# Copyright 2004-2006 Nanorex, Inc. See LICENSE file for details.
"""Vectors, Quaternions, and Trackballs
Vectors are a simplified interface to the Numeric arrays.
A relatively full implementation of Quaternions.
Trackball produces incremental quaternions using a mapping of the screen
onto a sphere, tracking the cursor on the sphere.
"""
import math, types
from math import *
from Numeric import *
from LinearAlgebra import *
intType = type(2)
floType = type(2.0)
numTypes = [intType, floType]
def V(*v): return array(v, Float)
def A(a): return array(a, Float)
def cross(v1, v2):
return V(v1[1]*v2[2] - v1[2]*v2[1],
v1[2]*v2[0] - v1[0]*v2[2],
v1[0]*v2[1] - v1[1]*v2[0])
def vlen(v1): return sqrt(dot(v1, v1))
def norm(v1):
lng = vlen(v1)
if lng:
return v1 / lng
# bruce 041012 optimized this by using lng instead of
# recomputing vlen(v1) -- code was v1 / vlen(v1)
else: return v1+0
# p1 and p2 are points, v1 is a direction vector from p1.
# return (dist, wid) where dist is the distance from p1 to p2
# measured in the direction of v1, and wid is the orthogonal
# distance from p2 to the p1-v1 line.
# v1 should be a unit vector.
def orthodist(p1, v1, p2):
dist = dot(v1, p2-p1)
wid = vlen(p1+dist*v1-p2)
return (dist, wid)
class Q:
"""Q(W, x, y, z) is the quaternion with axis vector x,y,z
and sin(theta/2) = W
(e.g. Q(1,0,0,0) is no rotation)
Q(x, y, z) where x, y, and z are three orthonormal vectors
is the quaternion that rotates the standard axes into that
reference frame. (the frame has to be right handed, or there's
no quaternion that can do it!)
Q(V(x,y,z), theta) is what you probably want.
Q(vector, vector) gives the quat that rotates between them
"""
def __init__(self, x, y=None, z=None, w=None):
# 4 numbers
if w != None: self.vec=V(x,y,z,w)
elif z: # three axis vectors
# Just use first two
a100 = V(1,0,0)
c1 = cross(a100,x)
if vlen(c1)<0.000001:
self.vec = Q(y,z).vec
return
ax1 = norm((a100+x)/2.0)
x2 = cross(ax1,c1)
a010 = V(0,1,0)
c2 = cross(a010,y)
if vlen(c2)<0.000001:
self.vec = Q(x,z).vec
return
ay1 = norm((a010+y)/2.0)
y2 = cross(ay1,c2)
axis = cross(x2, y2)
nw = sqrt(1.0 + x[0] + y[1] + z[2])/2.0
axis = norm(axis)*sqrt(1.0-nw**2)
self.vec = V(nw, axis[0], axis[1], axis[2])
elif type(y) in numTypes:
# axis vector and angle
v = (x / vlen(x)) * sin(y*0.5)
self.vec = V(cos(y*0.5), v[0], v[1], v[2])
elif y:
# rotation between 2 vectors
x = norm(x)
y = norm(y)
v = cross(x, y)
theta = acos(min(1.0,max(-1.0,dot(x, y))))
if dot(y, cross(x, v)) > 0.0:
theta = 2.0 * pi - theta
w=cos(theta*0.5)
vl = vlen(v)
# null rotation
if w==1.0: self.vec=V(1, 0, 0, 0)
# opposite pole
elif vl<0.000001:
ax1 = cross(x,V(1,0,0))
ax2 = cross(x,V(0,1,0))
if vlen(ax1)>vlen(ax2):
self.vec = norm(V(0, ax1[0],ax1[1],ax1[2]))
else:
self.vec = norm(V(0, ax2[0],ax2[1],ax2[2]))
