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--All rights reserved.
--Copyright (C) 2007-2009, Gabriel Dos Reis.
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)abbrev category QUATCAT QuaternionCategory
++ Author: Robert S. Sutor
++ Date Created: 23 May 1990
++ Change History:
++ 10 September 1990
++ Basic Operations: (Algebra)
++ abs, conjugate, imagI, imagJ, imagK, norm, quatern, rational,
++ rational?, real
++ Related Constructors: Quaternion, QuaternionCategoryFunctions2
++ Also See: DivisionRing
++ AMS Classifications: 11R52
++ Keywords: quaternions, division ring, algebra
++ Description:
++ \spadtype{QuaternionCategory} describes the category of quaternions
++ and implements functions that are not representation specific.
QuaternionCategory(R: CommutativeRing): Category ==
Join(Algebra R, FullyRetractableTo R, DifferentialExtension R,
FullyEvalableOver R, FullyLinearlyExplicitRingOver R) with
conjugate: $ -> $
++ conjugate(q) negates the imaginary parts of quaternion \spad{q}.
imagI: $ -> R
++ imagI(q) extracts the imaginary i part of quaternion \spad{q}.
imagJ: $ -> R
++ imagJ(q) extracts the imaginary j part of quaternion \spad{q}.
imagK: $ -> R
++ imagK(q) extracts the imaginary k part of quaternion \spad{q}.
norm: $ -> R
++ norm(q) computes the norm of \spad{q} (the sum of the
++ squares of the components).
quatern: (R,R,R,R) -> $
++ quatern(r,i,j,k) constructs a quaternion from scalars.
real: $ -> R
++ real(q) extracts the real part of quaternion \spad{q}.
if R has EntireRing then EntireRing
if R has OrderedSet then OrderedSet
if R has Field then DivisionRing
if R has ConvertibleTo InputForm then ConvertibleTo InputForm
if R has CharacteristicZero then CharacteristicZero
if R has CharacteristicNonZero then CharacteristicNonZero
if R has RealNumberSystem then
abs : $ -> R
++ abs(q) computes the absolute value of quaternion \spad{q}
++ (sqrt of norm).
if R has IntegerNumberSystem then
rational? : $ -> Boolean
++ rational?(q) returns {\it true} if all the imaginary
++ parts of \spad{q} are zero and the real part can be
++ converted into a rational number, and {\it false}
++ otherwise.
rational : $ -> Fraction Integer
++ rational(q) tries to convert \spad{q} into a
++ rational number. Error: if this is not
++ possible. If \spad{rational?(q)} is true, the
++ conversion will be done and the rational number returned.
rationalIfCan: $ -> Union(Fraction Integer, "failed")
++ rationalIfCan(q) returns \spad{q} as a rational number,
++ or "failed" if this is not possible.
++ Note: if \spad{rational?(q)} is true, the conversion
++ can be done and the rational number will be returned.
add
characteristic ==
characteristic$R
conjugate x ==
quatern(real x, - imagI x, - imagJ x, - imagK x)
map(fn, x) ==
quatern(fn real x, fn imagI x, fn imagJ x, fn imagK x)
norm x ==
real x * real x + imagI x * imagI x +
imagJ x * imagJ x + imagK x * imagK x
x = y ==
(real x = real y) and (imagI x = imagI y) and
(imagJ x = imagJ y) and (imagK x = imagK y)
x + y ==
quatern(real x + real y, imagI x + imagI y,
imagJ x + imagJ y, imagK x + imagK y)
x - y ==
quatern(real x - real y, imagI x - imagI y,
imagJ x - imagJ y, imagK x - imagK y)
- x ==
quatern(- real x, - imagI x, - imagJ x, - imagK x)
r:R * x:$ ==
quatern(r * real x, r * imagI x, r * imagJ x, r * imagK x)
n:Integer * x:$ ==
quatern(n * real x, n * imagI x, n * imagJ x, n * imagK x)
differentiate(x:$, d:R -> R) ==
quatern(d real x, d imagI x, d imagJ x, d imagK x)
coerce(r:R) ==
quatern(r,0$R,0$R,0$R)
coerce(n:Integer) ==
quatern(n :: R,0$R,0$R,0$R)
one? x ==
one? real x and zero? imagI x and
zero? imagJ x and zero? imagK x
zero? x ==
zero? real x and zero? imagI x and
zero? imagJ x and zero? imagK x
retract(x):R ==
not (zero? imagI x and zero? imagJ x and zero? imagK x) =>
error "Cannot retract quaternion."
