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Merge pull request #205 from RonaldoCMP/master
input data checks and addition of many new functions, test_quaternions updated
This commit is contained in:
Revar Desmera
GitHub
commit
e4cab05952
2 changed files with 522 additions and 77 deletions
483
quaternions.scad
483
quaternions.scad
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@ -20,59 +20,107 @@
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// Internal
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function _Quat(a,s,w) = [a[0]*s, a[1]*s, a[2]*s, w];
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function _Qvec(q) = [q.x,q.y,q.z];
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function _Qreal(q) = q[3];
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function _Qset(v,r) = concat( v, r );
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// normalizes without checking
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function _Qnorm(q) = q/norm(q);
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// Function: Q_is_quat()
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// Usage:
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// if(Q_is_quat(q)) a=0;
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// Description: Return true if q is a valid non-zero quaternion.
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// Arguments:
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// q = object to check.
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function Q_is_quat(q) = is_vector(q,4) && ! approx(norm(q),0) ;
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// Function: Quat()
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// Usage:
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// Quat(ax, ang);
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// Description: Create a new Quaternion from axis and angle of rotation.
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// Description: Create a normalized Quaternion from axis and angle of rotation.
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// Arguments:
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// ax = Vector of axis of rotation.
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// ang = Number of degrees to rotate around the axis counter-clockwise, when facing the origin.
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function Quat(ax=[0,0,1], ang=0) = _Quat(ax/norm(ax), sin(ang/2), cos(ang/2));
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function Quat(ax=[0,0,1], ang=0) =
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assert( is_vector(ax,3) && is_finite(ang), "Invalid input")
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let( n = norm(ax) )
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approx(n,0)
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? _Quat([0,0,0], sin(ang/2), cos(ang/2))
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: _Quat(ax/n, sin(ang/2), cos(ang/2));
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// Function: QuatX()
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// Usage:
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// QuatX(a);
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// Description: Create a new Quaternion for rotating around the X axis [1,0,0].
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// Description: Create a normalized Quaternion for rotating around the X axis [1,0,0].
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// Arguments:
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// a = Number of degrees to rotate around the axis counter-clockwise, when facing the origin.
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function QuatX(a=0) = Quat([1,0,0],a);
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function QuatX(a=0) =
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assert( is_finite(a), "Invalid angle" )
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Quat([1,0,0],a);
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// Function: QuatY()
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// Usage:
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// QuatY(a);
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// Description: Create a new Quaternion for rotating around the Y axis [0,1,0].
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// Description: Create a normalized Quaternion for rotating around the Y axis [0,1,0].
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// Arguments:
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// a = Number of degrees to rotate around the axis counter-clockwise, when facing the origin.
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function QuatY(a=0) = Quat([0,1,0],a);
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function QuatY(a=0) =
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assert( is_finite(a), "Invalid angle" )
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Quat([0,1,0],a);
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// Function: QuatZ()
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// Usage:
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// QuatZ(a);
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// Description: Create a new Quaternion for rotating around the Z axis [0,0,1].
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// Description: Create a normalized Quaternion for rotating around the Z axis [0,0,1].
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// Arguments:
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// a = Number of degrees to rotate around the axis counter-clockwise, when facing the origin.
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function QuatZ(a=0) = Quat([0,0,1],a);
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function QuatZ(a=0) =
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assert( is_finite(a), "Invalid angle" )
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Quat([0,0,1],a);
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// Function: QuatXYZ()
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// Usage:
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// QuatXYZ([X,Y,Z])
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// Description:
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// Creates a quaternion from standard [X,Y,Z] rotation angles in degrees.
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// Creates a normalized quaternion from standard [X,Y,Z] rotation angles in degrees.
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// Arguments:
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// a = The triplet of rotation angles, [X,Y,Z]
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function QuatXYZ(a=[0,0,0]) =
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assert( is_vector(a,3), "Invalid angles")
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let(
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qx = QuatX(a[0]),
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qy = QuatY(a[1]),
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qz = QuatZ(a[2])
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qx = QuatX(a[0]),
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qy = QuatY(a[1]),
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qz = QuatZ(a[2])
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)
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Q_Mul(qz, Q_Mul(qy, qx));
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// Function: Q_From_to()
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// Usage:
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// q = Q_From_to(v1, v2);
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// Description:
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// Returns the normalized quaternion that rotates the non zero 3D vector v1
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// to the non zero 3D vector v2.
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function Q_From_to(v1, v2) =
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assert( is_vector(v1,3) && is_vector(v2,3)
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&& ! approx(norm(v1),0) && ! approx(norm(v2),0)
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, "Invalid vector(s)")
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let( ax = cross(v1,v2),
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n = norm(ax) )
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approx(n, 0)
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? v1*v2>0 ? Q_Ident() : Quat([ v1.y, -v1.x, 0], 180)
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: Quat(ax, atan2( n , v1*v2 ));
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// Function: Q_Ident()
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// Description: Returns the "Identity" zero-rotation Quaternion.
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function Q_Ident() = [0, 0, 0, 1];
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@ -81,55 +129,85 @@ function Q_Ident() = [0, 0, 0, 1];
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// Function: Q_Add_S()
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// Usage:
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// Q_Add_S(q, s)
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// Description: Adds a scalar value `s` to the W part of a quaternion `q`.
