Methods
Constants
Zero = Quaternion(0)
One = Quaternion(1)
I = Quaternion(0,1)
J = Quaternion(0,0,1)
K = Quaternion(0,0,0,1)
Attributes
[R] im
[R] jm
[R] km
[R] re
Public Class methods
generic?(other)
# File lib/facets/more/quaternion.rb, line 247
  def Quaternion::generic?(other)
    return (other.kind_of?(Complex) or Complex.generic?(other));
  end
new(a=0,b=0,c=0,d=0)
# File lib/facets/more/quaternion.rb, line 250
  def initialize(a=0,b=0,c=0,d=0)
    raise "non numeric 1st arg `#{a.inspect}'" if !a.kind_of? Numeric;
    raise "non numeric 2nd arg `#{b.inspect}'" if !b.kind_of? Numeric;
    raise "non numeric 3rd arg `#{c.inspect}'" if !c.kind_of? Numeric;
    raise "non numeric 4th arg `#{d.inspect}'" if !d.kind_of? Numeric;
    @re=a; @im=b; @jm=c; @km=d
  end
polar(m,t1=0,t2=0,t3=0)
# File lib/facets/more/quaternion.rb, line 265
  def Quaternion::polar(m,t1=0,t2=0,t3=0)
    # q=
    #   m*cos(t1)
    #   +m*sin(t1)cos(t2)i
    #   +m*sin(t1)sin(t2)cos(t3)j
    #   +m*sin(t1)sin(t2)sin(t3)k
    # m is known as the magnitude,
    # t1 is the amplitude(or angle) of the quaternion,
    # t2 and t3 are the latitude (or co-latitude) and longitude respectively.
    if m.kind_of?(Array) and (m.size==4); t1=m[1]; t2=m[2]; t3=m[3]; m=m[0]; end;
    s=m
    r_part=s*Math.cos(t1); s=s*Math.sin(t1)
    i_part=s*Math.cos(t2); s=s*Math.sin(t2)
    j_part=s*Math.cos(t3); k_part=s*Math.sin(t3)
    new(r_part, i_part, j_part, k_part)
  end
rotation(v,t)
# File lib/facets/more/quaternion.rb, line 325
  def Quaternion::rotation(v,t)
    # t-rotatin along the 3-D vector v
    (Quaternion::vector(v).unit_vector) * Math::sin(t/2) + Math::cos(t/2)
  end
vector(v)
# File lib/facets/more/quaternion.rb, line 315
  def Quaternion::vector(v)
    # 3-D vector v=[x,y,z]
    Quaternion(0,v[0],v[1],v[2])
  end
Public Instance methods
%(other)
# File lib/facets/more/quaternion.rb, line 424
  def % other
    # right mod
    d,m=divmod(other); return m
  end
*(other)
# File lib/facets/more/quaternion.rb, line 370
  def * (other)
    if other.kind_of?(Quaternion)
      Quaternion(@re*other.re-@im*other.im-@jm*other.jm-@km*other.km,
           @re*other.im+@im*other.re+@jm*other.km-@km*other.jm,
           @re*other.jm-@im*other.km+@jm*other.re+@km*other.im,
           @re*other.km+@im*other.jm-@jm*other.im+@km*other.re)
    elsif other.kind_of?(Complex)
      Quaternion(@re*other.real - @im*other.image,
           @re*other.image + @im*other.real,
           @jm*other.real + @km*other.image,
           @km*other.real - @jm*other.image)
    elsif Complex.generic?(other)
      Quaternion(@re * other, @im * other, @jm * other, @km * other)
    else x , y = other.coerce(self); x * y
    end
  end
**(other)
# File lib/facets/more/quaternion.rb, line 475
  def ** other
    # q1^q2 = exp((log q1)*q2)
    if other.kind_of?(Quaternion); ((self.log)*other).exp
    elsif other.kind_of?(Complex); ((self.log)*other).exp
    elsif other.kind_of?(Integer);
      if other==0; return One;
      elsif other>0;
        x = self; q = x; n = other - 1
        while n != 0
          while (d, m = n.divmod(2); m == 0); x = x*x; n = d; end
          q *= x; n -= 1
        end
        return q
      else return self.inverse**(-other)
      end
    elsif Quaternion::generic?(other); ((self.log)*other).exp
    else x, y = other.coerce(self); x ** y
    end;
  end
+(other)
# File lib/facets/more/quaternion.rb, line 350
  def + (other)
    if other.kind_of?(Quaternion)
      Quaternion(@re+other.re,@im+other.im,@jm+other.jm,@km+other.km)
    elsif other.kind_of?(Complex)
      Quaternion(@re+other.real,@im+other.image, @jm, @km)
    elsif Complex.generic?(other)
      Quaternion(@re+other.real,@im, @jm, @km)
    else x , y = other.coerce(self); x + y
    end
  end
-(other)
# File lib/facets/more/quaternion.rb, line 360
  def - (other)
    if other.kind_of?(Quaternion)
      Quaternion(@re-other.re,@im-other.im,@jm-other.jm,@km-other.km)
    elsif other.kind_of?(Complex)
      Quaternion(@re-other.real,@im-other.image, @jm, @km)
    elsif Complex.generic?(other)
      Quaternion(@re-other.real,@im, @jm, @km)
    else x , y = other.coerce(self); x - y
    end
  end
/(other)
# File lib/facets/more/quaternion.rb, line 392
  def / other
    if other.kind_of?(Quaternion); self*other.conjugate/other.abs2
    elsif other.kind_of?(Complex); self*other.conjugate/other.abs2
    elsif Complex.generic?(other);
      Quaternion(@re/other, @im/other, @jm/other, @km/other )
    else x, y = other.coerce(self); x / y
    end
  end
<=>(other)
# File lib/facets/more/quaternion.rb, line 339
  def <=> (other); self.abs <=> other.abs; end
==(other)
# File lib/facets/more/quaternion.rb, line 340
  def == (other)
    if other.kind_of?(Quaternion)
      return (@re==other.re and @im==other.im and @jm==other.jm and @km==other.km)
    elsif other.kind_of?(Complex)
      @re==other.real and @im==other.image and @jm==0 and @km==0
    elsif Complex.generic?(other)
      @re==other and @im==0 and @jm==0 and @km==0
    else x , y = other.coerce(self); x == y
    end
  end
abs()
# File lib/facets/more/quaternion.rb, line 303
  def abs; Math.sqrt((@re*@re+@im*@im+@jm*@jm+@km*@km).to_f); end
abs2()
# File lib/facets/more/quaternion.rb, line 302
  def abs2; return @re*@re+@im*@im+@jm*@jm+@km*@km; end
acos()
# File lib/facets/more/quaternion.rb, line 522
  def acos
    # acos q = -u log(q+sqrt(q^2-1))
    q=self; u=unit_vector; -u*((q+(q*q-1).sqrt).log)
  end
amplitude()
# File lib/facets/more/quaternion.rb, line 281
  def amplitude; Math.atan2(Math.sqrt((@im*@im+@jm*@jm+@km*@km).to_f),@re.to_f); end
arg1()
# File lib/facets/more/quaternion.rb, line 284
  def arg1; return amplitude; end
arg2()
# File lib/facets/more/quaternion.rb, line 285
  def arg2; return latitude; end
arg3()
# File lib/facets/more/quaternion.rb, line 286
  def arg3; return longitude; end
asin()

