LBL 02
134 XEQ 06
135 RCL 35
136 REGMOVE
137 RCL 10
138 RCL 17
139 *
140 RCL 11
141 RCL 18
142 *
143 +
144 RCL 12
145 RCL 19
146 *
147 +
148 RCL 17
149 STO 20
150 X^2
151 RCL 18
152 STO 21
153 X^2
154 RCL 19
155 STO 22
156 X^2
157 +
158 +
159 /
160 ST* 20
161 ST* 21
162 ST* 22
163 RCL 20
164 ST- 10
165 RCL 21
166 ST- 11
167 RCL 22
168 ST- 12
169 XEQ 07
170 XEQ 05
171 CF 06
172 LBL 10
173 RCL 36
174 REGMOVE
175 CLX
176 STO 10
177 RCL 04
178 X=0?
179 ISG 10
180 CLX
181 RCL 10
182 RCL 24
183 *
184 RCL 30
185 +
186 STO 12
187 RCL 06
188 *
189 RCL 10
190 RCL 25
191 *
192 RCL 31
193 STO 14
194 +
195 RCL 05
196 *
197 -
198 RCL 10
199 ST* Y
200 RCL 23
201 ST+ 14
202 *
203 RCL 29
204 +
205 RCL 05
206 *
207 RCL 04
208 RCL 12
209 *
210 -
211 +
212 SIGN
213 RCL 14
214 RCL 27
215 +
216 1
217 FS? 06
218 CHS
219 -
220 2
221 /
222 ACOS
223 *
224 STO 00
225 FS?C 06
226 RTN
227 RCL 07
228 ABS
229 RCL 08
230 ABS
231 RCL 09
232 ABS
233 +
234 +
235 E-7
236 "ROT"
237 X<=Y?
238 "~*TRANS"
239 PROMPT
240 LBL 03
241 XEQ 05
242 FC? 03
243 GTO 00
244 RCL 36
245 REGMOVE
246 "CENTR SYM"
247 PROMPT
248 LBL 00
249 RCL 37
250 STO Y
251 1
252 XEQ "COMP"
253 XEQ 07
254 XEQ 06
255 SF 06
256 XEQ 10
257 RCL 01
258 RCL 04
259 *
260 RCL 02
261 RCL 05
262 *
263 +
264 RCL 03
265 RCL 06
266 *
267 +
268 STO 07
269 "ROT*REFL/P"
270 PROMPT
271 LBL 04
272 RCL 10
273 STO 13
274 RCL 07
275 STO 16
276 RCL 04
277 STO 15
278 RCL 01
279 STO 14
280 X#0?
281 RTN
282 SIGN
283 STO 15
284 RCL 08
285 STO 16
286 RCL 11
287 STO 13
288 RCL 05
289 X#0?
290 RTN
291 STO 15
292 SIGN
293 STO 16
294 RCL 12
295 STO 13
296 RTN
297 LBL 05
298 RCL 10
299 STO 14
300 RCL 11
301 STO 15
302 RCL 12
303 STO 16
304 RTN
305 LBL 06
306 RCL 07
307 STO 17
308 RCL 08
309 STO 18
310 1
311 CHS
312 STO 19
313 RCL 09
314 X=0?
315 RTN
316 STO 17
317 CLX
318 STO 18
319 STO 19
320 RCL 01
321 X=0?
322 RTN
323 STO 18
324 RCL 04
325 CHS
326 STO 17
327 RTN
328 LBL 07
329 1
330 CHS
331 ST+ 01
332 ST+ 05
333 ST+ 09
334 LBL 08
335 ST* 10
336 ST* 11
337 ST* 12
338 1.01203
339 E-7
340 XEQ "LS"
341 12
342 LBL 09
343 RCL IND X
344 ABS
345 E-7
346 X<=Y?
347 GTO 00
348 CLX
349 STO IND Z
350 LBL 00
351 X<> Z
352 DSE X
353 GTO 09
354 END
( 665 bytes / SIZE 038 )
a b c d
R01 R04 R07 R10
-In the following examples, the matrix M = a'
b' c' d' is given and it is to be stored
into R02 R05 R08 R11 respectively
before executing "MT3".