real x
retractIfCan(x):Union(R,"failed") ==
not (zero? imagI x and zero? imagJ x and zero? imagK x) =>
"failed"
real x
coerce(x:$):OutputForm ==
part,z : OutputForm
y : $
zero? x => (0$R) :: OutputForm
not zero?(real x) =>
y := quatern(0$R,imagI(x),imagJ(x),imagK(x))
zero? y => real(x) :: OutputForm
(real(x) :: OutputForm) + (y :: OutputForm)
-- we know that the real part is 0
not zero?(imagI(x)) =>
y := quatern(0$R,0$R,imagJ(x),imagK(x))
z :=
part := 'i::OutputForm
one? imagI(x) => part
(imagI(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part and i part are 0
not zero?(imagJ(x)) =>
y := quatern(0$R,0$R,0$R,imagK(x))
z :=
part := 'j::OutputForm
one? imagJ(x) => part
(imagJ(x) :: OutputForm) * part
zero? y => z
z + (y :: OutputForm)
-- we know that the real part and i and j parts are 0
part := 'k::OutputForm
one? imagK(x) => part
(imagK(x) :: OutputForm) * part
if R has Field then
inv x ==
norm x = 0 => error "This quaternion is not invertible."
(inv norm x) * conjugate x
if R has ConvertibleTo InputForm then
convert(x:$):InputForm ==
l : List InputForm := [convert("quatern" :: Symbol),
convert(real x)$R, convert(imagI x)$R, convert(imagJ x)$R,
convert(imagK x)$R]
convert(l)$InputForm
if R has OrderedSet then
x < y ==
real x = real y =>
imagI x = imagI y =>
imagJ x = imagJ y =>
imagK x < imagK y
imagJ x < imagJ y
imagI x < imagI y
real x < real y
if R has RealNumberSystem then
abs x == sqrt norm x
if R has IntegerNumberSystem then
rational? x ==
(zero? imagI x) and (zero? imagJ x) and (zero? imagK x)
rational x ==
rational? x => rational real x
error "Not a rational number"
rationalIfCan x ==
rational? x => rational real x
"failed"
)abbrev domain QUAT Quaternion
++ Author: Robert S. Sutor
++ Date Created: 23 May 1990
++ Change History:
++ 10 September 1990
++ Basic Operations: (Algebra)
++ abs, conjugate, imagI, imagJ, imagK, norm, quatern, rational,
++ rational?, real
++ Related Constructors: QuaternionCategoryFunctions2
++ Also See: QuaternionCategory, DivisionRing
++ AMS Classifications: 11R52
++ Keywords: quaternions, division ring, algebra
++ Description: \spadtype{Quaternion} implements quaternions over a
++ commutative ring. The main constructor function is \spadfun{quatern}
++ which takes 4 arguments: the real part, the i imaginary part, the j
++ imaginary part and the k imaginary part.
Quaternion(R:CommutativeRing): QuaternionCategory(R) == add
Rep := Record(r:R,i:R,j:R,k:R)
0 == [0,0,0,0]
1 == [1,0,0,0]
a,b,c,d : R
x,y : $
real x == x.r
imagI x == x.i
imagJ x == x.j
imagK x == x.k
quatern(a,b,c,d) == [a,b,c,d]
x * y == [x.r*y.r-x.i*y.i-x.j*y.j-x.k*y.k,
x.r*y.i+x.i*y.r+x.j*y.k-x.k*y.j,
x.r*y.j+x.j*y.r+x.k*y.i-x.i*y.k,
x.r*y.k+x.k*y.r+x.i*y.j-x.j*y.i]
)abbrev package QUATCT2 QuaternionCategoryFunctions2
++ Author: Robert S. Sutor
++ Date Created: 23 May 1990
++ Change History:
++ 23 May 1990
++ Basic Operations: map
++ Related Constructors: QuaternionCategory, Quaternion
++ Also See:
++ AMS Classifications: 11R52
++ Keywords: quaternions, division ring, map
++ Description:
++ \spadtype{QuaternionCategoryFunctions2} implements functions between
++ two quaternion domains. The function \spadfun{map} is used by
++ the system interpreter to coerce between quaternion types.
QuaternionCategoryFunctions2(QR,R,QS,S) : Exports ==
Implementation where
R : CommutativeRing
S : CommutativeRing
QR : QuaternionCategory R
QS : QuaternionCategory S
Exports == with
map: (R -> S, QR) -> QS
++ map(f,u) maps f onto the component parts of the quaternion
++ u.
Implementation == add
map(fn : R -> S, u : QR): QS ==
quatern(fn real u, fn imagI u, fn imagJ u, fn imagK u)$QS
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