Rotation in 3D with 4D digits
(tesseral arithmetic - digit by digit)
S.B.M. Bell and D.C. Mason: Tesseral Quaternions for the Octtree
THE COMPUTER JOURNAL, Vol. 33, No. 5, 1990, pp. 386-397
Confugure: "Arial Unicode MS" 16
The "qabc" (quaternion-abc)
qabc=:81
3$'o̥ḀB̥C̥D̥E̥F̥G̥H̥I̥J̥K̥L̥M̥N̥O̥P̥Q̥R̥S̥T̥U̥V̥W̥X̥Y̥Z̥o̟A̟B̟C̟D̟E̟F̟G̟H̟I̟J̟K̟L̟M̟N̟O̟P̟Q̟R̟S̟T̟U̟V̟W̟X̟Y̟Z̟o̠A̠B̠C̠D̠E̠F̠G̠H̠I̠J̠K̠L̠M̠N̠O̠P̠Q̠R̠S̠T̠U̠V̠W̠X̠Y̠Z̠
'
The "vabc" (vector-abc)
vabc=:'@ABCDEFGHIJKLMNOPQRSTUVWXYZ'
d=:3 :'q,"1 y,"1 q=.39{a.' NB. display of quaternion
AfC=: 3 : 0 NB. conversion to "alfa form" from "coordinate form"
A=.4 0$0
while.+./0~:y do.y=.3%~y-j-3*j=2[A=.A,.~|.j=.3|y end.
qabc{~3#.|:A
)
NB. conversion to "coordinate form" from "alfa form"
CfA=:3 :'3#.|.|:j-3*2=j=.3 3 3 3#:qabc i.y$~3,~3%~#y'
IfC=: 3 : 0 NB. conversion to "alfa-indexes" from "coordinate form"
A=.4 0$0
while.+./0~:y do.y=.3%~y-j-3*j=2[A=.A,.~|.j=.3|y end.
3#.|:A
)
NB.conversion to "balanced ternary form" from "index
form"
TfI=:3 : '|."1 j-3*2=j=.3 3 3 3#:y'
TfC=: 3 : 0 NB. conversion to "balanced ternary form" from "deci form"
A=.4 0$0
while.+./0~:y do.y=.3%~y-j-3*j=2[A=.A,.~|.j=.3|y end.
|."1 j-3*2=j=.|:A
)
TfA=: 3 : 0 NB. conversion to "balanced ternary form" from "alfa form"
A=.4 0$0[y=.CfA y
while.+./0~:y do.y=.3%~y-j-3*j=2[A=.A,.~|.j=.3|y end.
|."1 j-3*2=j=.|:A
)
afd=: 3 : 0 NB. conversion to "alfa form" from "deci form"
if. 0=+/|y do. A=.'@'
else.
A=.''
while.+./0~:y do.y=.3%~y-j-3*j=2[A=.A,~vabc{~3#.|.j=.3|y end.
end.
A
)
NB. conversion to "deci form" from "alfa form"
dfa=:3 :'3#.|.|:j-3*2=j=.3 3 3 3#:,,.vabc i.y'
HELMERT=: 4 : 0 NB. (x= vector) rotation and scaling (y=
quaternion)
print=: 1!:2&2
W=:0 0 0 NB. clear the accumulator
qt=:TfA y NB. quaternion in balanced ternary
form
qk=: _1 _1 _1 1*"1 qt NB. conjugated quaternion
V=:dfa x NB. vector in "deci form"
signums=:4 4$1 _1 1 1 1 1 _1 1 _1 1 1 1 _1 _1 _1 1
indexes=:4 4$3 2 1 0 2 3 0 1 1 0 3 2 0 1 2 3
N=: #qt NB. number of digits in quaternion
for_i. i.(2*N)-1 do. NB. interval to i-cycle
indent=:14+4*i NB. intend to partial products
Kf=:ij*0<ij=.i-N-1 NB. first value of i-cycle
Kl=:<.i%2 NB. last value of i-cycle
for_k. Kf}.i.Kl+1 do. NB. interval to k-cycle
l=: i-k NB. index of qk
h=: 3^((2*N)-i+2) NB. place value of partial product
KL=: k{qt NB. the left hand of quaternion's
digit
KR=: l{qk NB. the right hand of quaternion's
digit
KL_V=: (signums*indexes{KL) (+/ .*) V NB. multiplication V from
left
KL_V_KR=: }:(signums*indexes{KL_V) (+/ .*) KR NB. multiplication from
right
if.k~:l do. KL_V_KR=: 2*KL_V_KR end. NB. joint the offdiagonal
elements
W=: W+KL_V_KR*h NB. accumulation of partial products
WW=:afd KL_V_KR NB. partial product from deci to alfa form
WW=:(-indent){.WW NB. moving of partial product
print WW,(30$' '),5":KL_V_KR NB. write the partial product
end. NB. end of k-cycle
end. NB. end of i-cycle
if. 1<#qt do. NB. conversion of final product
SZ=:afd W NB. from deci form to alfa form
G=.(-indent){.F=.'-'#~2*#SZ NB. underscore of partial products
print G,(30$' '),F NB. write the underscore
print SW=:((-indent){.SZ),(30$' '),":x:W NB. write the final product
end.
