def torus_0():
x, y, z, r, R = symbols("x y z r R")
expr = (x * x + y * y + z * z + R * R -
r * r)**2 - 4 * R * R * (x * x + y * y)
ox, oy, oz, t, sx, sy, sz, SS, OO, OS = symbols(
"ox oy oz t sx sy sz SS OO OS")
ray = [(x, ox + t * sx), (y, oy + t * sy), (z, oz + t * sz)]
simp = [(sx**2 + sy**2 + sz**2, SS), (ox**2 + oy**2 + oz**2, OO),
(ox * sx + oy * sy + oz * sz, OS)]
expr2 = expr.subs(ray + simp)
x_expr2 = expand(expr2)
t_expr2 = collect(x_expr2, t)
c = get_coeff(expr2, t, 5)
subs_coeff(c, simp)
print_coeff(c)
return expr2, c
def cubic_1():
a,b,c = symbols("a b c")
x,y = symbols("x y")
ex = x**3 + a*x**2 + b*x + c
print "ex:%s " % ex
ey = expand(ex.subs([(x, y-a/3)]))
print "ey:%s " % ey
cy = get_coeff(ey, y, 4)
eq = cy[0]
ep = cy[1]
print "eq:%s " % eq
print "ep:%s " % ep
z,p,q = symbols("z p q")
simp = [
(eq*27, q*27),
(ep*3, p*3) ]
print_coeff(cy, "cy")
subs_coeff(cy, simp)
print_coeff(cy, "cy-simp")
ez = expr_coeff(cy, z)
print "ez:%s " % ez
def cubic_0():
a2,a1,a0 = symbols("a2 a1 a0")
x,y,z,l = symbols("x y z l")
ez = z**3 + a2*z**2 + a1*z + a0
print "ez: %r (cubic with top coeff 1)" % ez
ex = ez.subs(z, x-l ).subs(l, a2/3)
print "ex: %r (shift to kill **2 term)" % ex
print "ex: %r (shift to kill **2 term)" % expand(ex)
cx = get_coeff(ex, x, 4 )
assert cx[2] == 0 # **2 term is killed
p,q = symbols("p q", real=True)
eq = (9*a2*a1 - 2*a2**3 - 27*a0)/27
ep = ( 3*a1 - a2**2)/3
simp = [
(eq*27, q*27),
(ep*3, p*3) ]
print_coeff(cx, "cx")
subs_coeff(cx, simp)
print_coeff(cx, "c-simp")
ex2 = expr_coeff(cx, x)
w= symbols("w")
ew = expand(ex2.subs(x, w-p/(3*w) )) # Vieta's substitution
ew2 = expand(ew*w**3) # becomes quadratic in w**3
w3= symbols("w3")
ew3 = ew2.subs(w**3, w3)
cw3 = get_coeff(ew3,w3,3)
c,b,a = cw3
isc = b**2 - 4*a*c # quadratic, b^2 - 4ac , -b -sqrt() , -b+sqrt()
r1 = (-b + sqrt(isc))/(2*a)
r2 = (-b - sqrt(isc))/(2*a)
print "r1:%s " % r1
print "r2:%s " % r2
print "isc: %r " % isc
rr = solve(ew3, w3)
def quartic_0():
a0, a1, a2, a3, a4 = symbols("a0 a1 a2 a3 a4")
q0, q1, q2, q3, q4 = symbols("q0 q1 q2 q3 q4")
x, z, l = symbols("x z l")
ex_ = Pow(x, 4) + a3 * Pow(x, 3) + a2 * Pow(x, 2) + a1 * Pow(
x, 1) + a0 # setting a4 to 1
ex = ex_.subs(x, x - l).subs(l,
a3 / 4) # picking this l kills the x**3 term
c = get_coeff(ex, x, 5)
assert c[3] == 0 # x**3 term is killed
p, q, r = symbols("p q r")
simp = [(a2 - 3 * a3**2 / 8, p), (a1 - a2 * a3 / 2 + a3**3 / 8, q),
(-a1 * a3 / 4 + a2 * a3**2 / 16 - 3 * a3**4 / 256, r)]
cs = get_coeff_(ex, x, 5)
print_coeff(cs, "cs")
def torus_1():
"""
http://www.cosinekitty.com/raytrace/chapter13_torus.html
Symbolic manipulations matching cosinekitty example
* simplifying coeffs separately then putting
back together seems an effective approach
"""
x, y, z, A, B = symbols("x y z A B")
_sq_lhs = (x * x + y * y + z * z + A * A - B * B)
_rhs = 4 * A * A * (x * x + y * y)
ox, oy, oz, t, sx, sy, sz, SS, OO, OS = symbols(
"ox oy oz t sx sy sz SS OO OS")
ray = [(x, ox + t * sx), (y, oy + t * sy), (z, oz + t * sz)]
simp0 = [(sx**2 + sy**2 + sz**2, SS), (ox**2 + oy**2 + oz**2, OO),
(ox * sx + oy * sy + oz * sz, OS)]
G, H, I, J, K, L = symbols("G H I J K L")
exG = 4 * A * A * (sx * sx + sy * sy)
exH = 8 * A * A * (ox * sx + oy * sy)
exI = 4 * A * A * (ox * ox + oy * oy)
exJ = sx * sx + sy * sy + sz * sz
exK = 2 * (ox * sx + oy * sy + oz * sz)
exL = ox * ox + oy * oy + oz * oz + A * A - B * B
simp1 = [
(exG, G),
(exH, H),
(exI, I),
(exJ, J),
(exK / 2, K / 2), # fails to sub with 2 on other side
(exL, L)
]
sq_lhs = _sq_lhs.subs(ray + simp0)
rhs = _rhs.subs(ray + simp0)
cl = get_coeff(sq_lhs, t, 3)
cr = get_coeff(rhs, t, 3)
subs_coeff(cl, simp1)
subs_coeff(cr, simp1)
print_coeff(cl, "cl")
print_coeff(cr, "cr")
SQ_LHS = expr_coeff(cl, t)
RHS = expr_coeff(cr, t)
ex = Pow(SQ_LHS, 2) - RHS
c = get_coeff(ex, t, 5)
print_coeff(c, "c")
exc = range(5)
exc[4] = exJ**2
exc[3] = 2 * exJ * exK
exc[2] = -exG + 2 * exJ * exL + exK**2
exc[1] = -exH + 2 * exK * exL
exc[0] = -exI + exL**2
subs = {}
subs["O->X+"] = [(ox, 0), (oy, 0), (oz, 0), (sx, 1), (sy, 0), (sz, 0)]
subs["zenith->-Z"] = [(ox, 0), (oy, 0), (oz, A), (sx, 0), (sy, 0),
(sz, -1)]
subs["corner"] = [(ox, A), (oy, A), (oz, A), (sx, -isq3), (sy, -isq3),
(sz, -isq3)]
radii = [(A, 500), (B, 50)]
for key in subs:
print("\n\n", key)
for j, exc_ in enumerate(exc):
print("\n", j, exc_)
print(exc_.subs(subs[key]))
print(exc_.subs(subs[key]).subs(radii))
return ex, cl