def eqn_list_to_curve_plot(L, rat_pts):
xpoly_rng = PolynomialRing(QQ, 'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f + h**2 / 4
if len(g.real_roots()) == 0 and g(0) < 0:
return text(r"$X(\mathbb{R})=\emptyset$", (1, 1), fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()
] + [real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a, b = inflate_interval(min(X0), max(X0), 1.5)
groots = [a] + g.real_roots() + [b]
if b - a < 1e-7:
a = -3
b = 3
groots = [a, b]
ngints = len(groots) - 1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j + 1]
if g((c + d) / 2) < 0:
continue
(c, d) = inflate_interval(c, d, 1.1)
s = (d - c) / npts
u = c
yvals = []
for i in range(npts + 1):
v = g(u)
if v > 0:
v = sqrt(v)
w = -h(u) / 2
yvals.append(w + v)
yvals.append(w - v)
u += s
(m, M) = inflate_interval(min(yvals), max(yvals), 1.2)
plotzones.append((c, d, m, M))
x = var('x')
y = var('y')
plot = sum(
implicit_plot(y**2 + y * h(x) - f(x), (x, R[0], R[1]), (y, R[2], R[3]),
aspect_ratio='automatic',
plot_points=500,
zorder=1) for R in plotzones)
xmin = min([R[0] for R in plotzones])
xmax = max([R[1] for R in plotzones])
ymin = min([R[2] for R in plotzones])
ymax = max([R[3] for R in plotzones])
for P in rat_pts:
(x, y, z) = P
z = ZZ(z)
if z: # Do not attempt to plot points at infinity
x = ZZ(x) / z
y = ZZ(y) / z**3
if x >= xmin and x <= xmax and y >= ymin and y <= ymax:
plot += point((x, y), color='red', size=40, zorder=2)
return plot
def __init__(self, implicit_curve, xrange, yrange, plot_points=300):
ObjectGraph.__init__(self, implicit_curve)
GeometricImplicitCurve.__init__(self, implicit_curve.f)
self.implicit_curve = implicit_curve
self.implicit_plot = implicit_plot(self.f, xrange, yrange)
self.xrange = xrange
self.yrange = yrange
self.plot_points = plot_points
self.paths = get_paths_from_implicit_plot(self.implicit_plot)
self.parameters.color = "blue"
def eqn_list_to_curve_plot(L,rat_pts):
xpoly_rng = PolynomialRing(QQ,'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f+h**2/4
if len(g.real_roots())==0 and g(0)<0:
return text("$X(\mathbb{R})=\emptyset$",(1,1),fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()]+[real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a,b = inflate_interval(min(X0),max(X0),1.5)
groots = [a]+g.real_roots()+[b]
if b-a<1e-7:
a=-3
b=3
groots=[a,b]
ngints = len(groots)-1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j+1]
if g((c+d)/2)<0:
continue
(c,d) = inflate_interval(c,d,1.1)
s = (d-c)/npts
u = c
yvals = []
for i in range(npts+1):
v = g(u)
if v>0:
v = sqrt(v)
w = -h(u)/2
yvals.append(w+v)
yvals.append(w-v)
u += s
(m,M) = inflate_interval(min(yvals),max(yvals),1.2)
plotzones.append((c,d,m,M))
x = var('x')
y = var('y')
plot=sum(implicit_plot(y**2 + y*h(x) - f(x), (x,R[0],R[1]),(y,R[2],R[3]), aspect_ratio='automatic', plot_points=500, zorder=1) for R in plotzones)
xmin=min([R[0] for R in plotzones])
xmax=max([R[1] for R in plotzones])
ymin=min([R[2] for R in plotzones])
ymax=max([R[3] for R in plotzones])
for P in rat_pts:
(x,y,z)=eval(P.replace(':',','))
z=ZZ(z)
if z: # Do not attempt to plot points at infinity
x=ZZ(x)/z
y=ZZ(y)/z**3
if x >= xmin and x <= xmax and y >= ymin and y <= ymax:
plot += point((x,y),color='red',size=40,zorder=2)
return plot
def eqn_list_to_curve_plot(L):