Inverse trigonometric functions

# File lib/facets/more/quaternion.rb, line 518
  def asin
    # asin q = -u log(uq+sqrt(1-q^2))
    q=self; u=unit_vector; -u*((u*q+(1-q*q).sqrt).log)
  end
atan()
# File lib/facets/more/quaternion.rb, line 526
  def atan
    # atan q = u/2 log( (u+q)/(u-q)  )
    q=self; u=q.unit_vector; u*((u+q)/(u-q)).log/2
  end
coerce(other)

Arithmetic

# File lib/facets/more/quaternion.rb, line 333
  def coerce(other)
    if other.kind_of?(Complex); return Quaternion(other), self
    elsif Complex::generic?(other); return Quaternion(other), self
    else super
    end
  end
conjugate()
# File lib/facets/more/quaternion.rb, line 305
  def conjugate; Quaternion(@re,-@im,-@jm,-@km); end
cos()
# File lib/facets/more/quaternion.rb, line 505
  def cos
    # cos(r+uv)=cos r cosh v - u sin r sinh v
    vec=self.vector; v=vec.abs; if v==0; return Quaternion(Math::cos(@re)); end
    u = vec/v; e=Math::exp(v); er=1/e; c=e+er; s=e-er
    (Math::cos(@re)*c-u*Math::sin(@re)*s)/2
  end
cosh()
# File lib/facets/more/quaternion.rb, line 496
  def cosh; e=exp; return (e+e.inverse)/2; end
cross_product(other)
# File lib/facets/more/quaternion.rb, line 389
  def cross_product other
    -(self*other.conjugate).vector
  end
divmod(other)
# File lib/facets/more/quaternion.rb, line 408
  def divmod other
    # right divmod: q1=d*q2+m
    d=self.rdiv(other).round; m=self-d*other; return d,m
  end
divmod_D4(other)
# File lib/facets/more/quaternion.rb, line 412
  def divmod_D4 other
    # right divmod: q1=d*q2+m, d be D4
    d=self.rdiv(other).round_D4; m=self-d*other; return d,m
  end
dot_product(other)
# File lib/facets/more/quaternion.rb, line 386
  def dot_product other
    (self*other.conjugate).re
  end
exp()