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function Q_Add_S(q, s) = q+[0,0,0,s];
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// Description:
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// Adds a scalar value `s` to the W part of a quaternion `q`.
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// The returned quaternion is usually not normalized.
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function Q_Add_S(q, s) =
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assert( is_finite(s), "Invalid scalar" )
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q+[0,0,0,s];
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// Function: Q_Sub_S()
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// Usage:
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// Q_Sub_S(q, s)
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// Description: Subtracts a scalar value `s` from the W part of a quaternion `q`.
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function Q_Sub_S(q, s) = q-[0,0,0,s];
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// Description:
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// Subtracts a scalar value `s` from the W part of a quaternion `q`.
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// The returned quaternion is usually not normalized.
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function Q_Sub_S(q, s) =
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assert( is_finite(s), "Invalid scalar" )
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q-[0,0,0,s];
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// Function: Q_Mul_S()
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// Usage:
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// Q_Mul_S(q, s)
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// Description: Multiplies each part of a quaternion `q` by a scalar value `s`.
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function Q_Mul_S(q, s) = q*s;
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// Description:
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// Multiplies each part of a quaternion `q` by a scalar value `s`.
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// The returned quaternion is usually not normalized.
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function Q_Mul_S(q, s) =
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assert( is_finite(s), "Invalid scalar" )
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q*s;
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// Function: Q_Div_S()
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// Usage:
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// Q_Div_S(q, s)
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// Description: Divides each part of a quaternion `q` by a scalar value `s`.
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function Q_Div_S(q, s) = q/s;
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// Description:
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// Divides each part of a quaternion `q` by a scalar value `s`.
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// The returned quaternion is usually not normalized.
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function Q_Div_S(q, s) =
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assert( is_finite(s) && ! approx(s,0) , "Invalid scalar" )
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q/s;
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// Function: Q_Add()
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// Usage:
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// Q_Add(a, b)
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// Description: Adds each part of two quaternions together.
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function Q_Add(a, b) = a+b;
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// Description:
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// Adds each part of two quaternions together.
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// The returned quaternion is usually not normalized.
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function Q_Add(a, b) =
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assert( Q_is_quat(a) && Q_is_quat(a), "Invalid quaternion(s)")
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assert( ! approx(norm(a+b),0), "Quaternions cannot be opposed" )
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a+b;
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// Function: Q_Sub()
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// Usage:
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// Q_Sub(a, b)
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// Description: Subtracts each part of quaternion `b` from quaternion `a`.
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function Q_Sub(a, b) = a-b;
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// Description:
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// Subtracts each part of quaternion `b` from quaternion `a`.
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// The returned quaternion is usually not normalized.
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function Q_Sub(a, b) =
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assert( Q_is_quat(a) && Q_is_quat(a), "Invalid quaternion(s)")
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assert( ! approx(a,b), "Quaternions cannot be equal" )
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a-b;
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// Function: Q_Mul()
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// Usage:
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// Q_Mul(a, b)
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// Description: Multiplies quaternion `a` by quaternion `b`.
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function Q_Mul(a, b) = [
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a[3]*b.x + a.x*b[3] + a.y*b.z - a.z*b.y,
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a[3]*b.y - a.x*b.z + a.y*b[3] + a.z*b.x,
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a[3]*b.z + a.x*b.y - a.y*b.x + a.z*b[3],
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a[3]*b[3] - a.x*b.x - a.y*b.y - a.z*b.z,
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];
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// Description:
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// Multiplies quaternion `a` by quaternion `b`.
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// The returned quaternion is normalized if both `a` and `b` are normalized
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function Q_Mul(a, b) =
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assert( Q_is_quat(a) && Q_is_quat(b), "Invalid quaternion(s)")
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[
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a[3]*b.x + a.x*b[3] + a.y*b.z - a.z*b.y,
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a[3]*b.y - a.x*b.z + a.y*b[3] + a.z*b.x,
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a[3]*b.z + a.x*b.y - a.y*b.x + a.z*b[3],
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a[3]*b[3] - a.x*b.x - a.y*b.y - a.z*b.z,
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];
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// Function: Q_Cumulative()
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@ -139,6 +217,8 @@ function Q_Mul(a, b) = [
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// Given a list of Quaternions, cumulatively multiplies them, returning a list
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// of each cumulative Quaternion product. It starts with the first quaternion
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// given in the list, and applies successive quaternion rotations in list order.
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// The quaternion in the returned list are normalized if each quaternion in v
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// is normalized.
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function Q_Cumulative(v, _i=0, _acc=[]) =
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_i==len(v) ? _acc :
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Q_Cumulative(
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@ -154,42 +234,65 @@ function Q_Cumulative(v, _i=0, _acc=[]) =
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// Usage:
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// Q_Dot(a, b)
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// Description: Calculates the dot product between quaternions `a` and `b`.
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function Q_Dot(a, b) = a[0]*b[0] + a[1]*b[1] + a[2]*b[2] + a[3]*b[3];
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function Q_Dot(a, b) =
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assert( Q_is_quat(a) && Q_is_quat(b), "Invalid quaternion(s)" )
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a*b;
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// Function: Q_Neg()
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// Usage:
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// Q_Neg(q)
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// Description: Returns the negative of quaternion `q`.