a" b" c" d"
R03 R06 R09 R12
-The numbers a , a' , b , b' , ... etc ... in the solutions are of course different from the initial coefficients
-11 -18 -42 24
Example1: M = -2
-2 -7 4 XEQ
"MT3" >>>> "PROJ/P//V" ( 68 seconds ) Tf is
a projection onto a plane (P) in the direction of vector V
4 6 15 -8
(P):
ax + by + cz = d with a , b , c , d
in registers R01 R02 R03 R00 respectively
and V( a',b',c')
with a' , b' , c' ----------
R04 R05 R06
------------
-In this example: (P): x + 1.5 y + 3.5 z =
2 ( or if you prefer:
2x + 3y + 7z = 4 )
and V(3;1/2;-1)
( --------------: V'(6;1;-2) )
-23 -36 -84 48
Example2: M = -4
-5 -14 8 XEQ "MT3"
>>>> "SYM/P//V" ( 68s ) Tf is a symmetry with respect
to a plane (P) in the direction of vector V
8 12 29 -16
(P): ax + by + cz = d with a , b , c , d
in registers R01 R02 R03 R00 respectively
and V( a',b',c')
with a' , b' , c' ----------
R04 R05 R06
------------
-In this example: (P): x + 1.5 y + 3.5 z =
2 ( or if you prefer:
2x + 3y + 7z = 4 )
and V(3;1/2;-1)
( --------------: V'(6;1;-2) )
12 2 -4 -46
Example3: M = 18 3
-6 -74 XEQ "MT3" >>>> "PROJ/L//P"
( 70s ) Tf is a projection onto a line (L) in the
direction of a vectorial plane (P)
42 7 -14 -175
(L) is determined by one point A(xA,yA,zA)
in registers R01 R02 R03 and one vector U(a,b,c) in
registers R04 R05 R06
(P) is given by one of its equations: a'x + b'y + c'z = 0 where
a' , b' , c' are in R07 R08 R09
-Here, A(4,1,0) U(-2/7,-3/7,-1)
( or if you prefer: U'(2,3,7) )
and (P): x +y/6 - z/3 =
0
or 6x + y - 2z = 0
23 4 -8 -92
Example4: M = 36 5
-12 -148 XEQ "MT3" >>>> "SYM/L//P" ( 70s ) Tf
is a symmetry with respect to a line (L) in the direction of a vectorial
plane (P)
84 14 -29 -350
(L) is determined by one point A(xA,yA,zA)
in registers R01 R02 R03 and one vector U(a,b,c) in
registers R04 R05 R06
(P) is given by one of its equations: a'x + b'y + c'z = 0 where
a' , b' , c' are in R07 R08 R09
-Here, A(4,1,0) U(-2/7,-3/7,-1)
( or if you prefer: U'(2,3,7) )
and (P): x +y/6 - z/3 =
0
or 6x + y - 2z = 0
1 0 0 2
Example5: M = 0
1 0 3 XEQ "MT3"
>>>> "TRANS" ( 34s ) Tf is a translation
defined by a vector U(a,b,c) in registers R01 R02
R03
0 0 1 4
-In this example, U(2,3,4)
-If the last column is 0 0 0 ( instead of 2 3 4 )
, the HP-41 displays "ID"
0.64 -0.48 -0.6
1.08
Example6: M = -0.48
0.36 -0.8 1.44
XEQ "MT3" >>>> "REFL/P" ( 102s ) Tf is a reflexion
with respect to a plane (P)
-0.6 -0.8
0 1.8
(P): ax + by + cz = d with a , b , c , d in registers R01 R02 R03 R00 respectively
-Here, (P): x + 4y/3 +5z/3 = 3
( or 3x + 4y + 5z = 9 )
0.64 -0.48 -0.6
5.08
Example7: M = -0.48
0.36 -0.8 3.44
XEQ "MT3" >>>> "REFL/P*TRANS" ( 77s )
-0.6 -0.8
0 -2.2
