Re: [sage-support] Re: Computational geometry in the plane: in Sage?
Thanks. I've seen some of that, but it looks like they use the upper half-plane model for visualization (while my preference is for Poincare disk). Also, I want to do other geometric constructions/visualizations in Sage (see attached pics) for which commands to create segments, then get perpentidular bisecting lines or angle trisectors, would make life much easier. (I did this in Sage, but had to work from scratch.) On Thursday, April 24, 2014 5:43:57 PM UTC-4, Peter Bruin wrote: A certain amount of work on adding functionality for hyperbolic geometry to Sage has been done in recent years, see here: http://trac.sagemath.org/ticket/9439 There seem to be several different implementations by different authors; I am not sure about the status of all this work and how much of it is relevant to what you want to do. Peter Op donderdag 24 april 2014 18:54:32 UTC+1 schreef Ed Scheinerman: Glad to see this has gained some traction. Here is an illustration of the immediate issue for which this would have been helpful. I wanted to produce an illustration explaining lines in the hyperbolic plane using the Poincare disk model. It's the arc of a circle whose end points are on a given circle and the arc meets the given circle at right angles. (That's a lot to say but a picture makes it so much easier.) It took me a long time and a good deal of coding in Sage to get this how I wanted. My gut tells me there's go to be a better way! On Thu, Apr 24, 2014 at 9:31 AM, Volker Braun vbrau...@gmail.com wrote: We do have convex hull and lines. What would be lacking for your application are discs and their intersection with polyhedra. sage: line = Polyhedron(vertices=[(0,-1)], lines=[(1,1)]) sage: (triangle line).vertices() (A vertex at (8/5, 3/5), A vertex at (3/2, 1/2)) On Thursday, April 24, 2014 7:02:21 AM UTC+1, Dominique Laurain wrote: I need same packages that Ed asked for. Nowadays I do geometry coding myself basic functions in SAGE worksheets (to do intersection of lines etc). I do this using the Rational Trigonometry philosophy :-)..see njwilderger youtube videos and book. Previously I played a little with Tex tools (pst-tricks and pst-eucl)..check about pstricks too...at that time and until now, I was interested with tilings. In SAGE we have the basics tools (vectors,matplotlib...) but I feel there are not the intermediate tools : convex hull, rules and compass constructions, origami tools and so on. I can code some (for example, convex hull: I know an algorithm paquet cadeau ...but everytime I feel to reinvent wheel. What is not enough with Tex is that all is static : for example, you cannot have the generic solution of two lines AB and CD where A,B,C,D are 2D points with variables coordinates...because it is no more static geometry but symbolic computations (solving equations / inequations). -- You received this message because you are subscribed to a topic in the Google Groups sage-support group. To unsubscribe from this topic, visit https://groups.google.com/d/topic/sage-support/Y0t4ShFLXJ4/unsubscribe. To unsubscribe from this group and all its topics, send an email to sage-support...@googlegroups.com. To post to this group, send email to sage-s...@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout. -- Ed Scheinerman (e...@scheinerman.net) -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout. tangent-ring.pdf Description: Adobe PDF document morley.pdf Description: Adobe PDF document circum-out.pdf Description: Adobe PDF document
Re: [sage-support] Re: Computational geometry in the plane: in Sage?
Glad to see this has gained some traction. Here is an illustration of the immediate issue for which this would have been helpful. I wanted to produce an illustration explaining lines in the hyperbolic plane using the Poincare disk model. It's the arc of a circle whose end points are on a given circle and the arc meets the given circle at right angles. (That's a lot to say but a picture makes it so much easier.) It took me a long time and a good deal of coding in Sage to get this how I wanted. My gut tells me there's go to be a better way! On Thu, Apr 24, 2014 at 9:31 AM, Volker Braun vbraun.n...@gmail.com wrote: We do have convex hull and lines. What would be lacking for your application are discs and their intersection with polyhedra. sage: line = Polyhedron(vertices=[(0,-1)], lines=[(1,1)]) sage: (triangle line).vertices() (A vertex at (8/5, 3/5), A vertex at (3/2, 1/2)) On Thursday, April 24, 2014 7:02:21 AM UTC+1, Dominique Laurain wrote: I need same packages that Ed asked for. Nowadays I do geometry coding myself basic functions in SAGE worksheets (to do intersection of lines etc). I do this using the Rational Trigonometry philosophy :-)..see njwilderger youtube videos and book. Previously I played a little with Tex tools (pst-tricks and pst-eucl)..check about pstricks too...at that time and until now, I was interested with tilings. In SAGE we have the basics tools (vectors,matplotlib...) but I feel there are not the intermediate tools : convex hull, rules and compass constructions, origami tools and so on. I can code some (for example, convex hull: I know an algorithm paquet cadeau ...but everytime I feel to reinvent wheel. What is not enough with Tex is that all is static : for example, you cannot have the generic solution of two lines AB and CD where A,B,C,D are 2D points with variables coordinates...because it is no more static geometry but symbolic computations (solving equations / inequations). -- You received this message because you are subscribed to a topic in the Google Groups sage-support group. To unsubscribe from this topic, visit https://groups.google.com/d/topic/sage-support/Y0t4ShFLXJ4/unsubscribe. To unsubscribe from this group and all its topics, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout. -- Ed Scheinerman (e...@scheinerman.net) -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout. line-definition.pdf Description: Adobe PDF document
[sage-support] Computational geometry in the plane: in Sage?