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function Q_Neg(q) = -q;
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function Q_Neg(q) =
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assert( Q_is_quat(q), "Invalid quaternion" )
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-q;
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// Function: Q_Conj()
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// Usage:
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// Q_Conj(q)
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// Description: Returns the conjugate of quaternion `q`.
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function Q_Conj(q) = [-q.x, -q.y, -q.z, q[3]];
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function Q_Conj(q) =
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assert( Q_is_quat(q), "Invalid quaternion" )
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[-q.x, -q.y, -q.z, q[3]];
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// Function: Q_Inverse()
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// Usage:
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// qc = Q_Inverse(q)
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// Description: Returns the multiplication inverse of quaternion `q` that is normalized only if `q` is normalized.
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function Q_Inverse(q) =
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assert( Q_is_quat(q), "Invalid quaternion" )
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let(q = _Qnorm(q) )
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[-q.x, -q.y, -q.z, q[3]];
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// Function: Q_Norm()
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// Usage:
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// Q_Norm(q)
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// Description: Returns the `norm()` "length" of quaternion `q`.
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function Q_Norm(q) = norm(q);
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// Description:
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// Returns the `norm()` "length" of quaternion `q`.
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// Normalized quaternions have unitary norm.
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function Q_Norm(q) =
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assert( Q_is_quat(q), "Invalid quaternion" )
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norm(q);
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// Function: Q_Normalize()
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// Usage:
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// Q_Normalize(q)
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// Description: Normalizes quaternion `q`, so that norm([W,X,Y,Z]) == 1.
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function Q_Normalize(q) = q/norm(q);
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function Q_Normalize(q) =
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assert( Q_is_quat(q) , "Invalid quaternion" )
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q/norm(q);
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// Function: Q_Dist()
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// Usage:
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// Q_Dist(q1, q2)
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// Description: Returns the "distance" between two quaternions.
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function Q_Dist(q1, q2) = norm(q2-q1);
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function Q_Dist(q1, q2) =
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assert( Q_is_quat(q1) && Q_is_quat(q2), "Invalid quaternion(s)" )
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norm(q2-q1);
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// Function: Q_Slerp()
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@ -214,24 +317,24 @@ function Q_Dist(q1, q2) = norm(q2-q1);
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// for (q = Q_Slerp(a, b, [0:0.1:1]))
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// Qrot(q) right(80) cube([10,10,1]);
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// #sphere(r=80);
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function Q_Slerp(q1, q2, u) =
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assert(is_num(u) || is_num(u[0]))
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!is_num(u)? [for (uu=u) Q_Slerp(q1,q2,uu)] :
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let(
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q1 = Q_Normalize(q1),
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q2 = Q_Normalize(q2),
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dot = Q_Dot(q1, q2)
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) let(
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q2 = dot<0? Q_Neg(q2) : q2,
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dot = dot<0? -dot : dot
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) (dot>0.9995)? Q_Normalize(q1 + (u * (q2-q1))) :
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let(
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dot = constrain(dot,-1,1),
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theta_0 = acos(dot),
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theta = theta_0 * u,
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q3 = Q_Normalize(q2 - q1*dot),
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out = q1*cos(theta) + q3*sin(theta)
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) out;
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function Q_Slerp(q1, q2, u, _dot) =
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is_undef(_dot)
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? assert(is_finite(u) || is_range(u) || is_vector(u), "Invalid interpolation coefficient(s)")
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assert(Q_is_quat(q1) && Q_is_quat(q2), "Invalid quaternion(s)" )
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let(
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_dot = q1*q2,
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q1 = q1/norm(q1),
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q2 = _dot<0 ? -q2/norm(q2) : q2/norm(q2),
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dot = abs(_dot)
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)
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! is_finite(u) ? [for (uu=u) Q_Slerp(q1, q2, uu, dot)] :
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Q_Slerp(q1, q2, u, dot)
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: _dot>0.9995
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? _Qnorm(q1 + u*(q2-q1))
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: let( theta = u*acos(_dot),
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q3 = _Qnorm(q2 - _dot*q1)
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)
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_Qnorm(q1*cos(theta) + q3*sin(theta));
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// Function: Q_Matrix3()
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@ -239,11 +342,13 @@ function Q_Slerp(q1, q2, u) =
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// Q_Matrix3(q);
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// Description:
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// Returns the 3x3 rotation matrix for the given normalized quaternion q.
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function Q_Matrix3(q) = [
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[1-2*q[1]*q[1]-2*q[2]*q[2], 2*q[0]*q[1]-2*q[2]*q[3], 2*q[0]*q[2]+2*q[1]*q[3]],
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[ 2*q[0]*q[1]+2*q[2]*q[3], 1-2*q[0]*q[0]-2*q[2]*q[2], 2*q[1]*q[2]-2*q[0]*q[3]],
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[ 2*q[0]*q[2]-2*q[1]*q[3], 2*q[1]*q[2]+2*q[0]*q[3], 1-2*q[0]*q[0]-2*q[1]*q[1]]
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];
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function Q_Matrix3(q) =
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let( q = Q_Normalize(q) )
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[
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[1-2*q[1]*q[1]-2*q[2]*q[2], 2*q[0]*q[1]-2*q[2]*q[3], 2*q[0]*q[2]+2*q[1]*q[3]],
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[ 2*q[0]*q[1]+2*q[2]*q[3], 1-2*q[0]*q[0]-2*q[2]*q[2], 2*q[1]*q[2]-2*q[0]*q[3]],
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[ 2*q[0]*q[2]-2*q[1]*q[3], 2*q[1]*q[2]+2*q[0]*q[3], 1-2*q[0]*q[0]-2*q[1]*q[1]]
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];
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// Function: Q_Matrix4()
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@ -251,12 +356,14 @@ function Q_Matrix3(q) = [
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// Q_Matrix4(q);
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// Description:
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// Returns the 4x4 rotation matrix for the given normalized quaternion q.
