Class: Quaternion
Overview
Quaternion instances represents Quaternions, complex numbers with one real part and three imaginary parts. While the math is a bit obscure, they can be used to manipulate 3D rotations without “gimbal lock”, the problem of coincident viewing angle alignment problems at the boundaries, for example, where the difference between the x and y coordinates of the viewing vector are zero (ie. looking straight up or straight down.)
Interestingly, Quaternions were first conceptualized by Hamilton in 1843, well before the widespread use of vector notation. While mostly put on the shelf, they still solve certain problems elegantly, and in some cases more quickly than matrix-based 3D calulations.
The quaternion’s real part is accessed through the real instance variable, the three complex parts as the quaternion’s “axis” vector, a Point3.
Instance Attribute Summary collapse
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#axis ⇒ Object
complex axis vector part reader and writer.
-
#real ⇒ Object
real part reader and writer.
Class Method Summary collapse
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.from_axis_angle(axis, angle) ⇒ Object
Returns a new Quaternion-based encoding of a rotation.
Instance Method Summary collapse
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#==(quaternion) ⇒ Object
Returns true if the parts of the instance are equal to the parts of the given quaternion.
-
#abs ⇒ Object
Returns the absolute value of the instance.
-
#add(axis = origin, real = 0) ⇒ Object
(also: #+)
Returns a new Quaternion instance whose parts are the original instance’s with the given amounts added: * axis is a Quaternion, its parts are added.
-
#add!(axis = origin, real = 0) ⇒ Object
Returns the modified instance with the arguments added.
-
#approximately_equals?(quaternion, epsilon = Numeric.epsilon) ⇒ Boolean
(also: #=~)
Returns true if the parts of the instance are approximately equal to the parts of the given quaternion, each part less than a distance epsilon from the target.
-
#conjugate ⇒ Object
Returns a new Quaternion instance that is the conjugate of the original instance.
-
#conjugate! ⇒ Object
Returns the modified instance replaced with its conjugate.
-
#divide(axis = origin, real = 1) ⇒ Object
Returns a new Quaternion instance whose parts are the original instance’s divided by the given amounts: * axis is a Quaternion, the instance is divided by the axis’ parts.
-
#divide!(axis = origin, real = 1) ⇒ Object
Returns the modified instance divided by the arguments.
-
#initialize(axis = origin, real = 0) ⇒ Quaternion
constructor
Creates and returns a Quaternion instance.
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#inverse ⇒ Object
Returns a new Quaternion instance that is the inverse of the original instance.
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#inverse! ⇒ Object
Returns the modified instance replaced with its inverse.
-
#multiply(axis = origin, real = 1) ⇒ Object
(also: #*)
Returns a new Quaternion instance whose parts are the original instance’s multiplied by the given amounts: * axis is a Quaternion, the instance is multiplied by the axis’ parts.
-
#multiply!(axis = origin, real = 1) ⇒ Object
Returns the modified instance multiplied by the arguments.
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#norm ⇒ Object
Returns the norm of the instance.
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#quaternion(axis = self.axis, real = self.real) ⇒ Object
Returns a copy of a Quaternion with the given parts: * axis is a Quaternion, its parts are copied.
-
#scale(scalar = 1) ⇒ Object
Returns a new Quaternion instance whose parts are the original instance’s multiplied by the scalar: * scalar is a Quaternion, the instance is multiplied by the scalar’s parts.
-
#scale!(scalar = 1) ⇒ Object
Returns the modified instance multiplied by the scalar.
-
#set(axis = origin, real = 0) ⇒ Object
Sets the axis and real values of the instance.
-
#subtract(axis = origin, real = 0) ⇒ Object
(also: #-)
Returns a new Quaternion instance whose parts are the original instance’s with the given amounts subtracted: * axis is a Quaternion, its parts are subtracted.
-
#subtract!(axis = origin, real = 0) ⇒ Object
Returns the modified instance with the arguments subtracted.
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#to_axis_angle ⇒ Object
Returns a new Quaternion encoded into a rotation.
-
#to_s ⇒ Object
Returns a string representation of the instance.
Constructor Details
#initialize(axis = origin, real = 0) ⇒ Quaternion
Creates and returns a Quaternion instance. Sets the coordinates values using the set method.