Does Sage have, or is there a package I can add, that implements basic plane geometry objects and operations. The sort of thing I'd like to do is, given two points, construct the line that contains them. Or given two circles, intersect them yielding 0, 1, or 2 points. Given a set of points, determine which are on the convex hull, construct that polygon, and return its area. And, ultimately, after working with various geometric objects, perhaps visualize them on the screen (but that comes later). Extra bonus if I can do this in both the Euclidean plane and the Hyperbolic plane (and in the latter case visualize either in the Poincare disk or the upper half plane models). Thanks. -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
Re: [sage-support] Sage on Mavericks requesting installation of X
Yes. I reinstalled the same version of XQuartz as I had before, but it was the first such reinstall since upgrading to Mavericks. On Wednesday, October 30, 2013 12:41:23 AM UTC-4, yomcat wrote: On 30/10/2013, at 0945, Ed Scheinerman edward.sc...@gmail.comjavascript: wrote: Update: I re-installed XQuartz and the problem vanished. Was this the first time you re-installed KQuartz after installing Mavericks? Apple requires such a thing with every new major OS X upgrade, I do believe. -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Re: Sage on Mavericks requesting installation of X
Update: I re-installed XQuartz and the problem vanished. On Monday, October 28, 2013 5:44:42 PM UTC-4, Ed Scheinerman wrote: I have just upgraded my Mac to OS X version 10.9 Mavericks and am running Sage version 5.12. When I start sage from the command line an dialogue box opens asking me about opening Xquartz and that X11 is no longer supported etc. (See attached image.) I can dismiss past this dialogue box and all works fine. I have XQuartz installed and I get this error whether or not I already have Xquartz running. Anything I should do to make this annoyance stop? -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Sage on Mavericks requesting installation of X
I have just upgraded my Mac to OS X version 10.9 Mavericks and am running Sage version 5.12. When I start sage from the command line an dialogue box opens asking me about opening Xquartz and that X11 is no longer supported etc. (See attached image.) I can dismiss past this dialogue box and all works fine. I have XQuartz installed and I get this error whether or not I already have Xquartz running. Anything I should do to make this annoyance stop? -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out. attachment: needs-X.png
[sage-support] Trouble saving graphics
I'm using Sage 5.11 on a Mac
I've been used to creating graphs and then saving them as PDF's for
inclusion in papers. Using code like this:
sage: g = graphs.PetersenGraph()
sage: pic = g.plot()
sage: pic.save('pete.pdf')
But now it just throws a bunch of errors:
---
AssertionErrorTraceback (most recent call last)
ipython-input-56-d63f9d409a56 in module()
1 pic.save('pete.pdf')
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/sage/misc/decorators.pyc
in wrapper(*args, **kwds)
456 kwds[self.name + options] = suboptions
457
-- 458 return func(*args, **kwds)
459
460 #Add the options specified by @options to the signature of
the wrapped
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/sage/plot/graphics.pyc
in save(self, filename, **kwds)
2753 figure.savefig(filename, dpi=dpi,
bbox_inches='tight',