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function Q_Matrix4(q) = [
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[1-2*q[1]*q[1]-2*q[2]*q[2], 2*q[0]*q[1]-2*q[2]*q[3], 2*q[0]*q[2]+2*q[1]*q[3], 0],
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[ 2*q[0]*q[1]+2*q[2]*q[3], 1-2*q[0]*q[0]-2*q[2]*q[2], 2*q[1]*q[2]-2*q[0]*q[3], 0],
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[ 2*q[0]*q[2]-2*q[1]*q[3], 2*q[1]*q[2]+2*q[0]*q[3], 1-2*q[0]*q[0]-2*q[1]*q[1], 0],
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[ 0, 0, 0, 1]
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];
|
||||
function Q_Matrix4(q) =
|
||||
let( q = Q_Normalize(q) )
|
||||
[
|
||||
[1-2*q[1]*q[1]-2*q[2]*q[2], 2*q[0]*q[1]-2*q[2]*q[3], 2*q[0]*q[2]+2*q[1]*q[3], 0],
|
||||
[ 2*q[0]*q[1]+2*q[2]*q[3], 1-2*q[0]*q[0]-2*q[2]*q[2], 2*q[1]*q[2]-2*q[0]*q[3], 0],
|
||||
[ 2*q[0]*q[2]-2*q[1]*q[3], 2*q[1]*q[2]+2*q[0]*q[3], 1-2*q[0]*q[0]-2*q[1]*q[1], 0],
|
||||
[ 0, 0, 0, 1]
|
||||
];
|
||||
|
||||
|
||||
// Function: Q_Axis()
|
||||
|
|
@ -264,16 +371,28 @@ function Q_Matrix4(q) = [
|
|||
// Q_Axis(q)
|
||||
// Description:
|
||||
// Returns the axis of rotation of a normalized quaternion `q`.
|
||||
function Q_Axis(q) = let(d = sqrt(1-(q[3]*q[3]))) (d==0)? [0,0,1] : [q[0]/d, q[1]/d, q[2]/d];
|
||||
|
||||
// The input doesn't need to be normalized.
|
||||
function Q_Axis(q) =
|
||||
assert( Q_is_quat(q) , "Invalid quaternion" )
|
||||
let( d = norm(_Qvec(q)) )
|
||||
approx(d,0)? [0,0,1] : _Qvec(q)/d;
|
||||
|
||||
// Function: Q_Angle()
|
||||
// Usage:
|
||||
// Q_Angle(q)
|
||||
// a = Q_Angle(q)
|
||||
// a12 = Q_Angle(q1,q2);
|
||||
// Description:
|
||||
// Returns the angle of rotation (in degrees) of a normalized quaternion `q`.
|
||||
function Q_Angle(q) = 2 * acos(q[3]);
|
||||
|
||||
// If only q1 is given, returns the angle of rotation (in degrees) of that quaternion.
|
||||
// If both q1 and q2 are given, returns the angle (in degrees) between them.
|
||||
// The input quaternions don't need to be normalized.
|
||||
function Q_Angle(q1,q2) =
|
||||
assert(Q_is_quat(q1) && (is_undef(q2) || Q_is_quat(q2)), "Invalid quaternion(s)" )
|
||||
let( n1 = is_undef(q2)? norm(_Qvec(q1)): norm(q1) )
|
||||
is_undef(q2)
|
||||
? 2 * atan2(n1,_Qreal(q1))
|
||||
: let( q1 = q1/norm(q1),
|
||||
q2 = q2/norm(q2) )
|
||||
4 * atan2(norm(q1 - q2), norm(q1 + q2));
|
||||
|
||||
// Function&Module: Qrot()
|
||||
// Usage: As Module
|
||||
|
|
@ -305,9 +424,9 @@ module Qrot(q) {
|
|||
}
|
||||
|
||||
function Qrot(q,p) =
|
||||
is_undef(p)? Q_Matrix4(q) :
|
||||
is_vector(p)? Qrot(q,[p])[0] :
|
||||
apply(Q_Matrix4(q), p);
|
||||
is_undef(p)? Q_Matrix4(q) :
|
||||
is_vector(p)? Qrot(q,[p])[0] :
|
||||
apply(Q_Matrix4(q), p);
|
||||
|
||||
|
||||
// Module: Qrot_copies()
|
||||
|
|
@ -327,4 +446,214 @@ function Qrot(q,p) =
|
|||
module Qrot_copies(quats) for (q=quats) Qrot(q) children();
|
||||
|
||||
|
||||
// Function: Q_Rotation()
|
||||
// Usage:
|
||||
// Q_Rotation(R)
|
||||
// Description:
|
||||
// Returns a normalized quaternion corresponding to the rotation matrix R.
|
||||
// R may be a 3x3 rotation matrix or a homogeneous 4x4 rotation matrix.
|
||||
// The last row and last column of R are ignored for 4x4 matrices.
