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Removed quaternions.scad
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Garth Minette
parent
5780671578
commit
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2 changed files with 0 additions and 662 deletions
661
quaternions.scad
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quaternions.scad
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///////////////////////////////////////////
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// LibFile: quaternions.scad
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// Support for Quaternions.
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// Includes:
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// include <BOSL2/std.scad>
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// FileGroup: Math
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// FileSummary: Quaternion based rotations that avoid gimbal lock issues.
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// FileFootnotes: STD=Included in std.scad
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///////////////////////////////////////////
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// Section: Quaternions
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// Quaternions are fast methods of storing and calculating arbitrary rotations.
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// Quaternions contain information on both axis of rotation, and rotation angle.
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// You can chain multiple rotation together by multiplying quaternions together.
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// They don't suffer from the gimbal-lock issues that `[X,Y,Z]` rotation angles do.
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// Quaternions are stored internally as a 4-value vector:
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// `[X,Y,Z,W]`, where the quaternion formula is `W+Xi+Yj+Zk`
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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: is_quaternion()
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// Usage:
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// if(is_quaternion(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 is_quaternion(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 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) =
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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: quat_x()
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// Usage:
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// quat_x(a);
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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 quat_x(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: quat_y()
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// Usage:
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// quat_y(a);
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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 quat_y(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: quat_z()
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// Usage:
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// quat_z(a);
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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 quat_z(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: quat_xyz()
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// Usage:
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// quat_xyz([X,Y,Z])
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// Description:
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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 quat_xyz(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 = quat_x(a[0]),
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qy = quat_y(a[1]),
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qz = quat_z(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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// Function: q_add_s()
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// Usage:
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// q_add_s(q, 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:
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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:
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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:
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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:
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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( is_quaternion(a) && is_quaternion(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:
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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( is_quaternion(a) && is_quaternion(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:
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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( is_quaternion(a) && is_quaternion(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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// Usage:
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// q_cumulative(v);
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// Description:
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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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v, _i+1,
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concat(
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_acc,
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[_i==0 ? v[_i] : q_mul(v[_i], last(_acc))]
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)
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);
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// Function: q_dot()
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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) =
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assert( is_quaternion(a) && is_quaternion(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) =
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assert( is_quaternion(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) =
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assert( is_quaternion(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( is_quaternion(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:
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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( is_quaternion(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) =
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assert( is_quaternion(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) =
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assert( is_quaternion(q1) && is_quaternion(q2), "Invalid quaternion(s)" )
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norm(q2-q1);
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// Function: q_slerp()
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// Usage:
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// q_slerp(q1, q2, u);
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// Description:
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// Returns a quaternion that is a spherical interpolation between two quaternions.
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// Arguments:
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// q1 = The first quaternion. (u=0)
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// q2 = The second quaternion. (u=1)
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// u = The proportional value, from 0 to 1, of what part of the interpolation to return.
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// Example(3D): Giving `u` as a Scalar
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// a = quat_y(-135);
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// b = quat_xyz([0,-30,30]);
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// for (u=[0:0.1:1])
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// q_rot(q_slerp(a, b, u))
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// right(80) cube([10,10,1]);
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// #sphere(r=80);
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// Example(3D): Giving `u` as a Range
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// a = quat_z(-135);
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// b = quat_xyz([90,0,-45]);
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// for (q = q_slerp(a, b, [0:0.1:1]))
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// q_rot(q) right(80) cube([10,10,1]);
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// #sphere(r=80);
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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(is_quaternion(q1) && is_quaternion(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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// Usage:
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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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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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// Usage:
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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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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], 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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];
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// Function: q_axis()
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// Usage:
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// q_axis(q)
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// Description:
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// Returns the axis of rotation of a normalized quaternion `q`.
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// The input doesn't need to be normalized.
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function q_axis(q) =
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assert( is_quaternion(q) , "Invalid quaternion" )
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let( d = norm(_qvec(q)) )
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approx(d,0)? [0,0,1] : _qvec(q)/d;
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// Function: q_angle()
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// Usage:
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// a = q_angle(q)
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// a12 = q_angle(q1,q2);
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// Description:
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// If only q1 is given, returns the angle of rotation (in degrees) of that quaternion.
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// If both q1 and q2 are given, returns the angle (in degrees) between them.
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// The input quaternions don't need to be normalized.
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function q_angle(q1,q2) =
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assert(is_quaternion(q1) && (is_undef(q2) || is_quaternion(q2)), "Invalid quaternion(s)" )
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let( n1 = is_undef(q2)? norm(_qvec(q1)): norm(q1) )
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is_undef(q2)
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? 2 * atan2(n1,_qreal(q1))
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: let( q1 = q1/norm(q1),
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q2 = q2/norm(q2) )
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4 * atan2(norm(q1 - q2), norm(q1 + q2));
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// Function&Module: q_rot()
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// Usage: As Module
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// q_rot(q) ...