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# File 'lib/quaternion.rb', line 28 def initialize(axis=origin,real=0) set axis, real end |
Instance Attribute Details
#axis ⇒ Object
complex axis vector part reader and writer
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# File 'lib/quaternion.rb', line 22 def axis @axis end |
#real ⇒ Object
real part reader and writer
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# File 'lib/quaternion.rb', line 24 def real @real end |
Class Method Details
.from_axis_angle(axis, angle) ⇒ Object
Returns a new Quaternion-based encoding of a rotation.
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# File 'lib/quaternion.rb', line 79 def Quaternion.from_axis_angle(axis,angle) half_angle = angle/2.0 Quaternion.new Point3.new(axis).scale(Math.sin(half_angle)), Math.cos(half_angle) end |
Instance Method Details
#==(quaternion) ⇒ Object
Returns true if the parts of the instance are equal to the parts of the given quaternion.
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# File 'lib/quaternion.rb', line 56 def ==(quaternion) (@axis == quaternion.axis)&&(@real == quaternion.real) end |
#abs ⇒ Object
Returns the absolute value of the instance.
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# File 'lib/quaternion.rb', line 205 def abs Math.sqrt(norm) end |
#add(axis = origin, real = 0) ⇒ Object Also known as: +
Returns a new Quaternion instance whose parts are the original instance’s with the given amounts added:
-
axis is a Quaternion, its parts are added.
-
axis responds like a Point3, the arguments are added.
-
otherwise a TypeError is raised.
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# File 'lib/quaternion.rb', line 99 def add(axis=origin,real=0) quaternion.add!(axis,real) end |
#add!(axis = origin, real = 0) ⇒ Object
Returns the modified instance with the arguments added.
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# File 'lib/quaternion.rb', line 106 def add!(axis=origin,real=0) if axis.kind_of? Quaternion add!(axis.axis,axis.real) elsif axis.point3_like? set(@axis+axis,@real+real) else raise_no_conversion axis end self end |
#approximately_equals?(quaternion, epsilon = Numeric.epsilon) ⇒ Boolean Also known as: =~
Returns true if the parts of the instance are approximately equal to the parts of the given quaternion, each part less than a distance epsilon from the target.
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# File 'lib/quaternion.rb', line 63 def approximately_equals?(quaternion,epsilon=Numeric.epsilon) (@axis.approximately_equals?(quaternion.axis,epsilon)&& @real.approximately_equals?(quaternion.real,epsilon)) end |
#conjugate ⇒ Object
Returns a new Quaternion instance that is the conjugate of the original instance.
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# File 'lib/quaternion.rb', line 190 def conjugate quaternion.conjugate! end |
#conjugate! ⇒ Object
Returns the modified instance replaced with its conjugate.
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# File 'lib/quaternion.rb', line 195 def conjugate! set(@axis.mirror,@real) end |
#divide(axis = origin, real = 1) ⇒ Object
Returns a new Quaternion instance whose parts are the original instance’s divided by the given amounts:
-
axis is a Quaternion, the instance is divided by the axis’ parts.
-
axis responds like a Point3, the instance is divided by the arguments.
-
otherwise a TypeError is raised.
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# File 'lib/quaternion.rb', line 226 def divide(axis=origin,real=1) quaternion.divide!(axis,origin) end |
#divide!(axis = origin, real = 1) ⇒ Object
Returns the modified instance divided by the arguments.
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# File 'lib/quaternion.rb', line 231 def divide!(axis=origin,real=1) if axis.kind_of? Quaternion divide!(axis.axis,axis.real) elsif axis.point3_like? multiply!(quaternion(axis,real).inverse!) else raise_no_conversion axis end self end |
#inverse ⇒ Object
Returns a new Quaternion instance that is the inverse of the original instance.
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# File 'lib/quaternion.rb', line 211 def inverse quaternion.inverse! end |
#inverse! ⇒ Object
Returns the modified instance replaced with its inverse.
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# File 'lib/quaternion.rb', line 216 def inverse! conjugate! scale!(1.0/norm) end |
#multiply(axis = origin, real = 1) ⇒ Object Also known as: *
Returns a new Quaternion instance whose parts are the original instance’s multiplied by the given amounts:
-
axis is a Quaternion, the instance is multiplied by the axis’ parts.