|
||||
// It doesn't check whether R is in fact a rotation matrix.
|
||||
// If R is not a rotation, the returned quaternion is an unpredictable quaternion .
|
||||
function Q_Rotation(R) =
|
||||
assert( is_matrix(R,3,3) || is_matrix(R,4,4) ,
|
||||
"Matrix is neither 3x3 nor 4x4")
|
||||
let( tr = R[0][0]+R[1][1]+R[2][2] ) // R trace
|
||||
tr>0
|
||||
? let( r = 1+tr )
|
||||
_Qnorm( _Qset([ R[1][2]-R[2][1], R[2][0]-R[0][2], R[0][1]-R[1][0] ], -r ) )
|
||||
: let( i = max_index([ R[0][0], R[1][1], R[2][2] ]),
|
||||
r = 1 + 2*R[i][i] -R[0][0] -R[1][1] -R[2][2] )
|
||||
i==0 ? _Qnorm( _Qset( [ 4*r, (R[1][0]+R[0][1]), (R[0][2]+R[2][0]) ], (R[2][1]-R[1][2])) ):
|
||||
i==1 ? _Qnorm( _Qset( [ (R[1][0]+R[0][1]), 4*r, (R[2][1]+R[1][2]) ], (R[0][2]-R[2][0])) ):
|
||||
_Qnorm( _Qset( [ (R[2][0]+R[0][2]), (R[1][2]+R[2][1]), 4*r ], (R[1][0]-R[0][1])) ) ;
|
||||
|
||||
|
||||
// Function&Module: Q_Rotation_path(q1, n, [q2])
|
||||
// Usage: As a function
|
||||
// path = Q_Rotation_path(q1, n, q2);
|
||||
// path = Q_Rotation_path(q1, n);
|
||||
// Usage: As a module
|
||||
// Q_Rotation_path(q1, n, q2) ...
|
||||
// Description:
|
||||
// If q2 is undef and it is called as a function, the path, with length n+1 (n>=1), will be the
|
||||
// cumulative multiplications of the matrix rotation of q1 by itself.
|
||||
// If q2 is defined and it is called as a function, returns a rotation matrix path of length n+1 (n>=1)
|
||||
// that interpolates two given rotation quaternions. The first matrix of the sequence is the
|
||||
// matrix rotation of q1 and the last one, the matrix rotation of q2. The intermediary matrix
|
||||
// rotations are an uniform interpolation of the path extreme matrices.
|
||||
// When called as a module, applies to its children() each rotation of the sequence computed
|
||||
// by the function.
|
||||
// The input quaternions don't need to be normalized.
|
||||
// Arguments:
|
||||
// q1 = The quaternion of the first rotation.
|
||||
// q2 = The quaternion of the last rotation.
|
||||
// n = An integer defining the path length ( path length = n+1).
|
||||
// Example(3D): as a function
|
||||
// a = QuatY(-135);
|
||||
// b = QuatXYZ([0,-30,30]);
|
||||
// for (M=Q_Rotation_path(a, 10, b))
|
||||
// multmatrix(M)
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
// Example(3D): as a module
|
||||
// a = QuatY(-135);
|
||||
// b = QuatXYZ([0,-30,30]);
|
||||
// Q_Rotation_path(a, 10, b)
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
// Example(3D): as a function
|
||||
// a = QuatY(5);
|
||||
// for (M=Q_Rotation_path(a, 10))
|
||||
// multmatrix(M)
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
// Example(3D): as a module
|
||||
// a = QuatY(5);
|
||||
// Q_Rotation_path(a, 10)
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
function Q_Rotation_path(q1, n=1, q2) =
|
||||
assert( Q_is_quat(q1) && (is_undef(q2) || Q_is_quat(q2) ), "Invalid quaternion(s)" )
|
||||
assert( is_finite(n) && n>=1 && n==floor(n), "Invalid integer" )
|
||||
assert( is_undef(q2) || ! approx(norm(q1+q2),0), "Quaternions cannot be opposed" )
|
||||
is_undef(q2)
|
||||
? [for( i=0, dR=Q_Matrix4(q1), R=dR; i<=n; i=i+1, R=dR*R ) R]
|
||||
: let( q2 = Q_Normalize( q1*q2<0 ? -q2: q2 ),
|
||||
dq = Q_pow( Q_Mul( q2, Q_Inverse(q1) ), 1/n ),
|
||||
dR = Q_Matrix4(dq) )
|
||||
[for( i=0, R=Q_Matrix4(q1); i<=n; i=i+1, R=dR*R ) R];
|
||||
|
||||
module Q_Rotation_path(q1, n=1, q2) {
|
||||
for(Mi=Q_Rotation_path(q1, n, q2))
|
||||
multmatrix(Mi)
|
||||
children();
|
||||
}
|
||||
|
||||
|
||||
// Function: Q_Nlerp()
|
||||
// Usage:
|
||||
// q = Q_Nlerp(q1, q2, u);
|
||||
// Description:
|
||||
// Returns a quaternion that is a normalized linear interpolation between two quaternions
|
||||
// when u is a number.
|
||||
// If u is a list of numbers, computes the interpolations for each value in the
|
||||
// list and returns the interpolated quaternions in a list.
|
||||
// The input quaternions don't need to be normalized.
