Hi
I've made an interact on elliptic curve point addition, see below.
When I run it in notebook() mode I get an error:
Traceback (click to the left for traceback)
...
AttributeError: 'SymbolicEquation' object has no attribute
'_fast_float_'
It worked in sage 3.1.1. I can't figure out how to fix it. Help is
much appreciated since I want to add this to the wiki and my thesis
presentation.
______________
def point_txt(P,name,rgbcolor):
if (P.xy()[1]) < 0:
r = text(name,[float(P.xy()[0]),float(P.xy()
[1])-1],rgbcolor=rgbcolor)
elif P.xy()[1] == 0:
r = text(name,[float(P.xy()[0]),float(P.xy()[1])
+1],rgbcolor=rgbcolor)
else:
r = text(name,[float(P.xy()[0]),float(P.xy()[1])
+1],rgbcolor=rgbcolor)
return r
E = EllipticCurve('37a')
list_of_points = E.integral_points()
html("Graphical addition of two points $P$ and $Q$ on the curve $ E:
%s $"%latex(E))
@interact
def _(P=selector(list_of_points,label='Point P'),Q=selector
(list_of_points,label='Point Q'), marked_points = checkbox
(default=True,label = 'Points'), Lines = selector([0..2],nrows=1),
Axes=True):
curve = plot(E,rgbcolor = (0,0,1),xmin=25,xmax=25)
#Q = list_of_points[Q]
#P = list_of_points[P]
R = P + Q
Rneg = -R
#html("$P=(%s:%s:%s)$, $Q=(%s:%s:%s)$\n"%(latex(P[0]),latex(P
[1]),latex(P[2]),latex(Q[0]),latex(Q[1]),latex(Q[2])))
#html("Graphical addition of points $P$ and $Q$ \non the curve $ E:
%s $ \n\n$P + Q = (%s:%s:%s)$"%(latex(E),latex(R[0]),latex(R[1]),latex
(R[2]))) # $P + Q$ = $%s + %s = %s"%(P,Q,R)
l1 = line_from_curve_points(E,P,Q)
l2 = line_from_curve_points(E,R,Rneg,style='--')
p1 = point(P,rgbcolor=(1,0,0),pointsize=40)
p2 = plot(Q,rgbcolor=(1,0,0),pointsize=40)
p3 = plot(R,rgbcolor=(1,0,0),pointsize=40)
p4 = plot(Rneg,rgbcolor=(1,0,0),pointsize=40)
textp1 = point_txt(P,"$P$",rgbcolor=(0,0,0))
textp2 = point_txt(Q,"$Q$",rgbcolor=(0,0,0))
textp3 = point_txt(R,"$P+Q$",rgbcolor=(0,0,0))
if Lines==0:
g=curve
elif Lines ==1:
g=curve+l1
elif Lines == 2:
g=curve+l1+l2
if marked_points:
g=g+p1+p2+p3+p4
if P != Q:
g=g+textp1+textp2+textp3
else:
g=g+textp1+textp3
g.axes_range(xmin=-5,xmax=5,ymin=-13,ymax=13)
show(g,axes = Axes)
def line_from_curve_points(E,P,Q,style='-',rgb=(1,0,0),length=25):
"""
P,Q two points on an elliptic curve.
Output is a graphic representation of the straight line intersecting
with P,Q.
"""
# The function tangent to P=Q on E
if P == Q:
if P[2]==0:
return
line([(1,-length),(1,length)],linestyle=style,rgbcolor=rgb)
else:
# Compute slope of the curve E in P
l=-(3*P[0]^2 + 2*E.a2()*P[0] + E.a4() -
E.a1()*P[1])/((-2)*P[1] -
E.a1()*P[0] - E.a3())
f(x) = l * (x - P[0]) + P[1]
return
plot(f(x),-length,length,linestyle=style,rgbcolor=rgb)
# Trivial case of P != R where P=O or R=O then we get the vertical
line from the other point
elif P[2] == 0:
return line([(Q[0],-length),(Q
[0],length)],linestyle=style,rgbcolor=rgb)
elif Q[2] == 0:
return line([(P[0],-length),(P
[0],length)],linestyle=style,rgbcolor=rgb)
# Non trivial case where P != R
else:
# Case where x_1 = x_2 return vertical line evaluated in Q
if P[0] == Q[0]:
return line([(P[0],-length),(P
[0],length)],linestyle=style,rgbcolor=rgb)
#Case where x_1 != x_2 return line trough P,R evaluated in Q"
l=(Q[1]-P[1])/(Q[0]-P[0])
f(x) = l * (x - P[0]) + P[1]
return plot(f(x),-length,length,linestyle=style,rgbcolor=rgb)
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