2754 bbox_extra_artists=self._bbox_extra_artists,
- 2755 transparent=transparent)
2756 else:
2757 figure.savefig(filename, dpi=dpi,
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/figure.pyc
in savefig(self, *args, **kwargs)
1368 kwargs.setdefault('edgecolor',
rcParams['savefig.edgecolor'])
1369
- 1370 self.canvas.print_figure(*args, **kwargs)
1371
1372 if transparent:
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/backend_bases.pyc
in print_figure(self, filename, dpi, facecolor, edgecolor, orientation,
format, **kwargs)
2053 orientation=orientation,
2054 dryrun=True,
- 2055 **kwargs)
2056 renderer = self.figure._cachedRenderer
2057 bbox_inches = self.figure.get_tightbbox(renderer)
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/backend_bases.pyc
in print_pdf(self, *args, **kwargs)
1846 from backends.backend_pdf import FigureCanvasPdf # lazy
import
1847 pdf = self.switch_backends(FigureCanvasPdf)
- 1848 return pdf.print_pdf(*args, **kwargs)
1849
1850 def print_pgf(self, *args, **kwargs):
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/backends/backend_pdf.py
in print_pdf(self, filename, **kwargs)
2299 width, height, image_dpi,
RendererPdf(file, image_dpi),
2300
bbox_inches_restore=_bbox_inches_restore)
- 2301 self.figure.draw(renderer)
2302 renderer.finalize()
2303 finally:
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/artist.pyc
in draw_wrapper(artist, renderer, *args, **kwargs)
52 def draw_wrapper(artist, renderer, *args, **kwargs):
53 before(artist, renderer)
--- 54 draw(artist, renderer, *args, **kwargs)
55 after(artist, renderer)
56
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/figure.pyc
in draw(self, renderer)
1004 dsu.sort(key=itemgetter(0))
1005 for zorder, a, func, args in dsu:
- 1006 func(*args)
1007
1008 renderer.close_group('figure')
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/artist.pyc
in draw_wrapper(artist, renderer, *args, **kwargs)
52 def draw_wrapper(artist, renderer, *args, **kwargs):
53 before(artist, renderer)
--- 54 draw(artist, renderer, *args, **kwargs)
55 after(artist, renderer)
56
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/axes.pyc
in draw(self, renderer, inframe)
2084
2085 for zorder, a in dsu:
- 2086 a.draw(renderer)
2087
2088 renderer.close_group('axes')
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/artist.pyc
in draw_wrapper(artist, renderer, *args, **kwargs)
52 def draw_wrapper(artist, renderer, *args, **kwargs):
53 before(artist, renderer)
--- 54 draw(artist, renderer, *args, **kwargs)
55 after(artist, renderer)
56
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/collections.pyc
in draw(self, renderer)
693 (np.sqrt(x) * self.figure.dpi / 72.0))
694 for x in self._sizes]
-- 695 return Collection.draw(self, renderer)
696
697 class PolyCollection(Collection):
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/artist.pyc
in draw_wrapper(artist, renderer, *args, **kwargs)
52 def draw_wrapper(artist, renderer, *args, **kwargs):
53 before(artist, renderer)
--- 54
[sage-support] Re: Trouble saving graphics
I found a temporary workaround. Save as .eps (which works fine) and then
use epstopdf to convert to .pdf.
On Saturday, September 7, 2013 12:01:40 PM UTC-4, P Purkayastha wrote:
It seems saving to pdf is broken on sage-5.10 onwards. Works till
sage-5.9. The problem is only in graphs. Normal plots work fine.
I have opened http://trac.sagemath.org/ticket/15173 to track this.