|
||||
// Arguments:
|
||||
// q1 = The first quaternion. (u=0)
|
||||
// q2 = The second quaternion. (u=1)
|
||||
// u = A value (or a list of values), between 0 and 1, of the proportion(s) of each quaternion in the interpolation.
|
||||
// Example(3D): Giving `u` as a Scalar
|
||||
// a = QuatY(-135);
|
||||
// b = QuatXYZ([0,-30,30]);
|
||||
// for (u=[0:0.1:1])
|
||||
// Qrot(Q_Nlerp(a, b, u))
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
// Example(3D): Giving `u` as a Range
|
||||
// a = QuatZ(-135);
|
||||
// b = QuatXYZ([90,0,-45]);
|
||||
// for (q = Q_Nlerp(a, b, [0:0.1:1]))
|
||||
// Qrot(q) right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
function Q_Nlerp(q1,q2,u) =
|
||||
assert(is_finite(u) || is_range(u) || is_vector(u) ,
|
||||
"Invalid interpolation coefficient(s)" )
|
||||
assert(Q_is_quat(q1) && Q_is_quat(q2), "Invalid quaternion(s)" )
|
||||
assert( ! approx(norm(q1+q2),0), "Quaternions cannot be opposed" )
|
||||
let( q1 = Q_Normalize(q1),
|
||||
q2 = Q_Normalize(q2) )
|
||||
is_num(u)
|
||||
? _Qnorm((1-u)*q1 + u*q2 )
|
||||
: [for (ui=u) _Qnorm((1-ui)*q1 + ui*q2 ) ];
|
||||
|
||||
|
||||
// Function: Q_Squad()
|
||||
// Usage:
|
||||
// qn = Q_Squad(q1,q2,q3,q4,u);
|
||||
// Description:
|
||||
// Returns a quaternion that is a cubic spherical interpolation of the quaternions
|
||||
// q1 and q4 taking the other two quaternions, q2 and q3, as parameter of a cubic
|
||||
// on the sphere similar to the control points of a Bezier curve.
|
||||
// If u is a number, usually between 0 and 1, returns the quaternion that results
|
||||
// from the interpolation.
|
||||
// If u is a list of numbers, computes the interpolations for each value in the
|
||||
// list and returns the interpolated quaternions in a list.
|
||||
// The input quaternions don't need to be normalized.
|
||||
// Arguments:
|
||||
// q1 = The start quaternion. (u=0)
|
||||
// q1 = The first intermediate quaternion.
|
||||
// q2 = The second intermediate quaternion.
|
||||
// q4 = The end quaternion. (u=1)
|
||||
// u = A value (or a list of values), of the proportion(s) of each quaternion in the cubic interpolation.
|
||||
// Example(3D): Giving `u` as a Scalar
|
||||
// a = QuatY(-135);
|
||||
// b = QuatXYZ([-50,-50,120]);
|
||||
// c = QuatXYZ([-50,-40,30]);
|
||||
// d = QuatY(-45);
|
||||
// color("red"){
|
||||
// Qrot(b) right(80) cube([10,10,1]);
|
||||
// Qrot(c) right(80) cube([10,10,1]);
|
||||
// }
|
||||
// for (u=[0:0.05:1])
|
||||
// Qrot(Q_Squad(a, b, c, d, u))
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
// Example(3D): Giving `u` as a Range
|
||||
// a = QuatY(-135);
|
||||
// b = QuatXYZ([-50,-50,120]);
|
||||
// c = QuatXYZ([-50,-40,30]);
|
||||
// d = QuatY(-45);
|
||||
// for (q = Q_Squad(a, b, c, d, [0:0.05:1]))
|
||||
// Qrot(q) right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
function Q_Squad(q1,q2,q3,q4,u) =
|
||||
assert(is_finite(u) || is_range(u) || is_vector(u) ,
|
||||
"Invalid interpolation coefficient(s)" )
|
||||
is_num(u)
|
||||
? Q_Slerp( Q_Slerp(q1,q4,u), Q_Slerp(q2,q3,u), 2*u*(1-u))
|
||||
: [for(ui=u) Q_Slerp( Q_Slerp(q1,q4,ui), Q_Slerp(q2,q3,ui), 2*ui*(1-ui) ) ];
|
||||
|
||||
|
||||
// Function: Q_exp()
|
||||
// Usage:
|
||||
// q2 = Q_exp(q);
|
||||
// Description:
|
||||
// Returns the quaternion that is the exponential of the quaternion q in base e
|
||||
// The returned quaternion is usually not normalized.
|
||||
function Q_exp(q) =
|
||||
assert( is_vector(q,4), "Input is not a valid quaternion")
|
||||
let( nv = norm(_Qvec(q)) ) // q may be equal to zero here!
|
||||
exp(_Qreal(q))*Quat(_Qvec(q),2*nv);
|
||||
|
||||
|
||||
// Function: Q_ln()
|
||||
// Usage:
|
||||
// q2 = Q_ln(q);
|
||||
// Description:
|
||||
// Returns the quaternion that is the natural logarithm of the quaternion q.
|
||||
// The returned quaternion is usually not normalized and may be zero.
