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#!/usr/bin/python
# 3dproj.py
#
"""
Various transforms used for by the 3D code
"""
from matplotlib.collections import LineCollection
from matplotlib.patches import Circle
import numpy as np
import numpy.linalg as linalg
def line2d(p0, p1):
"""
Return 2D equation of line in the form ax+by+c = 0
"""
# x + x1 = 0
x0, y0 = p0[:2]
x1, y1 = p1[:2]
#
if x0 == x1:
a = -1
b = 0
c = x1
elif y0 == y1:
a = 0
b = 1
c = -y1
else:
a = (y0-y1)
b = (x0-x1)
c = (x0*y1 - x1*y0)
return a, b, c
def line2d_dist(l, p):
"""
Distance from line to point
line is a tuple of coefficients a,b,c
"""
a, b, c = l
x0, y0 = p
return abs((a*x0 + b*y0 + c)/np.sqrt(a**2+b**2))
def line2d_seg_dist(p1, p2, p0):
"""distance(s) from line defined by p1 - p2 to point(s) p0
p0[0] = x(s)
p0[1] = y(s)
intersection point p = p1 + u*(p2-p1)
and intersection point lies within segement if u is between 0 and 1
"""
x21 = p2[0] - p1[0]
y21 = p2[1] - p1[1]
x01 = np.asarray(p0[0]) - p1[0]
y01 = np.asarray(p0[1]) - p1[1]
u = (x01*x21 + y01*y21)/float(abs(x21**2 + y21**2))
u = np.clip(u, 0, 1)
d = np.sqrt((x01 - u*x21)**2 + (y01 - u*y21)**2)
return d
def test_lines_dists():
import pylab
ax = pylab.gca()
xs, ys = (0,30), (20,150)
pylab.plot(xs, ys)
points = zip(xs, ys)
p0, p1 = points
xs, ys = (0,0,20,30), (100,150,30,200)
pylab.scatter(xs, ys)
dist = line2d_seg_dist(p0, p1, (xs[0], ys[0]))
dist = line2d_seg_dist(p0, p1, np.array((xs, ys)))
for x, y, d in zip(xs, ys, dist):
c = Circle((x, y), d, fill=0)
ax.add_patch(c)
pylab.xlim(-200, 200)
pylab.ylim(-200, 200)
pylab.show()
def mod(v):
"""3d vector length"""
return np.sqrt(v[0]**2+v[1]**2+v[2]**2)
def world_transformation(xmin, xmax,
ymin, ymax,
zmin, zmax):
dx, dy, dz = (xmax-xmin), (ymax-ymin), (zmax-zmin)
return np.array([
[1.0/dx,0,0,-xmin/dx],
[0,1.0/dy,0,-ymin/dy],
[0,0,1.0/dz,-zmin/dz],
[0,0,0,1.0]])
def test_world():
xmin, xmax = 100, 120
ymin, ymax = -100, 100
zmin, zmax = 0.1, 0.2
M = world_transformation(xmin, xmax, ymin, ymax, zmin, zmax)
print M
def view_transformation(E, R, V):
n = (E - R)
## new
# n /= mod(n)
# u = np.cross(V,n)
# u /= mod(u)
# v = np.cross(n,u)
# Mr = np.diag([1.]*4)
# Mt = np.diag([1.]*4)
# Mr[:3,:3] = u,v,n
# Mt[:3,-1] = -E
## end new
## old
n = n / mod(n)
u = np.cross(V, n)
u = u / mod(u)
v = np.cross(n, u)
Mr = [[u[0],u[1],u[2],0],
[v[0],v[1],v[2],0],
[n[0],n[1],n[2],0],
[0, 0, 0, 1],
]
#
Mt = [[1, 0, 0, -E[0]],
[0, 1, 0, -E[1]],
[0, 0, 1, -E[2]],
[0, 0, 0, 1]]
## end old
return np.dot(Mr, Mt)
def persp_transformation(zfront, zback):
a = (zfront+zback)/(zfront-zback)
b = -2*(zfront*zback)/(zfront-zback)
return np.array([[1,0,0,0],
[0,1,0,0],
[0,0,a,b],
[0,0,-1,0]
])
def proj_transform_vec(vec, M):
vecw = np.dot(M, vec)
w = vecw[3]