else:
s=sqrt(1-w**2)/vl
self.vec=V(w, v[0]*s, v[1]*s, v[2]*s)
elif type(x) in numTypes:
# just one number
self.vec=V(1, 0, 0, 0)
else:
self.vec=V(x[0], x[1], x[2], x[3])
self.counter = 50
def __getattr__(self, name):
if name == 'w':
return self.vec[0]
elif name in ('x', 'i'):
return self.vec[1]
elif name in ('y', 'j'):
return self.vec[2]
elif name in ('z', 'k'):
return self.vec[3]
elif name == 'angle':
if -1.0<self.vec[0]<1.0: return 2.0*acos(self.vec[0])
else: return 0.0
elif name == 'axis':
return V(self.vec[1], self.vec[2], self.vec[3])
elif name == 'matrix':
# this the transpose of the normal form
# so we can use it on matrices of row vectors
self.__dict__['matrix'] = array([\
[1.0 - 2.0*(self.y**2 + self.z**2),
2.0*(self.x*self.y + self.z*self.w),
2.0*(self.z*self.x - self.y*self.w)],
[2.0*(self.x*self.y - self.z*self.w),
1.0 - 2.0*(self.z**2 + self.x**2),
2.0*(self.y*self.z + self.x*self.w)],
[2.0*(self.z*self.x + self.y*self.w),
2.0*(self.y*self.z - self.x*self.w),
1.0 - 2.0 * (self.y**2 + self.x**2)]])
return self.__dict__['matrix']
else:
raise AttributeError, 'No "%s" in Quaternion' % name
def __getitem__(self, num):
return self.vec[num]
def setangle(self, theta):
"""Set the quaternion's rotation to theta (destructive modification).
(In the same direction as before.)
"""
theta = remainder(theta/2.0, pi)
self.vec[1:] = norm(self.vec[1:]) * sin(theta)
self.vec[0] = cos(theta)
self.__reset()
return self
def __reset(self):
if self.__dict__.has_key('matrix'):
del self.__dict__['matrix']
def __setattr__(self, name, value):
if name=="w": self.vec[0] = value
elif name=="x": self.vec[1] = value
elif name=="y": self.vec[2] = value
elif name=="z": self.vec[3] = value
else: self.__dict__[name] = value
def __len__(self):
return 4
def __add__(self, q1):
"""Q + Q1 is the quaternion representing the rotation achieved
by doing Q and then Q1.
"""
return Q(q1.w*self.w - q1.x*self.x - q1.y*self.y - q1.z*self.z,
q1.w*self.x + q1.x*self.w + q1.y*self.z - q1.z*self.y,
q1.w*self.y - q1.x*self.z + q1.y*self.w + q1.z*self.x,
q1.w*self.z + q1.x*self.y - q1.y*self.x + q1.z*self.w)
def __iadd__(self, q1):
"""this is self += q1
"""
temp=V(q1.w*self.w - q1.x*self.x - q1.y*self.y - q1.z*self.z,
q1.w*self.x + q1.x*self.w + q1.y*self.z - q1.z*self.y,
q1.w*self.y - q1.x*self.z + q1.y*self.w + q1.z*self.x,
q1.w*self.z + q1.x*self.y - q1.y*self.x + q1.z*self.w)
self.vec=temp
self.counter -= 1
if self.counter <= 0:
self.counter = 50
self.normalize()
self.__reset()
return self
def __sub__(self, q1):
return self + (-q1)
def __isub__(self, q1):
return __iadd__(self, -q1)
def __mul__(self, n):
"""multiplication by a scalar, i.e. Q1 * 1.3, defined so that
e.g. Q1 * 2 == Q1 + Q1, or Q1 = Q1*0.5 + Q1*0.5
Python syntax makes it hard to do n * Q, unfortunately.