|
||||
function Q_ln(q) =
|
||||
assert(Q_is_quat(q), "Input is not a valid quaternion")
|
||||
let( nq = norm(q),
|
||||
nv = norm(_Qvec(q)) )
|
||||
approx(nv,0) ? _Qset([0,0,0] , ln(nq) ) :
|
||||
_Qset(_Qvec(q)*atan2(nv,_Qreal(q))/nv, ln(nq));
|
||||
|
||||
|
||||
// Function: Q_pow()
|
||||
// Usage:
|
||||
// q2 = Q_pow(q, r);
|
||||
// Description:
|
||||
// Returns the quaternion that is the power of the quaternion q to the real exponent r.
|
||||
// The returned quaternion is normalized if `q` is normalized.
|
||||
function Q_pow(q,r=1) =
|
||||
assert( Q_is_quat(q) && is_finite(r),
|
||||
"Invalid inputs")
|
||||
let( theta = 2*atan2(norm(_Qvec(q)),_Qreal(q)) )
|
||||
Quat(_Qvec(q), r*theta); // Q_exp(r*Q_ln(q));
|
||||
|
||||
|
||||
|
||||
// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
|
||||
|
|
|
|||
|
|
@ -8,6 +8,7 @@ function rec_cmp(a,b,eps=1e-9) =
|
|||
is_list(a)? len(a)==len(b) && all([for (i=idx(a)) rec_cmp(a[i],b[i],eps=eps)]) :
|
||||
a == b;
|
||||
|
||||
function Qstandard(q) = sign([for(qi=q) if( ! approx(qi,0)) qi,0 ][0])*q;
|
||||
|
||||
module verify_f(actual,expected) {
|
||||
if (!rec_cmp(actual,expected)) {
|
||||
|
|
@ -22,6 +23,15 @@ module verify_f(actual,expected) {
|
|||
}
|
||||
|
||||
|
||||
module test_is_quat() {
|
||||
verify_f(Q_is_quat([0]),false);
|
||||
verify_f(Q_is_quat([0,0,0,0]),false);
|
||||
verify_f(Q_is_quat([1,0,2,0]),true);
|
||||
verify_f(Q_is_quat([1,0,2,0,0]),false);
|
||||
}
|
||||
test_is_quat();
|
||||
|
||||
|
||||
module test_Quat() {
|
||||
verify_f(Quat(UP,0),[0,0,0,1]);
|
||||
verify_f(Quat(FWD,0),[0,0,0,1]);
|
||||
|
|
@ -92,6 +102,15 @@ module test_QuatXYZ() {
|
|||
test_QuatXYZ();
|
||||
|
||||
|
||||
module test_Q_From_to() {
|
||||
verify_f(Q_Mul(Q_From_to([1,2,3], [4,5,2]),Q_From_to([4,5,2], [1,2,3])), Q_Ident());
|
||||
verify_f(Q_Matrix4(Q_From_to([1,2,3], [4,5,2])), rot(from=[1,2,3],to=[4,5,2]));
|
||||
verify_f(Qrot(Q_From_to([1,2,3], -[1,2,3]),[1,2,3]), -[1,2,3]);
|
||||
verify_f(unit(Qrot(Q_From_to([1,2,3], [4,5,2]),[1,2,3])), unit([4,5,2]));
|
||||
}
|
||||
test_Q_From_to();
|
||||
|
||||
|
||||
module test_Q_Ident() {
|
||||
verify_f(Q_Ident(), [0,0,0,1]);
|
||||
}
|
||||
|
|
@ -207,6 +226,16 @@ module test_Q_Conj() {
|
|||
test_Q_Conj();
|
||||
|
||||
|
||||
module test_Q_Inverse() {
|
||||
|
||||
verify_f(Q_Inverse([1,0,0,1]),[-1,0,0,1]/sqrt(2));
|
||||
verify_f(Q_Inverse([0,1,1,0]),[0,-1,-1,0]/sqrt(2));
|
||||
verify_f(Q_Inverse(QuatXYZ([23,45,67])),Q_Conj(QuatXYZ([23,45,67])));
|
||||
verify_f(Q_Mul(Q_Inverse(QuatXYZ([23,45,67])),QuatXYZ([23,45,67])),Q_Ident());
|
||||
}
|
||||
test_Q_Inverse();
|
||||
|
||||
|
||||
module test_Q_Norm() {
|
||||
verify_f(Q_Norm([1,0,0,1]),1.414213562);
|
||||
verify_f(Q_Norm([0,1,1,0]),1.414213562);
|
||||
|
|
@ -276,6 +305,10 @@ module test_Q_Angle() {
|
|||
verify_f(Q_Angle(QuatY(-37)),37);
|
||||
verify_f(Q_Angle(QuatZ(37)),37);
|
||||
verify_f(Q_Angle(QuatZ(-37)),37);
|
||||
|
||||
verify_f(Q_Angle(QuatZ(-37),QuatZ(-37)), 0);
|
||||
verify_f(Q_Angle(QuatZ( 37.123),QuatZ(-37.123)), 74.246);
|
||||
verify_f(Q_Angle(QuatX( 37),QuatY(-37)), 51.86293283);
|
||||
}
|
||||
test_Q_Angle();
|
||||
|
||||
|
|
@ -288,4 +321,87 @@ module test_Qrot() {
|
|||
test_Qrot();
|
||||
|
||||
|
||||
module test_Q_Rotation() {
|
||||
verify_f(Qstandard(Q_Rotation(Q_Matrix3(Quat([12,34,56],33)))),Qstandard(Quat([12,34,56],33)));
|
||||
verify_f(Q_Matrix3(Q_Rotation(Q_Matrix3(QuatXYZ([12,34,56])))),
|
||||
Q_Matrix3(QuatXYZ([12,34,56])));
|
||||
}
|
||||
test_Q_Rotation();
|
||||
|
||||
|
||||
module test_Q_Rotation_path() {
|
||||
|
||||
verify_f(Q_Rotation_path(QuatX(135), 5, QuatY(13.5))[0] , Q_Matrix4(QuatX(135)));