' '
)
Example No. 1 (with abstract data)
dfa 'LY' NB. x y z s - 3D vector in 4D (quaternion)
form
1 2 2 0 NB. converted into 3D
(alphabetic) form
afd 1 2 2 0 NB. in order to be rotated and
scaled
LY NB. (back-and-forth
conversion)
CfA 'I̥D̟T̠' NB. X Y Z S - 4D vector (quaternion) in deci
form
2 3 8 2 NB. converted into 4D
(alphabetic) form
,AfC 2 3 8 2 NB. contains the transformation
parameters:
I̥D̟T̠ NB. rotation-axis, -angle and
magnification
'LY' HELMERT 'I̥D̟T̠' 1 2 2 rotation 2 3 8 2
OW _1 _2 2
LX@ 0 6 6
RMH 2 2 _6
A@@ 9 0 0
TT@ _12 0 _12
IFY 1
_4 8
------------------
------------------------------
[EMAIL PROTECTED] _17 14
242
dfa '[EMAIL PROTECTED]' NB. x y z s - in 3D (alphabetic) form
_17 14 242 0 NB. converted into 4D (decimal) form.
afd _17 14 242 0 NB. It is the transformed vector obtained
[EMAIL PROTECTED] NB. as the sum of partial products.
NB.============================================================
Example No. 2 (with abstract data)
afd 17 _22 4 0 NB. x y z s - 3D vector in 4D (quaternion)
form
GEOQ NB. converted into 3D
(alphabetic) form
dfa 'GEOQ' NB. in order to be rotated and
scaled
17 _22 4 0 NB. (back-and-forth conversion)
,AfC _28 12 _6 11 NB. X Y Z S - 4D vector (quaternion) in deci form
B̥U̟L̟B̠ NB. converted into 4D
(alphabetic) form
CfA 'B̥U̟L̟B̠' NB. contains the transformation
parameters:
_28 12 _6 11 NB. rotation-axis, -angle and magnification
'GEOQ' HELMERT 'B̥U̟L̟B̠' 17 _22 4 rotation _28 12 _6 11
DHUW 17 22 _4
[EMAIL PROTECTED] 52 _26 78
[EMAIL PROTECTED] 36 _26 10
HMXRY _53 _64 14
RMDKS 34 36 _52
FVEIWT 50 _132 _76
[EMAIL PROTECTED] 54
52 104
RDPUQ 35 20
_74
BPLDG _50
_16 36
RJPBM 34
_8 _44
-------------------------------
------------------------------------
JSIUPGMJFM 26605 430 14860
dfa 'JSIUPGMJFM' NB. x y z s - in 3D (alphabetic)
form
26605 430 14860 0 NB. converted into 4D (decimal) form.
afd 26605 430 14860 0 NB. It is the transformed vector obtained
JSIUPGMJFM NB. as the sum of partial products.
NB.============================================================
Example No. 3 (with abstract data)
dfa 'CCDXI' NB. x y z s - 3D vector in 4D
(quaternion) form
9 114 _2 0 NB. converted into 3D
(alphabetic) form
afd 9 114 _2 0 NB. in order to be
rotated and scaled
CCDXI NB. (back-and-forth
conversion)
CfA 'A̟Z̥o̥X̠o̟M̥' NB. X Y Z S - 4D vector
(quaternion) in deci form
163 _89 _89 237 NB. converted into 4D
(alphabetic) form
,AfC 163 _89 _89 237 NB. contains the transformation
parameters:
A̟Z̥o̥X̠o̟M̥ NB. rotation-axis, -angle and
magnification
'CCDXI' HELMERT 'A̟Z̥o̥X̠o̟M̥' 9 114 _2 rotation 163 _89 _89 237
[EMAIL PROTECTED] 18 4 228
UFCKIZ _10 188 _232
@ 0
0 0
JFHLRZ 215 _100 248
[EMAIL PROTECTED] _10 _268
_224
@
0 0 0
[EMAIL PROTECTED]
18
232 224
[EMAIL PROTECTED] _44
32 684
@ 0 0 0
OCFSRM 10
_188 232
[EMAIL PROTECTED]
232
_22 _210
@ 0 0 0
TMGEUDZ _430
200 _496
@ 0 0 0
NFXXDQ
_241 128 208
@ 0 0 0
[EMAIL PROTECTED]
214 _246 22
[EMAIL PROTECTED]
44 _32 _684
CCDXI 9 114 _2
[EMAIL PROTECTED]
_232 22 210
JFHLRZ 215 _100 248
--------------------------------
-------------------------------------------
[EMAIL PROTECTED] 2246072 2856440
10673204
dfa '[EMAIL PROTECTED]' NB. x y z s - in 3D
(alphabetic) form
2246072 2856440 10673204 0 NB. converted into 4D (decimal)
form.
afd 2246072 2856440 10673204 0 NB. It is the transformed vector
obtained
[EMAIL PROTECTED] NB. as the sum of partial products.
Istvan Kadar
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