def biconic_grad(test_vector):
"""
Computation of biconic gradient equals explicit calculation
"""
coordinate_system = LocalCoordinates.p(name="root")
radiusx = 100.0
radiusy = 120.0
conic_constantx = -0.5
conic_constanty = -1.7
cx = 1./radiusx
cy = 1./radiusy
alpha2 = 1e-3
alpha4 = -1e-6
alpha6 = 1e-8
beta2 = 0.1
beta4 = -0.6
beta6 = 0.2
maxradiusx = (math.sqrt(1./((1+conic_constantx)*cx**2))
if conic_constantx > -1 else abs(radiusx))
maxradiusy = (math.sqrt(1./((1+conic_constanty)*cy**2))
if conic_constanty > -1 else abs(radiusy))
maxradius = min([maxradiusx, maxradiusy]) # choose minimal radius
values = (2*test_vector - 1)*maxradius
x = values[0]
y = values[1]
shape = Biconic.p(coordinate_system, curvx=cx, curvy=cy, ccx=conic_constantx, ccy=conic_constanty,
coefficients=[(alpha2, beta2), (alpha4, beta4), (alpha6, beta6)])
gradient = shape.getGrad(x, y)
comparison = np.zeros_like(gradient)
comparison[2, :] = 1.
comparison[0, :] = (
-alpha2*(-2*beta2*x + 2*x)
- alpha4*(-4*beta4*x + 4*x)*(-beta4*(x**2 - y**2) + x**2 + y**2)
- alpha6*(-6*beta6*x + 6*x)*(-beta6*(x**2 - y**2) + x**2 + y**2)**2
- cx**2*x*(conic_constantx + 1)*(cx*x**2 + cy*y**2)/((np.sqrt(-cx**2*x**2*(conic_constantx + 1) - cy**2*y**2*(conic_constanty + 1) + 1) + 1)**2*np.sqrt(-cx**2*x**2*(conic_constantx + 1) - cy**2*y**2*(conic_constanty + 1) + 1)) - 2*cx*x/(np.sqrt(-cx**2*x**2*(conic_constantx + 1) - cy**2*y**2*(conic_constanty + 1) + 1) + 1))
comparison[1, :] = (
-alpha2*(2*beta2*y + 2*y)
- alpha4*(4*beta4*y + 4*y)*(-beta4*(x**2 - y**2) + x**2 + y**2)
- alpha6*(6*beta6*y + 6*y)*(-beta6*(x**2 - y**2) + x**2 + y**2)**2
- cy**2*y*(conic_constanty + 1)*(cx*x**2 + cy*y**2)/((np.sqrt(-cx**2*x**2*(conic_constantx + 1) - cy**2*y**2*(conic_constanty + 1) + 1) + 1)**2*np.sqrt(-cx**2*x**2*(conic_constantx + 1) - cy**2*y**2*(conic_constanty + 1) + 1)) - 2*cy*y/(np.sqrt(-cx**2*x**2*(conic_constantx + 1) - cy**2*y**2*(conic_constanty + 1) + 1) + 1)
)
assert np.allclose(gradient, comparison)
def biconic_sag(test_vector):
"""
Computation of biconic sag equals explicit calculation
"""
coordinate_system = LocalCoordinates.p(name="root")
radiusx = 100.0
radiusy = 120.0
conic_constantx = -0.5
conic_constanty = -1.7
cx = 1. / radiusx
cy = 1. / radiusy
alpha2 = 1e-3
alpha4 = -1e-6
alpha6 = 1e-8
beta2 = 0.1
beta4 = -0.6
beta6 = 0.2
maxradiusx = (math.sqrt(1. / ((1 + conic_constantx) * cx**2))
if conic_constantx > -1 else abs(radiusx))
maxradiusy = (math.sqrt(1. / ((1 + conic_constanty) * cy**2))
if conic_constanty > -1 else abs(radiusy))
maxradius = min([maxradiusx, maxradiusy]) # choose minimal radius
values = (2 * test_vector - 1) * maxradius
x_coordinate = values[0]
y_coordinate = values[1]
shape = Biconic.p(coordinate_system,
curvx=cx,
curvy=cy,
ccx=conic_constantx,
ccy=conic_constanty,
coefficients=[(alpha2, beta2), (alpha4, beta4),
(alpha6, beta6)])
sag = shape.getSag(x_coordinate, y_coordinate)
# comparison with explicitly entered formula
comparison = (
(cx * x_coordinate**2 + cy * y_coordinate**2) /
(1. + np.sqrt(1. - (1. + conic_constantx) * cx**2 * x_coordinate**2 -
(1. + conic_constanty) * cy**2 * y_coordinate**2)) +
alpha2 * (x_coordinate**2 + y_coordinate**2 - beta2 *
(x_coordinate**2 - y_coordinate**2)) + alpha4 *
(x_coordinate**2 + y_coordinate**2 - beta4 *
(x_coordinate**2 - y_coordinate**2))**2 + alpha6 *
(x_coordinate**2 + y_coordinate**2 - beta6 *
(x_coordinate**2 - y_coordinate**2))**3)
assert np.allclose(sag, comparison)
tiltThenDecenter=False),
refname=lc0.name)
lcimage = s.addLocalCoordinateSystem(LocalCoordinates.p(
name="image",
decy=-30.892,
decz=43.083,
tiltx=-si * 50.668 * degree,
tiltThenDecenter=False),
refname=lc0.name)
objsurf = Surface.p(lc0)
D1surf = Surface.p(lcD1)
S1surf = Surface.p(lcS1,
shape=Biconic.p(lcS1,
curvy=si * 1. / 108.187,
curvx=si * 1. / 73.105,
coefficients=[(0., 0.),
(-si * 5.542e-7, -0.08),
(-si * 8.176e-11, -1.379)]),
aperture=CircularAperture.p(lcS1, maxradius=40.0))
D1Psurf = Surface.p(lcD1prime)
D2surf = Surface.p(lcD2)
S2surf = Surface.p(lcS2,
shape=Biconic.p(lcS2,
curvy=si * 1. / 69.871,
curvx=si * 1. / 60.374,
ccy=-0.1368,
ccx=-0.123,
coefficients=[(0., 0.),
(si * 7.233e-11, 29.075),
(si * 4.529e-12, -2.085)]),
aperture=CircularAperture.p(lcS2, maxradius=40.0))