|
||||
verify_f(Q_Rotation_path(QuatX(135), 11, QuatY(13.5))[11] , yrot(13.5));
|
||||
verify_f(Q_Rotation_path(QuatX(135), 16, QuatY(13.5))[8] , Q_Rotation_path(QuatX(135), 8, QuatY(13.5))[4]);
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verify_f(Q_Rotation_path(QuatX(135), 16, QuatY(13.5))[7] ,
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Q_Rotation_path(QuatY(13.5),16, QuatX(135))[9]);
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verify_f(Q_Rotation_path(QuatX(11), 5)[0] , xrot(11));
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verify_f(Q_Rotation_path(QuatX(11), 5)[4] , xrot(55));
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|
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}
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test_Q_Rotation_path();
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||||
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||||
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module test_Q_Nlerp() {
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verify_f(Q_Nlerp(QuatX(45),QuatY(30),0.0),QuatX(45));
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verify_f(Q_Nlerp(QuatX(45),QuatY(30),0.5),[0.1967063121, 0.1330377423, 0, 0.9713946602]);
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verify_f(Q_Rotation_path(QuatX(135), 16, QuatY(13.5))[8] , Q_Matrix4(Q_Nlerp(QuatX(135), QuatY(13.5),0.5)));
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verify_f(Q_Nlerp(QuatX(45),QuatY(30),1.0),QuatY(30));
|
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}
|
||||
test_Q_Nlerp();
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|
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|
||||
module test_Q_Squad() {
|
||||
verify_f(Q_Squad(QuatX(45),QuatZ(30),QuatX(90),QuatY(30),0.0),QuatX(45));
|
||||
verify_f(Q_Squad(QuatX(45),QuatZ(30),QuatX(90),QuatY(30),1.0),QuatY(30));
|
||||
verify_f(Q_Squad(QuatX(0),QuatX(30),QuatX(90),QuatX(120),0.5),
|
||||
Q_Slerp(QuatX(0),QuatX(120),0.5));
|
||||
verify_f(Q_Squad(QuatY(0),QuatY(0),QuatX(120),QuatX(120),0.3),
|
||||
Q_Slerp(QuatY(0),QuatX(120),0.3));
|
||||
}
|
||||
test_Q_Squad();
|
||||
|
||||
|
||||
module test_Q_exp() {
|
||||
verify_f(Q_exp(Q_Ident()), exp(1)*Q_Ident());
|
||||
verify_f(Q_exp([0,0,0,33.7]), exp(33.7)*Q_Ident());
|
||||
verify_f(Q_exp(Q_ln(Q_Ident())), Q_Ident());
|
||||
verify_f(Q_exp(Q_ln([1,2,3,0])), [1,2,3,0]);
|
||||
verify_f(Q_exp(Q_ln(QuatXYZ([31,27,34]))), QuatXYZ([31,27,34]));
|
||||
let(q=QuatXYZ([12,23,34]))
|
||||
verify_f(Q_exp(q+Q_Inverse(q)),Q_Mul(Q_exp(q),Q_exp(Q_Inverse(q))));
|
||||
|
||||
}
|
||||
test_Q_exp();
|
||||
|
||||
|
||||
module test_Q_ln() {
|
||||
verify_f(Q_ln([1,2,3,0]), [24.0535117721, 48.1070235442, 72.1605353164, 1.31952866481]);
|
||||
verify_f(Q_ln(Q_Ident()), [0,0,0,0]);
|
||||
verify_f(Q_ln(5.5*Q_Ident()), [0,0,0,ln(5.5)]);
|
||||
verify_f(Q_ln(Q_exp(QuatXYZ([13,37,43]))), QuatXYZ([13,37,43]));
|
||||
verify_f(Q_ln(QuatXYZ([12,23,34]))+Q_ln(Q_Inverse(QuatXYZ([12,23,34]))), [0,0,0,0]);
|
||||
}
|
||||
test_Q_ln();
|
||||
|
||||
|
||||
module test_Q_pow() {
|
||||
q = Quat([1,2,3],77);
|
||||
verify_f(Q_pow(q,1), q);
|
||||
verify_f(Q_pow(q,0), Q_Ident());
|
||||
verify_f(Q_pow(q,-1), Q_Inverse(q));
|
||||
verify_f(Q_pow(q,2), Q_Mul(q,q));
|
||||
verify_f(Q_pow(q,3), Q_Mul(q,Q_pow(q,2)));
|
||||
verify_f(Q_Mul(Q_pow(q,0.456),Q_pow(q,0.544)), q);
|
||||
verify_f(Q_Mul(Q_pow(q,0.335),Q_Mul(Q_pow(q,.552),Q_pow(q,.113))), q);
|
||||
}
|
||||
test_Q_pow();
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap
|
||||
|
|
|
|||
Loading…
Reference in a new issue