Tf is the product of a reflexion with respect to a plane (P) and a translation defined by a vector U parallel to (P)
(P):
ax + by + cz = d with a , b , c , d
in registers R01 R02 R03 R00 respectively
and U(a',b',c')
with a' , b' , c' in R04 R05 R06
-Here, (P): x + 4y/3 +5z/3 = 3
( or 3x + 4y + 5z = 9 )
and U(4,2,-4)
-0.56672 -0.16896 0.8064
3.64032
Example8: M =
0.79104 -0.38528 0.4752
2.57376 XEQ "MT3" >>>>
"ROT" ( 66s ) Tf is a rotation by an angle µ
around an axis (L)
0.2304 0.9072
0.352 -3.1824
-The angle µ is in register R00
-(L) is defined by one point A(xA,yA,zA)
in registers R01 R02 R03 and one vector U(a,b,c) in
registers R04 R05 R06
-Here, µ = -143.1301024° ( if the HP-41 is in DEG mode ) A(2,3,0) and U(-0.45 , -0.6 , -1 )
or if you prefer, after multiplying U by -20: U'(9,12,20) and µ = 143.1301024°
Note: Don't forget to change
the sign of µ if you multiply the coordinates of U
by a negative number!
-0.56672 -0.16896 0.8064
5.89032
Example9: M =
0.79104 -0.38528 0.4752
5.57376 XEQ "MT3" >>>>
"ROT*TRANS" ( 66s )
0.2304 0.9072
0.352 1.8176
Tf is the product of a rotation by an angle µ around an axis (L) and a translation defined by a vector V parallel to (L)
-The angle µ is in register R00
-(L) is defined by one point A(xA,yA,zA)
in registers R01 R02 R03 and one vector U(a,b,c) in
registers R04 R05 R06
-The vector V(a',b',c') is in registers R07 R08
R09
-Here, µ = -143.1301024°
(L) is
determined by A(2,3,0) and U(-0.45 , -0.6
, -1 )
and V(2.25;3;5)
-1 0 0 4
Example10: M =
0 -1 0 6 XEQ
"MT3" >>>> "CENTR SYM" ( 70s ) Tf
is a central symmetry with respect to A
0 0 -1 8
-The point A(xA,yA,zA) is in registers R01 R02 R03
-Here, we have A(-2;-3;-4)
0.38528 -0.1344
0.91296 -2.0192
Example11: M =
-0.9024 0.152
0.4032 2.736
XEQ "MT3" >>>> "ROT*REFL/P"
( 92s )
0.19296 0.9792
0.06272 0.4256
Tf is the product of a rotation by an angle µ around an axis (L) and a reflexion with respect to a plane (P) orthogonal to (L)
-The angle µ is in register R00
-(L) is defined by the point A(xA,yA,zA)
in registers R01 R02 R03 and one vector U(a,b,c) in
registers R04 R05 R06
-(P) is defined by the equation ax + by + cz = d
( d is in R07 )
-Note that A is the unique "fixed point" of the transformation
( Tf(A) = A ):
A is the intersection of (P) and (L)
-In this example: µ = 36.86989765
A(2;3;4) U(3/4;15/16;-1)
(P): 3x/4 + 15y/16 - z = 5/16
or if you prefer: U'(12;15;-16)
(P): 12x + 15y - 16z = 5
1 1 1 4
Example12:
M = 1 2 3
1 XEQ "MT3" >>>>
"TF NOT FOUND" ( 8s ) Tf is neither a projection, nor a
symmetry nor an isometry.
1 3 6 8
Remarks: -Examples 5 to 11 illustrate
all the possible isometries in a 3-dimensional Euclidean space.
-An axial symmetry is equivalent to a rotation by 180° around this
axis.
