def _pointsOnEllipse(c, smaa, smia, ang, n):
"""Generates points lying on the ellipse with center c,
semi-major/semi-minor axis lengths smaa/smia, and major axis
angle (E of N) ang.
"""
points = []
c = g.cartesianUnitVector(c)
north, east = g.northEast(c)
sa = math.sin(math.radians(ang))
ca = math.cos(math.radians(ang))
smaa = math.radians(smaa * g.DEG_PER_ARCSEC)
smia = math.radians(smia * g.DEG_PER_ARCSEC)
aoff = random.uniform(0.0, 2.0 * math.pi)
for i in xrange(n):
a = aoff + 2.0 * i * math.pi / n
x = smaa * math.cos(a)
y = smia * math.sin(a)
# rotate x, y by a
nc = ca * x - sa * y
ec = sa * x + ca * y
cc = math.sqrt(abs(1.0 - nc * nc - ec * ec))
points.append(g.normalize((cc * c[0] + nc * north[0] + ec * east[0],
cc * c[1] + nc * north[1] + ec * east[1],
cc * c[2] + nc * north[2] + ec * east[2])))
return points
def _pointsOnEllipse(c, smaa, smia, ang, n):
"""Generates points lying on the ellipse with center c,
semi-major/semi-minor axis lengths smaa/smia, and major axis
angle (E of N) ang.
"""
points = []
c = g.cartesianUnitVector(c)
north, east = g.northEast(c)
sa = math.sin(math.radians(ang))
ca = math.cos(math.radians(ang))
smaa = math.radians(smaa * g.DEG_PER_ARCSEC)
smia = math.radians(smia * g.DEG_PER_ARCSEC)
aoff = random.uniform(0.0, 2.0 * math.pi)
for i in xrange(n):
a = aoff + 2.0 * i * math.pi / n
x = smaa * math.cos(a)
y = smia * math.sin(a)
# rotate x, y by a
nc = ca * x - sa * y
ec = sa * x + ca * y
cc = math.sqrt(abs(1.0 - nc * nc - ec * ec))
points.append(
g.normalize((cc * c[0] + nc * north[0] + ec * east[0],
cc * c[1] + nc * north[1] + ec * east[1],
cc * c[2] + nc * north[2] + ec * east[2])))
return points
def _hemPoints(v, n):
"""Randomly generates a list of n points in the hemisphere
given by the plane with normal v.
"""
v = g.normalize(v)
north, east = g.northEast(v)
points = []
for i in xrange(n):
z = -1.0
while z < 0.0:
x = random.uniform(-1.0 + 1.0e-7, 1.0 - 1.0e-7)
y = random.uniform(-1.0 + 1.0e-7, 1.0 - 1.0e-7)
z = 1.0 - x * x - y * y
z = math.sqrt(z)
p = (z * v[0] + x * north[0] + y * east[0],
z * v[1] + x * north[1] + y * east[1],
z * v[2] + x * north[2] + y * east[2])
points.append(g.normalize(p))
return points
def _hemPoints(v, n):
"""Randomly generates a list of n points in the hemisphere
given by the plane with normal v.
"""
v = g.normalize(v)
north, east = g.northEast(v)
points = []
for i in xrange(n):
z = -1.0
while z < 0.0:
x = random.uniform(-1.0 + 1.0e-7, 1.0 - 1.0e-7)
y = random.uniform(-1.0 + 1.0e-7, 1.0 - 1.0e-7)
z = 1.0 - x * x - y * y
z = math.sqrt(z)
p = (z * v[0] + x * north[0] + y * east[0],
z * v[1] + x * north[1] + y * east[1],
z * v[2] + x * north[2] + y * east[2])
points.append(g.normalize(p))
return points
def _pointsOnCircle(c, r, n, clockwise=False):
"""Generates an n-gon lying on the circle with center c and
radius r. Vertices are equi-spaced.
"""
c = g.cartesianUnitVector(c)
north, east = g.northEast(c)
points = []
sr = math.sin(math.radians(r))
cr = math.cos(math.radians(r))
aoff = random.uniform(0.0, 2.0 * math.pi)
for i in xrange(n):
a = 2.0 * i * math.pi / n
if not clockwise:
a = -a
sa = math.sin(a + aoff)
ca = math.cos(a + aoff)
p = (ca * north[0] + sa * east[0], ca * north[1] + sa * east[1],
ca * north[2] + sa * east[2])
points.append(
g.normalize((cr * c[0] + sr * p[0], cr * c[1] + sr * p[1],
cr * c[2] + sr * p[2])))
return points
def _pointsOnPeriodicHypotrochoid(c, R, r, d, n):
assert isinstance(r, int) and isinstance(R, int)
assert R > r
assert (2 * r) % (R - r) == 0
points = []
c = g.cartesianUnitVector(c)
north, east = g.northEast(c)
scale = 1.0 / (R - r + d + 1)
period = (2 * r) / (R - r)
if period % 2 != 0:
period *= 2
for i in xrange(n):
theta = i * period * math.pi / n
x = (R - r) * math.cos(theta) + d * math.cos((R - r) * theta / r)
y = (R - r) * math.sin(theta) - d * math.sin((R - r) * theta / r)
x *= scale
y *= scale
z = math.sqrt(1.0 - x * x - y * y)
points.append(g.normalize((z * c[0] + x * north[0] + y * east[0],
z * c[1] + x * north[1] + y * east[1],
z * c[2] + x * north[2] + y * east[2])))
return points
def _pointsOnPeriodicHypotrochoid(c, R, r, d, n):
assert isinstance(r, int) and isinstance(R, int)
assert R > r
assert (2 * r) % (R - r) == 0
points = []
c = geom.cartesianUnitVector(c)
north, east = geom.northEast(c)
scale = 1.0 / (R - r + d + 1)
period = (2 * r) / (R - r)
if period % 2 != 0:
period *= 2
for i in range(n):
theta = i * period * math.pi / n
x = (R - r) * math.cos(theta) + d * math.cos((R - r) * theta / r)
y = (R - r) * math.sin(theta) - d * math.sin((R - r) * theta / r)
x *= scale
y *= scale
z = math.sqrt(1.0 - x * x - y * y)
points.append(geom.normalize((z * c[0] + x * north[0] + y * east[0],
z * c[1] + x * north[1] + y * east[1],
z * c[2] + x * north[2] + y * east[2])))
return points
def _pointsOnCircle(c, r, n, clockwise=False):
"""Generates an n-gon lying on the circle with center c and
radius r. Vertices are equi-spaced.
"""
c = g.cartesianUnitVector(c)
north, east = g.northEast(c)
points = []
sr = math.sin(math.radians(r))
cr = math.cos(math.radians(r))
aoff = random.uniform(0.0, 2.0 * math.pi)
for i in xrange(n):
a = 2.0 * i * math.pi / n
if not clockwise:
a = -a
sa = math.sin(a + aoff)
ca = math.cos(a + aoff)
p = (ca * north[0] + sa * east[0],
ca * north[1] + sa * east[1],
ca * north[2] + sa * east[2])
points.append(g.normalize((cr * c[0] + sr * p[0],
cr * c[1] + sr * p[1],
cr * c[2] + sr * p[2])))
return points