"""
if type(n) in numTypes:
nq = +self
nq.setangle(n*self.angle)
return nq
else:
raise MulQuat
def __imul__(self, q2):
if type(n) in numTypes:
self.setangle(n*self.angle)
self.__reset()
return self
else:
raise MulQuat
def __div__(self, q2):
return self*q2.conj()*(1.0/(q2*q2.conj()).w)
def __repr__(self):
return 'Q(%g, %g, %g, %g)' % (self.w, self.x, self.y, self.z)
def __str__(self):
a= "<q:%6.2f @ " % (2.0*acos(self.w)*180/pi)
l = sqrt(self.x**2 + self.y**2 + self.z**2)
if l:
z=V(self.x, self.y, self.z)/l
a += "[%4.3f, %4.3f, %4.3f] " % (z[0], z[1], z[2])
else: a += "[%4.3f, %4.3f, %4.3f] " % (self.x, self.y, self.z)
a += "|%8.6f|>" % vlen(self.vec)
return a
def __pos__(self):
return Q(self.w, self.x, self.y, self.z)
def __neg__(self):
return Q(self.w, -self.x, -self.y, -self.z)
def conj(self):
return Q(self.w, -self.x, -self.y, -self.z)
def normalize(self):
w=self.vec[0]
v=V(self.vec[1],self.vec[2],self.vec[3])
length = vlen(v)
if length:
s=sqrt(1.0-w**2)/length
self.vec = V(w, v[0]*s, v[1]*s, v[2]*s)
else: self.vec = V(1,0,0,0)
return self
def unrot(self,v):
return matrixmultiply(self.matrix,v)
def vunrot(self,v):
# for use with row vectors
return matrixmultiply(v,transpose(self.matrix))
def rot(self,v):
return matrixmultiply(v,self.matrix)
def twistor(axis, pt1, pt2):
"""return the quaternion that, rotating around axis, will bring
pt1 closest to pt2.
"""
q = Q(axis, V(0,0,1))
pt1 = q.rot(pt1)
pt2 = q.rot(pt2)
a1 = atan2(pt1[1],pt1[0])
a2 = atan2(pt2[1],pt2[0])
theta = a2-a1
return Q(axis, theta)
# project a point from a tangent plane onto a unit sphere
def proj2sphere(x, y):
d = sqrt(x*x + y*y)
theta = pi * 0.5 * d
s=sin(theta)
if d>0.0001: return V(s*x/d, s*y/d, cos(theta))
else: return V(0.0, 0.0, 1.0)
class Trackball:
'''A trackball object. The current transformation matrix
can be retrieved using the "matrix" attribute.'''
def __init__(self, wide, high):
'''Create a Trackball object.
"size" is the radius of the inner trackball
sphere. '''
self.w2=wide/2.0
self.h2=high/2.0
self.scale = 1.1 / min(wide/2.0, high/2.0)
self.quat = Q(1,0,0,0)
self.oldmouse = None
def rescale(self, wide, high):
self.w2=wide/2.0
self.h2=high/2.0
self.scale = 1.1 / min(wide/2.0, high/2.0)
def start(self, px, py):
self.oldmouse=proj2sphere((px-self.w2)*self.scale,
(self.h2-py)*self.scale)
def update(self, px, py, uq=None):
newmouse = proj2sphere((px-self.w2)*self.scale,
(self.h2-py)*self.scale)
if self.oldmouse and not uq:
quat = Q(self.oldmouse, newmouse)
elif self.oldmouse and uq:
quat = uq + Q(self.oldmouse, newmouse) - uq
else:
quat = Q(1,0,0,0)
self.oldmouse = newmouse
return quat
def ptonline(xpt, lpt, ldr):
"""return the point on a line (point lpt, direction ldr)
nearest to point xpt
"""
ldr = norm(ldr)
return dot(xpt-lpt,ldr)*ldr + lpt
def planeXline(ppt, pv, lpt, lv):
"""find the intersection of a line (point lpt, vector lv)
with a plane (point ppt, normal pv)
return None if (almost) parallel
"""
d=dot(lv,pv)
if abs(d)<0.000001: return None
return lpt+lv*(dot(ppt-lpt,pv)/d)
def cat(a,b):
"""concatenate two arrays (the NumPy version is a mess)
"""
if not a: return b
if not b: return a
r1 = shape(a)
r2 = shape(b)
if len(r1) == len(r2): return concatenate((a,b))
if len(r1)<len(r2):
return concatenate((reshape(a,(1,)+r1), b))
else: return concatenate((a,reshape(b,(1,)+r2)))
def Veq(v1, v2):
"tells if v1 is all equal to v2"
return logical_and.reduce(v1==v2)
__author__ = "Josh"