-If you want to detemine the characteristics of a similarity:
1°) Key in 3 XEQ "SIM?"
if Flag F02 is still set at the end, X = k = the ratio of magnification
2°) Press ST/ 01 ST/ 02
ST/ 03 ST/ 04 ST/ 05 ST/ 06 ST/ 07 ST/ 08
ST/ 09 ST/ 10 ST/ 11 ST/ 12
3°) XEQ "MT3"
-If the HP-41 finds an isometry Is , the original transformation
Tf verifies
Tf = hoIs where h is an homothecy ( homothetic center
= the origin O ; k = ratio of magnification )
-You may also replace line 20 by XEQ "SIM?"
add ST/ 01 ST/ 02 ST/ 03 ST/ 04
ST/ 05 ST/ 06 ST/ 07 ST/ 08 ST/ 09 ST/ 10
ST/ 11 ST/ 12
ST/ 23
ST/ 24 ST/ 25 ST/ 26 ST/ 27 ST/ 28 ST/ 29
ST/ 30 ST/ 31 ST/ 32 ST/ 33 ST/ 34 STO 38
3 XEQ "PS?" after line 91
and modify the text lines if k # 1
-However, choosing the homothetic center = O is not necessarily the
best choice!
-In "MT2" and "MT3", the coefficients are replaced by zero if they are
smaller than 10-7 ( in absolute value )
-You can replace E-7 by another small positive number.
c) n-Dimensional Space
-The general case is extremely complex and we give only one ( easy ) example:
x' = 9x + 12y + 16z + 28t - 36
Tf is defined by y' = 4x + 7y + 8z +
14t - 18
find the characteristics of this transformation.
z' = 20x + 30y + 41z + 70t - 90
t' = -16x - 24y - 32z - 55t + 72
1°) Store these 20 coefficients ( in column order ) into registers
R01 to R20
2°) 4 XEQ "PS?" >>>> when the program stops,
Flag F00 is clear , Flag F03 is set: Tf is a symmetry
3°) 4 XEQ "ISO?" >>>> ------------------------
Flag F02 is clear: Tf is not an isometry
4°) To find the fixed points, subtract 1 to registers R01
R06 R11 R16 ( the diagonal elements )
multiply registers R17 R18 R19 R20 by -1 ( the last column )
and we solve this system: 1.02004
ENTER^ E-7 XEQ "LS"
( cf "linear and non-linear systems for the HP-41" )
1 1.5 2 3.5
4.5
The array is now:
0 0 0
0 0
whence the set of fixed points is the hyperplane (H): x + 1.5y +
2z + 3.5t = 4.5
0 0 0
0 0
or 2x + 3y + 4z + 7t = 9
0 0 0
0 0
5°) Since Tf is a symmetry, let's find an eigenvector corresponding
to the eigenvalue -1 ( we would choose 0 if Tf were a projection
)
We restore the 16 original coefficients
into R01 to R16 ( the last column is unused here or may be
replaced by 0 )
we subtract -1 ( i-e we add 1 ) to the
diagonal elements: 1 ST+ 01 ST+ 06 ST+ 11
ST+ 16
and 1.01604
ENTER^ E-7 XEQ "LS"
1 0 0 0.5
x + 0.5t = 0
the matrix becomes: 0
1 0 0.25 meaning
y + 0.25t = 0 for any eigenvector V(x,y,z,t)
corresponding to -1
0 0 1 1.25
z + 1.25t = 0
0 0 0 0
-Setting arbitrarily t = -4 we get x = 2 ; y = 1 ; z = 5
-Therefore, Tf is the symmetry with respect to the hyperplane (H): 2x + 3y +4z + 7t = 9 in the direction of the vector V(2,1,5,-4)
Note: There are so many other cases
that writing a general "MTN" program seems almost impossible...
Reference: J.M. Ferrard , "Mathez la
HP-48 G/GX" , D3I Diffusion , ISBN 2-908791-12-9
( in French )
Go back to the HP-41 software library
Go back to the general software library
Go
back to the main exhibit hall