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// Usage: As Function
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// pts = q_rot(q,p);
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// Description:
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// When called as a module, rotates all children by the rotation stored in quaternion `q`.
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// When called as a function with a `p` argument, rotates the point or list of points in `p` by the rotation stored in quaternion `q`.
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// When called as a function without a `p` argument, returns the affine3d rotation matrix for the rotation stored in quaternion `q`.
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// Example(FlatSpin,VPD=225,VPT=[71,-26,16]):
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// module shape() translate([80,0,0]) cube([10,10,1]);
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// q = quat_xyz([90,-15,-45]);
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// q_rot(q) shape();
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// #shape();
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// Example(NORENDER):
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// q = quat_xyz([45,35,10]);
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// mat4x4 = q_rot(q);
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// Example(NORENDER):
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// q = quat_xyz([45,35,10]);
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// pt = q_rot(q, p=[4,5,6]);
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||||
// Example(NORENDER):
|
||||
// q = quat_xyz([45,35,10]);
|
||||
// pts = q_rot(q, p=[[2,3,4], [4,5,6], [9,2,3]]);
|
||||
module q_rot(q) {
|
||||
multmatrix(q_matrix4(q)) {
|
||||
children();
|
||||
}
|
||||
}
|
||||
|
||||
function q_rot(q,p) =
|
||||
is_undef(p)? q_matrix4(q) :
|
||||
is_vector(p)? q_rot(q,[p])[0] :
|
||||
apply(q_matrix4(q), p);
|
||||
|
||||
|
||||
// Module: q_rot_copies()
|
||||
// Usage:
|
||||
// q_rot_copies(quats) ...
|
||||
// Description:
|
||||
// For each quaternion given in the list `quats`, rotates to that orientation and creates a copy
|
||||
// of all children. This is equivalent to `for (q=quats) q_rot(q) ...`.
|
||||
// Arguments:
|
||||
// quats = A list containing all quaternions to rotate to and create copies of all children for.
|
||||
// Example:
|
||||
// a = quat_z(-135);
|
||||
// b = quat_xyz([0,-30,30]);
|
||||
// q_rot_copies(q_slerp(a, b, [0:0.1:1]))
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
module q_rot_copies(quats) for (q=quats) q_rot(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()
|
||||
// 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 = quat_y(-135);
|
||||
// b = quat_xyz([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 = quat_y(-135);
|
||||
// b = quat_xyz([0,-30,30]);
|
||||
// q_rotation_path(a, 10, b)
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
// Example(3D): as a function
|
||||
// a = quat_y(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 = quat_y(5);
|
||||
// q_rotation_path(a, 10)
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
function q_rotation_path(q1, n=1, q2) =
|
||||
assert( is_quaternion(q1) && (is_undef(q2) || is_quaternion(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 = quat_y(-135);
|
||||
// b = quat_xyz([0,-30,30]);
|
||||
// for (u=[0:0.1:1])
|
||||
// q_rot(q_nlerp(a, b, u))
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
// Example(3D): Giving `u` as a Range
|
||||
// a = quat_z(-135);
|
||||
// b = quat_xyz([90,0,-45]);
|
||||
// for (q = q_nlerp(a, b, [0:0.1:1]))
|
||||
// q_rot(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(is_quaternion(q1) && is_quaternion(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 = quat_y(-135);
|
||||
// b = quat_xyz([-50,-50,120]);
|
||||
// c = quat_xyz([-50,-40,30]);
|
||||
// d = quat_y(-45);
|
||||
// color("red"){
|
||||
// q_rot(b) right(80) cube([10,10,1]);
|
||||
// q_rot(c) right(80) cube([10,10,1]);
|
||||
// }
|
||||
// for (u=[0:0.05:1])
|
||||
// q_rot(q_squad(a, b, c, d, u))
|
||||
// right(80) cube([10,10,1]);
|
||||
// #sphere(r=80);
|
||||
// Example(3D): Giving `u` as a Range
|
||||
// a = quat_y(-135);
|
||||
// b = quat_xyz([-50,-50,120]);
|
||||
// c = quat_xyz([-50,-40,30]);
|
||||
// d = quat_y(-45);
|
||||
// for (q = q_squad(a, b, c, d, [0:0.05:1]))
|
||||
// q_rot(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(is_quaternion(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( is_quaternion(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
|
||||
1
std.scad
1
std.scad
|
|
@ -27,7 +27,6 @@ include <math.scad>
|
|||
include <linalg.scad>
|
||||
include <trigonometry.scad>
|
||||
include <vectors.scad>
|
||||
include <quaternions.scad>
|
||||
include <affine.scad>
|
||||
include <coords.scad>
|
||||
include <geometry.scad>
|
||||
|
|
|
|||
Loading…
Reference in a new issue