xpoly_rng = PolynomialRing(QQ, 'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f + h**2 / 4
if len(g.real_roots()) == 0 and g(0) < 0:
return text("$X(\mathbb{R})=\emptyset$", (1, 1), fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()
] + [real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a, b = inflate_interval(min(X0), max(X0), 1.5)
groots = [a] + g.real_roots() + [b]
if b - a < 1e-7:
a = -3
b = 3
groots = [a, b]
ngints = len(groots) - 1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j + 1]
if g((c + d) / 2) < 0:
continue
(c, d) = inflate_interval(c, d, 1.1)
s = (d - c) / npts
u = c
yvals = []
for i in range(npts + 1):
v = g(u)
if v > 0:
v = sqrt(v)
w = -h(u) / 2
yvals.append(w + v)
yvals.append(w - v)
u += s
(m, M) = inflate_interval(min(yvals), max(yvals), 1.2)
plotzones.append((c, d, m, M))
x = var('x')
y = var('y')
return sum(
implicit_plot(
y**2 + y * h(x) - f(x), (x, R[0], R[1]), (y, R[2], R[3]),
aspect_ratio='automatic',
plot_points=500) for R in plotzones)
def eqn_list_to_curve_plot(L):
xpoly_rng = PolynomialRing(QQ, 'x')
poly_tup = [xpoly_rng(tup) for tup in L]
f = poly_tup[0]
h = poly_tup[1]
g = f + h**2 / 4
if len(g.real_roots()) == 0 and g(0) < 0:
return text("$X(\mathbb{R})=\emptyset$", (1, 1), fontsize=50)
X0 = [real(z[0]) for z in g.base_extend(CC).roots()
] + [real(z[0]) for z in g.derivative().base_extend(CC).roots()]
a, b = inflate_interval(min(X0), max(X0), 1.5)
groots = [a] + g.real_roots() + [b]
if b - a < 1e-7:
a = -3
b = 3
groots = [a, b]
ngints = len(groots) - 1
plotzones = []
npts = 100
for j in range(ngints):
c = groots[j]
d = groots[j + 1]
if g((c + d) / 2) < 0:
continue
(c, d) = inflate_interval(c, d, 1.1)
s = (d - c) / npts
u = c
yvals = []
for i in range(npts + 1):
v = g(u)
if v > 0:
v = sqrt(v)
w = -h(u) / 2
yvals.append(w + v)
yvals.append(w - v)
u += s
(m, M) = inflate_interval(min(yvals), max(yvals), 1.2)
plotzones.append((c, d, m, M))
x = var('x')
y = var('y')
return sum(
implicit_plot(y**2 + y * h(x) - f(x), (x, R[0], R[1]), (y, R[2], R[3]),
aspect_ratio='automatic',
plot_points=500) for R in plotzones)
def EC_R_plot(ainvs, xmin, xmax, ymin, ymax, colour, legend):
x = var('x')
y = var('y')
c = (xmin + xmax) / 2
d = (xmax - xmin)
return implicit_plot(
y**2 + ainvs[0] * x * y + ainvs[2] * y - x**3 - ainvs[1] * x**2 -
ainvs[3] * x - ainvs[4], (x, xmin, xmax), (y, ymin, ymax),
plot_points=500,
aspect_ratio="automatic",
color=colour
) + plot(
0,
xmin=c - 1e-5 * d,
xmax=c + 1e-5 * d,
ymin=ymin,
ymax=ymax,
aspect_ratio="automatic",
color=colour,
legend_label=legend
) # Add an extra plot outside the visible frame because implicit plots are buggy: their legend does not show (http://trac.sagemath.org/ticket/15903)
def EC_R_plot(ainvs, xmin, xmax, ymin, ymax, colour, legend):
x = var("x")
y = var("y")
c = (xmin + xmax) / 2
d = xmax - xmin
return implicit_plot(
y ** 2 + ainvs[0] * x * y + ainvs[2] * y - x ** 3 - ainvs[1] * x ** 2 - ainvs[3] * x - ainvs[4],
(x, xmin, xmax),
(y, ymin, ymax),
plot_points=1000,
aspect_ratio="automatic",
color=colour,
) + plot(
0,
xmin=c - 1e-5 * d,
xmax=c + 1e-5 * d,
ymin=ymin,
ymax=ymax,
aspect_ratio="automatic",
color=colour,
legend_label=legend,
) # Add an extra plot outside the visible frame because implicit plots are buggy: their legend does not show (http://trac.sagemath.org/ticket/15903)
