- %
- *
- **
- +
- -
- /
- <=>
- ==
- abs
- abs2
- acos
- amplitude
- arg1
- arg2
- arg3
- asin
- atan
- coerce
- conjugate
- cos
- cosh
- cross_product
- divmod
- divmod_D4
- dot_product
- exp
- gcd
- generic?
- hash
- image
- inspect
- inverse
- is_complex?
- is_quaternion?
- is_real?
- is_unit_vector?
- is_vector?
- latitude
- ldiv
- ldivmod
- ldivmod_D4
- lmod
- lmod_D4
- log
- longitude
- magnitude
- new
- orthogonal_split
- polar
- polar
- rdiv
- real
- real_part
- rmod
- rmod_D4
- rotate
- rotate_angle
- rotation
- round
- round_D4
- sin
- sinh
- sqrt
- tan
- tanh
- to_a
- to_c
- to_c2
- to_s
- to_v
- unit_vector
- vector
- vector
Zero | = | Quaternion(0) |
One | = | Quaternion(1) |
I | = | Quaternion(0,1) |
J | = | Quaternion(0,0,1) |
K | = | Quaternion(0,0,0,1) |
[R] | im | |
[R] | jm | |
[R] | km | |
[R] | re |
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# File lib/facets/more/quaternion.rb, line 247 def Quaternion::generic?(other) return (other.kind_of?(Complex) or Complex.generic?(other)); end
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# 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
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# 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
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# 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
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# 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
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# File lib/facets/more/quaternion.rb, line 424 def % other # right mod d,m=divmod(other); return m end
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# 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
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# 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
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# 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
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# 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
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# 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
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# File lib/facets/more/quaternion.rb, line 339 def <=> (other); self.abs <=> other.abs; end
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# 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
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# File lib/facets/more/quaternion.rb, line 303 def abs; Math.sqrt((@re*@re+@im*@im+@jm*@jm+@km*@km).to_f); end
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# File lib/facets/more/quaternion.rb, line 302 def abs2; return @re*@re+@im*@im+@jm*@jm+@km*@km; end
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# 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
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# 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
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# File lib/facets/more/quaternion.rb, line 284 def arg1; return amplitude; end
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# File lib/facets/more/quaternion.rb, line 285 def arg2; return latitude; end
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# File lib/facets/more/quaternion.rb, line 286 def arg3; return longitude; end
Inverse trigonometric functions
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# 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
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# 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
Arithmetic
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# 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
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# File lib/facets/more/quaternion.rb, line 305 def conjugate; Quaternion(@re,-@im,-@jm,-@km); end
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# 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
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# File lib/facets/more/quaternion.rb, line 496 def cosh; e=exp; return (e+e.inverse)/2; end
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# File lib/facets/more/quaternion.rb, line 389 def cross_product other -(self*other.conjugate).vector end
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# 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
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# 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
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# File lib/facets/more/quaternion.rb, line 386 def dot_product other (self*other.conjugate).re end
Exponential and logarithmic functions
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# 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
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# 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
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# File lib/facets/more/quaternion.rb, line 532 def hash; @re^@im^@jm^@km; end
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# File lib/facets/more/quaternion.rb, line 241 def image; return @im; end
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# 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
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# File lib/facets/more/quaternion.rb, line 306 def inverse; conjugate/abs2; end
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# File lib/facets/more/quaternion.rb, line 309 def is_complex?; @jm==0 and @km==0; end
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# File lib/facets/more/quaternion.rb, line 310 def is_quaternion?; not(is_complex?); end
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# File lib/facets/more/quaternion.rb, line 308 def is_real?; @im==0 and @jm==0 and @km==0; end
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# File lib/facets/more/quaternion.rb, line 324 def is_unit_vector?; @re==0 and abs2==1; end
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# File lib/facets/more/quaternion.rb, line 313 def is_vector?; @re==0; end
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# File lib/facets/more/quaternion.rb, line 282 def latitude; Math.atan2(Math.sqrt((@jm*@jm+@km*@km).to_f),@im.to_f); end
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# File lib/facets/more/quaternion.rb, line 404 def ldiv other # left division: 1/q1 * q2 (self.conjugate)*other/self.abs2 end
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# 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
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# 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
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# File lib/facets/more/quaternion.rb, line 436 def lmod other # left mod d,m=ldivmod(other); return m end
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# File lib/facets/more/quaternion.rb, line 440 def lmod_D4 other # left mod with D4 d,m=ldivmod_D4(other); return m end
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# 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
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# File lib/facets/more/quaternion.rb, line 283 def longitude; Math.atan2( @km.to_f, @jm.to_f); end
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# File lib/facets/more/quaternion.rb, line 304 def magnitude; return abs; end
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# 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
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# File lib/facets/more/quaternion.rb, line 287 def polar; [magnitude, amplitude, latitude, longitude]; end
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# File lib/facets/more/quaternion.rb, line 400 def rdiv other # right division: q1/q2 self/other end
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# File lib/facets/more/quaternion.rb, line 243 def real; return @re; end
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# File lib/facets/more/quaternion.rb, line 242 def real_part; return @re; end
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# File lib/facets/more/quaternion.rb, line 428 def rmod other # right mod(same as %) d,m=divmod(other); return m end
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# File lib/facets/more/quaternion.rb, line 432 def rmod_D4 other # right mod with D4 d,m=divmod_D4(other); return m end
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# File lib/facets/more/quaternion.rb, line 329 def rotate(r); r * self * r.conjugate / r.abs2; end
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# File lib/facets/more/quaternion.rb, line 330 def rotate_angle; amplitude/2; end
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# File lib/facets/more/quaternion.rb, line 289 def round; Quaternion(@re.round,@im.round,@jm.round,@km.round);end
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# 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
Trigonometric functions
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# 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
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# File lib/facets/more/quaternion.rb, line 495 def sinh; e=exp; return (e-e.inverse)/2; end
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# File lib/facets/more/quaternion.rb, line 494 def sqrt; self**(0.5); end
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# 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
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# File lib/facets/more/quaternion.rb, line 497 def tanh; e=exp; e=e*e; return (e-1)/(e+1); end
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# File lib/facets/more/quaternion.rb, line 246 def to_a; return [@re, @im, @jm, @km]; end
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# File lib/facets/more/quaternion.rb, line 244 def to_c; return Complex(@re,@im); end
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# File lib/facets/more/quaternion.rb, line 245 def to_c2; return Complex(@jm,@km); end
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# 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
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# File lib/facets/more/quaternion.rb, line 314 def to_v; return [@im, @jm, @km]; end
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# 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
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# File lib/facets/more/quaternion.rb, line 312 def vector; Quaternion(0,@im,@jm,@km); end