Exponential and logarithmic functions

# File lib/facets/more/quaternion.rb, line 459
  def exp
    # e^(r+uv)=exp(r)(cos(v)+u*sin(v))
    if is_real?; return Quaternion(Math::exp(@re)); end
    vec=self.vector; v=vec.abs; u = vec/v;
    Math::exp(@re)*(Math::cos(v)+u*Math::sin(v))
  end
gcd(other)
# File lib/facets/more/quaternion.rb, line 444
  def gcd other
    a=self; b=other
    while true
      if b==0 ; return a;end
      a=a.rmod_D4(b)
      if a==0 ; return b;end
      b=a.lmod_D4(b)
    end
  end
hash()
# File lib/facets/more/quaternion.rb, line 532
  def hash; @re^@im^@jm^@km; end
image()
# File lib/facets/more/quaternion.rb, line 241
  def image; return @im; end
inspect()
# File lib/facets/more/quaternion.rb, line 534
  def inspect
    sprintf("Quaternion(%s,%s,%s,%s)",
        @re.inspect, @im.inspect, @jm.inspect, @km.inspect)
  end
inverse()
# File lib/facets/more/quaternion.rb, line 306
  def inverse; conjugate/abs2; end
is_complex?()
# File lib/facets/more/quaternion.rb, line 309
  def is_complex?; @jm==0 and @km==0; end
is_quaternion?()
# File lib/facets/more/quaternion.rb, line 310
  def is_quaternion?; not(is_complex?); end
is_real?()
# File lib/facets/more/quaternion.rb, line 308
  def is_real?; @im==0 and @jm==0 and @km==0; end
is_unit_vector?()
# File lib/facets/more/quaternion.rb, line 324
  def is_unit_vector?; @re==0 and abs2==1; end
is_vector?()
# File lib/facets/more/quaternion.rb, line 313
  def is_vector?; @re==0; end
latitude()
# File lib/facets/more/quaternion.rb, line 282
  def latitude; Math.atan2(Math.sqrt((@jm*@jm+@km*@km).to_f),@im.to_f); end
ldiv(other)
# File lib/facets/more/quaternion.rb, line 404
  def ldiv other
    # left division: 1/q1 * q2
    (self.conjugate)*other/self.abs2
  end
ldivmod(other)
# File lib/facets/more/quaternion.rb, line 416
  def ldivmod other
    # left divmod: q2=q1*d+m
    d=self.ldiv(other).round; m=other-self*d; return d,m
  end
ldivmod_D4(other)
# File lib/facets/more/quaternion.rb, line 420
  def ldivmod_D4 other
    # left divmod: q2=q1*d+m, d be D4
    d=self.ldiv(other).round_D4; m=other-self*d; return d,m
  end
lmod(other)
# File lib/facets/more/quaternion.rb, line 436
  def lmod other
    # left mod
    d,m=ldivmod(other); return m
  end
lmod_D4(other)
# File lib/facets/more/quaternion.rb, line 440
  def lmod_D4 other
    # left mod with D4
    d,m=ldivmod_D4(other); return m
  end
log()
# File lib/facets/more/quaternion.rb, line 465
  def log
    # log(r+uv)=1/2 log(r^2+v^2)+u atan(v/r)
    if is_real?;
      if @re>=0; return Quaternion(Math::log(@re));
      else return Quaternion(Math::log(-@re),Math::PI,0,0);
      end
    end
    vec=self.vector; v=vec.abs; u = vec/v;
    Math::log(self.abs2.to_f)/2+u*Math::atan2( v, @re)
  end
longitude()
# File lib/facets/more/quaternion.rb, line 283
  def longitude; Math.atan2( @km.to_f, @jm.to_f); end
magnitude()
# File lib/facets/more/quaternion.rb, line 304
  def magnitude; return abs; end
orthogonal_split(o)
# File lib/facets/more/quaternion.rb, line 453
  def orthogonal_split(o)
    # [q1,q2]. q = q1 + q2 such that q1 parallel to o, and q2 orthogonal to o.
    q1 = o * dot_product(o); q2=self-q1; return q1,q2
  end
polar()
# File lib/facets/more/quaternion.rb, line 287
  def polar; [magnitude, amplitude, latitude, longitude]; end
rdiv(other)
# File lib/facets/more/quaternion.rb, line 400
  def rdiv other
    # right division: q1/q2
    self/other
  end
real()
# File lib/facets/more/quaternion.rb, line 243
  def real; return @re; end
real_part()
# File lib/facets/more/quaternion.rb, line 242
  def real_part; return @re; end
rmod(other)
# File lib/facets/more/quaternion.rb, line 428
  def rmod other
    # right mod(same as %)
    d,m=divmod(other); return m
  end
rmod_D4(other)
# File lib/facets/more/quaternion.rb, line 432
  def rmod_D4 other
    # right mod with D4
    d,m=divmod_D4(other); return m
  end
rotate(r)
# File lib/facets/more/quaternion.rb, line 329
  def rotate(r); r * self * r.conjugate / r.abs2; end
rotate_angle()
# File lib/facets/more/quaternion.rb, line 330
  def rotate_angle; amplitude/2; end
round()
# File lib/facets/more/quaternion.rb, line 289
  def round; Quaternion(@re.round,@im.round,@jm.round,@km.round);end
round_D4()
# File lib/facets/more/quaternion.rb, line 290
  def round_D4
    # round to D4 lattice
    r1=@re.round; a1=@im.round; b1=@jm.round; c1=@km.round;
    q1=Quaternion(r1,a1,b1,c1); d1=(q1-self).abs2
    if d1<=1/4; return q1; end
    if @re<r1; r2=r1-1/2; else r2=r1+1/2; end
    if @im<r1; a2=a1-1/2; else a2=a1+1/2; end
    if @jm<r1; b2=b1-1/2; else b2=b1+1/2; end
    if @km<r1; c2=c1-1/2; else c2=c1+1/2; end
    q2=Quaternion(r2,a2,b2,c2); d2=(q2-self).abs2
    if d1<=d2; return q1; else return q2; end
  end
sin()