# clip here..
txs, tys, tzs = vecw[0]/w, vecw[1]/w, vecw[2]/w
return txs, tys, tzs
def proj_transform_vec_clip(vec, M):
vecw = np.dot(M, vec)
w = vecw[3]
# clip here..
txs, tys, tzs = vecw[0]/w, vecw[1]/w, vecw[2]/w
tis = (vecw[0] >= 0) * (vecw[0] <= 1) * (vecw[1] >= 0) * (vecw[1] <= 1)
if np.sometrue(tis):
tis = vecw[1] < 1
return txs, tys, tzs, tis
def inv_transform(xs, ys, zs, M):
iM = linalg.inv(M)
vec = vec_pad_ones(xs, ys, zs)
vecr = np.dot(iM, vec)
try:
vecr = vecr/vecr[3]
except OverflowError:
pass
return vecr[0], vecr[1], vecr[2]
def vec_pad_ones(xs, ys, zs):
try:
try:
vec = np.array([xs,ys,zs,np.ones(xs.shape)])
except (AttributeError,TypeError):
vec = np.array([xs,ys,zs,np.ones((len(xs)))])
except TypeError:
vec = np.array([xs,ys,zs,1])
return vec
def proj_transform(xs, ys, zs, M):
"""
Transform the points by the projection matrix
"""
vec = vec_pad_ones(xs, ys, zs)
return proj_transform_vec(vec, M)
def proj_transform_clip(xs, ys, zs, M):
"""
Transform the points by the projection matrix
and return the clipping result
returns txs,tys,tzs,tis
"""
vec = vec_pad_ones(xs, ys, zs)
return proj_transform_vec_clip(vec, M)
transform = proj_transform
def proj_points(points, M):
return zip(*proj_trans_points(points, M))
def proj_trans_points(points, M):
xs, ys, zs = zip(*points)
return proj_transform(xs, ys, zs, M)
def proj_trans_clip_points(points, M):
xs, ys, zs = zip(*points)
return proj_transform_clip(xs, ys, zs, M)
def test_proj_draw_axes(M, s=1):
import pylab
xs, ys, zs = [0, s, 0, 0], [0, 0, s, 0], [0, 0, 0, s]
txs, tys, tzs = proj_transform(xs, ys, zs, M)
o, ax, ay, az = (txs[0], tys[0]), (txs[1], tys[1]), \
(txs[2], tys[2]), (txs[3], tys[3])
lines = [(o, ax), (o, ay), (o, az)]
ax = pylab.gca()
linec = LineCollection(lines)
ax.add_collection(linec)
for x, y, t in zip(txs, tys, ['o', 'x', 'y', 'z']):
pylab.text(x, y, t)
def test_proj_make_M(E=None):
# eye point
E = E or np.array([1, -1, 2]) * 1000
#E = np.array([20,10,20])
R = np.array([1, 1, 1]) * 100
V = np.array([0, 0, 1])
viewM = view_transformation(E, R, V)
perspM = persp_transformation(100, -100)
M = np.dot(perspM, viewM)
return M
def test_proj():
import pylab
M = test_proj_make_M()
ts = ['%d' % i for i in [0,1,2,3,0,4,5,6,7,4]]
xs, ys, zs = [0,1,1,0,0, 0,1,1,0,0], [0,0,1,1,0, 0,0,1,1,0], \
[0,0,0,0,0, 1,1,1,1,1]
xs, ys, zs = [np.array(v)*300 for v in (xs, ys, zs)]
#
test_proj_draw_axes(M, s=400)
txs, tys, tzs = proj_transform(xs, ys, zs, M)
ixs, iys, izs = inv_transform(txs, tys, tzs, M)
pylab.scatter(txs, tys, c=tzs)
pylab.plot(txs, tys, c='r')
for x, y, t in zip(txs, tys, ts):
pylab.text(x, y, t)
pylab.xlim(-0.2, 0.2)
pylab.ylim(-0.2, 0.2)
pylab.show()
def rot_x(V, alpha):
cosa, sina = np.cos(alpha), np.sin(alpha)
M1 = np.array([[1,0,0,0],
[0,cosa,-sina,0],
[0,sina,cosa,0],
[0,0,0,0]])
return np.dot(M1, V)
def test_rot():
V = [1,0,0,1]
print rot_x(V, np.pi/6)
V = [0,1,0,1]
print rot_x(V, np.pi/6)
if __name__ == "__main__":
test_proj()