-
axis responds like a Point3, the instance is multiplied by the arguments.
-
otherwise a TypeError is raised.
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# File 'lib/quaternion.rb', line 145 def multiply(axis=origin,real=1) quaternion.multiply!(axis,real) end |
#multiply!(axis = origin, real = 1) ⇒ Object
Returns the modified instance multiplied by the arguments.
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# File 'lib/quaternion.rb', line 152 def multiply!(axis=origin,real=1) if axis.kind_of? Quaternion multiply!(axis.axis,axis.real) elsif axis.point3_like? r = (@real*real)-@axis.dot(axis) p3 = axis.scale r qp3 = @axis.scale real cross = @axis.cross axis a = p3.add!(qp3).add!(cross) set(a,r) else raise_no_conversion axis end self end |
#norm ⇒ Object
Returns the norm of the instance.
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# File 'lib/quaternion.rb', line 200 def norm (@real * @real) + @axis.modulus end |
#quaternion(axis = self.axis, real = self.real) ⇒ Object
Returns a copy of a Quaternion with the given parts:
-
axis is a Quaternion, its parts are copied.
-
axis responds like a Point3, the arguments are copied.
-
otherwise a TypeError is raised.
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# File 'lib/quaternion.rb', line 74 def quaternion(axis=self.axis,real=self.real) Quaternion.new(axis,real) end |
#scale(scalar = 1) ⇒ Object
Returns a new Quaternion instance whose parts are the original instance’s multiplied by the scalar:
-
scalar is a Quaternion, the instance is multiplied by the scalar’s parts.
-
axis is a Numeric, the instance’s parts are multiplied by the scalar.
-
otherwise a TypeError is raised.
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# File 'lib/quaternion.rb', line 173 def scale(scalar=1) quaternion.scale!(scalar) end |
#scale!(scalar = 1) ⇒ Object
Returns the modified instance multiplied by the scalar.
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# File 'lib/quaternion.rb', line 178 def scale!(scalar=1) if (scalar.kind_of? Quaternion) multiply!(scalar) elsif (scalar.kind_of? Numeric) set(@axis.scale(scalar),@real*scalar) else raise_no_conversion scalar, "Numeric" end end |
#set(axis = origin, real = 0) ⇒ Object
Sets the axis and real values of the instance. When
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axis is a Quaternion, the arguments are interpretted as the parts of a quaternion.
-
axis reponds like a Point3, the axis and real are used to set the quaternion’s values.
-
otherwise a TypeError is raised.
The modified instance is returned.
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# File 'lib/quaternion.rb', line 42 def set(axis=origin,real=0) if axis.kind_of? Quaternion set axis.axis, axis.real elsif axis.point3_like? (@axis ||= point3).set(axis) @real = real else raise_no_conversion(self.class.name,axis) end self end |
#subtract(axis = origin, real = 0) ⇒ Object Also known as: -
Returns a new Quaternion instance whose parts are the original instance’s with the given amounts subtracted:
-
axis is a Quaternion, its parts are subtracted.
-
axis responds like a Point3, the arguments are subtracted.
-
otherwise a TypeError is raised.
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# File 'lib/quaternion.rb', line 122 def subtract(axis=origin,real=0) quaternion.subtract!(axis,real) end |
#subtract!(axis = origin, real = 0) ⇒ Object
Returns the modified instance with the arguments subtracted.
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# File 'lib/quaternion.rb', line 129 def subtract!(axis=origin,real=0) if axis.kind_of? Quaternion subtract!(axis.axis,axis.real) elsif axis.point3_like? set(@axis-axis,@real-real) else raise_no_conversion axis end self end |
#to_axis_angle ⇒ Object
Returns a new Quaternion encoded into a rotation.
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# File 'lib/quaternion.rb', line 86 def to_axis_angle half_angle = Math.acos(@real) sin_half_angle = Math.sin(half_angle) axis = (sin_half_angle.abs < 0.00000001)? Point3.new(1,0,0) : Point3.new(@axis).scale!(1.0/sin_half_angle) Quaternion.new(axis,2.0*half_angle) end |
#to_s ⇒ Object
Returns a string representation of the instance.
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# File 'lib/quaternion.rb', line 33 def to_s "Quaternion: axis: x #{axis.x} y #{axis.y} z #{axis.z} real #{real}" end |