Trigonometric functions

# File lib/facets/more/quaternion.rb, line 499
  def sin
    # sin(r+uv)=sin r cosh v + u cos r sinh v
    vec=self.vector; v=vec.abs; if v==0; return Quaternion(Math::sin(@re)); end
    u = vec/v; e=Math::exp(v); er=1/e; c=e+er; s=e-er
    (Math::sin(@re)*c+u*Math::cos(@re)*s)/2
  end
sinh()
# File lib/facets/more/quaternion.rb, line 495
  def sinh; e=exp; return (e-e.inverse)/2; end
sqrt()
# File lib/facets/more/quaternion.rb, line 494
  def sqrt; self**(0.5); end
tan()
# File lib/facets/more/quaternion.rb, line 511
  def tan
    vec=self.vector; v=vec.abs; if v==0; return Quaternion(Math::tan(@re)); end
    u = vec/v; e=Math::exp(v); er=1/e; c=e+er; s=e-er
    co=Math::cos(@re); si=Math::sin(@re);  (si*c+u*co*s)/(co*c-u*si*s)
  end
tanh()
# File lib/facets/more/quaternion.rb, line 497
  def tanh; e=exp; e=e*e; return (e-1)/(e+1); end
to_a()
# File lib/facets/more/quaternion.rb, line 246
  def to_a; return [@re, @im, @jm, @km]; end
to_c()
# File lib/facets/more/quaternion.rb, line 244
  def to_c; return Complex(@re,@im); end
to_c2()
# File lib/facets/more/quaternion.rb, line 245
  def to_c2; return Complex(@jm,@km); end
to_s()
# File lib/facets/more/quaternion.rb, line 538
  def to_s
    s=""
    if @re!=0; s=@re.to_s; end
    if @im!=0;
      if s==""; s=sprintf("%si", @im);
      else if @im>0; s=sprintf("%s+%si",s,@im); else s=sprintf("%s-%si",s,-@im); end
      end
    end
    if @jm!=0;
      if s==""; s=sprintf("%sj", @jm);
      else if @jm>0; s=sprintf("%s+%sj",s,@jm); else s=sprintf("%s-%sj",s,-@jm); end
      end
    end
    if @km!=0;
      if s==""; s=sprintf("%sk", @km);
      else if @km>0; s=sprintf("%s+%sk",s,@km); else s=sprintf("%s-%sk",s,-@km); end
      end
    end
    if s=="" ; s="0"; end;
    return s
  end
to_v()
# File lib/facets/more/quaternion.rb, line 314
  def to_v; return [@im, @jm, @km]; end
unit_vector()
# File lib/facets/more/quaternion.rb, line 319
  def unit_vector
    if is_real?; return Quaternion(0,1); end
    m=Math::sqrt((@im*@im+@jm*@jm+@km*@km).to_f)
    Quaternion(0,@im/m,@jm/m,@km/m);
  end
vector()
# File lib/facets/more/quaternion.rb, line 312
  def vector; Quaternion(0,@im,@jm,@km); end