Overview
-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 ) and recalling
registers R13 to R16, we get: 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 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 and we recall registers R14
thru R17 to get 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
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 and we recall registers R17
thru R20 to get 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 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
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
( these 5 registers are to be initialized before executing "PS2" )
• R03 = a' • R04 = b'
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 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 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
R05 = a ; R07 = b ; R09 = c
program stops: 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
R13 = a ; R16 = b ; R19 = c ; R22 = d
program
R14 = a' ; R17 = b' ; R20 = c' ; R23 = d'
stops:
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 ]
= R14 R17
R20 R23 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 by the 7 instructions:
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 =
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
-Thus, this symmetry may be defined by:
x'
= -3x - 8y - 12z - 24t + 40
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
-If you need, say 2 extra registers: 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
-If you need 2n extra registers: 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 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 CLA 70 END |
( 118 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 - 0.4
-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 ]
= 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" )
-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 )
| 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
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 |
Example: Tf is defined
by
x' = x + y + z + 4
y' = x + 2y +3z + 1
Find the reciprocal transformation.
z' = x + 3y + 6z + 8
-Store
1 1 1 4
R01 R04 R07 R10
1 2 3
1 in
R02 R05 R08 R11
respectively
1 3 6
8
R03 R06 R09 R12
-Then 3 XEQ "INVTF" >>>> ( 28 seconds ) -1 and we get
3 -3
1 -17
-3 5 -2
23 in the same registers ( and R00 = det
A = 1 ). Thus:
1 -2
1 -10
x = 3x' - 3y + z - 17
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
If flag F00 is still set when the program stops, Tf is a projection or
a constant mapping.
F03 If flag F03 is still
set when the program stops, 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 20 RCL a 21 X^2 22 RCL 00 23 X<=Y? 24 GTO 01 25 RCL IND 00 26 STO P 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 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
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
-We store
9
12 28 -16
R01 R04 R07 R10
-16 -23 -56
32 in 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:
x' = 9x + 12y -28
store 9 12
-28 in
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 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 0.8 -0.6
1 in R01 R03 R05
( in fact, registers R05 & R06 are unused )
y' = 0.6.x + 0.8.y +2
store 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 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 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 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 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
-We store
7.68
-6.4 -0.24
9.28
R01 R04 R07 R10
-4.24
-4.8 -7.68 -3.04
into 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" )
-The append character is denoted ~
-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" 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" 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
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
store 9 12 -28
in 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
store 17 24 -56
in 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 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
y' = -9.6x - 2.8y
+ 58.8
1°) Store these 6 numbers into R01
to R06
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: ( Registers R01 thru R12 are to be initialized before executing "MT3" )
• R01 = a • R04 = b • R07 = c
• R10 = d
• 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 denoted ~ in this listing.
-If Tf is a constant mapping ( a = a' = a" = b = b' = b" = c = c' =
c" = 0 ), the HP-41 displays "DATA ERROR" ( line 38 )
-Lines 37-38 may be replaced by
X<=Y? GTO 00 RCL 34 STO 03 RCL 33
STO 02 RCL 32 STO 01 "CST" PROMPT LBL 00
-But is it really worthwhile?
| 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 39 X=Y? 40 CF 07 41 X=Y? 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 )
-In the following examples, the matrix
a b c d
is given and R01 R04 R07
R10
M = a' b' c'
d' it is to be
R02 R05 R08 R11 respectively before executing
"MT3".
a" b" c" d" stored into
R03 R06 R09 R12
-The numbers a , a' , b , b' , ... etc ... in the solutions are
of course different from the initial coefficients
Example1:
-11 -18
-42 24
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' in registers 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)
( or if you prefer: V'(6;1;-2) )
Example2:
-23
-36 -84 48
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' in registers 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)
( or if you prefer: V'(6;1;-2) )
Example3:
12 2 -4 -46
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
Example4:
23 4 -8 -92
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
Example5:
1 0 0 2
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"
Example6:
0.64 -0.48 -0.6
1.08
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 )
Example7:
0.64 -0.48 -0.6
5.08
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)
Example8:
-0.56672 -0.16896 0.8064
3.64032
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!
Example9:
-0.56672 -0.16896 0.8064
5.89032
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)
Example10:
-1 0 0 4
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)
Example11:
0.38528 -0.1344
0.91296 -2.0192
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
Example12:
1 1 1 4
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:
Tf is defined by
x' = 9x + 12y + 16z + 28t - 36
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?" >>>> when the program stops,
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" )
The array is now
1 1.5
2 3.5 4.5
0
0 0 0
0 whence the set of fixed points
is the hyperplane (H): x + 1.5y + 2z + 3.5t = 4.5
or 2x + 3y + 4z + 7t = 9
0
0 0 0
0
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" the matrix becomes
1 0 0
0.5
x + 0.5t = 0
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:
[1] J.M. Ferrard , "Mathez la HP-48 G/GX" , D3I Diffusion ,
ISBN 2-908791-12-9 ( in French )