On 09/07/2013 09:06 PM, Ed Scheinerman wrote:
I'm using Sage 5.11 on a Mac
I've been used to creating graphs and then saving them as PDF's for
inclusion in papers. Using code like this:
sage: g = graphs.PetersenGraph()
sage: pic = g.plot()
sage: pic.save('pete.pdf')
But now it just throws a bunch of errors:
---
AssertionErrorTraceback (most recent call
last)
ipython-input-56-d63f9d409a56 in module()
1 pic.save('pete.pdf')
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/sage/misc/decorators.pyc
in wrapper(*args, **kwds)
456 kwds[self.name + options] = suboptions
457
-- 458 return func(*args, **kwds)
459
460 #Add the options specified by @options to the signature
of the wrapped
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/sage/plot/graphics.pyc
in save(self, filename, **kwds)
2753 figure.savefig(filename, dpi=dpi,
bbox_inches='tight',
2754
bbox_extra_artists=self._bbox_extra_artists,
- 2755 transparent=transparent)
2756 else:
2757 figure.savefig(filename, dpi=dpi,
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/figure.pyc
in savefig(self, *args, **kwargs)
1368 kwargs.setdefault('edgecolor',
rcParams['savefig.edgecolor'])
1369
- 1370 self.canvas.print_figure(*args, **kwargs)
1371
1372 if transparent:
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/backend_bases.pyc
in print_figure(self, filename, dpi, facecolor, edgecolor, orientation,
format, **kwargs)
2053 orientation=orientation,
2054 dryrun=True,
- 2055 **kwargs)
2056 renderer = self.figure._cachedRenderer
2057 bbox_inches =
self.figure.get_tightbbox(renderer)
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/backend_bases.pyc
in print_pdf(self, *args, **kwargs)
1846 from backends.backend_pdf import FigureCanvasPdf # lazy
import
1847 pdf = self.switch_backends(FigureCanvasPdf)
- 1848 return pdf.print_pdf(*args, **kwargs)
1849
1850 def print_pgf(self, *args, **kwargs):
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/backends/backend_pdf.py
in print_pdf(self, filename, **kwargs)
2299 width, height,
image_dpi, RendererPdf(file, image_dpi),
2300
bbox_inches_restore=_bbox_inches_restore)
- 2301 self.figure.draw(renderer)
2302 renderer.finalize()
2303 finally:
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/artist.pyc
in draw_wrapper(artist, renderer, *args, **kwargs)
52 def draw_wrapper(artist, renderer, *args, **kwargs):
53 before(artist, renderer)
--- 54 draw(artist, renderer, *args, **kwargs)
55 after(artist, renderer)
56
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/figure.pyc
in draw(self, renderer)
1004 dsu.sort(key=itemgetter(0))
1005 for zorder, a, func, args in dsu:
- 1006 func(*args)
1007
1008 renderer.close_group('figure')
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/artist.pyc
in draw_wrapper(artist, renderer, *args, **kwargs)
52 def draw_wrapper(artist, renderer, *args, **kwargs):
53 before(artist, renderer)
--- 54 draw(artist, renderer, *args, **kwargs)
55 after(artist, renderer)
56
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/axes.pyc
in draw(self, renderer, inframe)
2084
2085 for zorder, a in dsu:
- 2086 a.draw(renderer)
2087
2088 renderer.close_group('axes')
/Users/ers/share/sage-5.11/local/lib/python2.7/site-packages/matplotlib/artist.pyc
in draw_wrapper(artist, renderer, *args, **kwargs)
52 def draw_wrapper(artist, renderer
[sage-support] Limit with gamma function gives incorrect answer
When I try this: sage: y = gamma(x+1/2)/gamma(x)/sqrt(x) sage: limit(y,x=oo) 0 sage: plot(y,x,1,50) I observe that the limit appears to be 1, not 0. Indeed, Mathematica confirms: In[1]:= y = Gamma[x + 1/2]/(Gamma[x]*Sqrt[x]) Out[1]= Gamma[1/2 + x]/(Sqrt[x] Gamma[x]) In[2]:= Limit[y, x - Infinity] Out[2]= 1 -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Error summing inverses of binomial coefficients
I am looking to evaluate $\sum_{k=0}^n 1/\binom{n}{k}$ so I type:
sage: n = var('n')
sage: k = var('k')
sage: sum(1/binomial(n,k),k,0,n)
(n + 1)*2^(-n)
and that answer is wrong.
For example, with n=10 we get
sage: sum(1/binomial(10,k) for k in range(11))
1433/630
but the alleged answer of (n+1)*2^(-n) is 11/1024.
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[sage-support] Problems upgrading to 5.9 on Ubunutu using apt-get
I'm using Ubuntu 13.04 and am having trouble upgrading to 5.9. I get this error message: W: Failed to fetch http://ppa.launchpad.net/aims/sagemath/ubuntu/dists/raring/main/binary-amd64/Packages Ideas? Thanks. -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
Re: [sage-support] Problems upgrading to 5.9 on Ubunutu using apt-get
Working great. Thank you!! On Monday, May 6, 2013 9:39:31 AM UTC-4, Jan Groenewald wrote: Hi These 64bit packages are now copied aqnd available in quantal and raring from the PPA. Please test and let us know. Regards, Jan On 6 May 2013 13:42, Jan Groenewald j...@aims.ac.za javascript: wrote: Hi There is only a binary built on 12.04. This binary works on 12.04 and 12.10, and I hope on 13.04. I've uploaded 5.9 64bit to the PPA for 12.04, but not yet the other versions. I've just now copied the packages on the PPA over to 12.10 and 13.04 -- this might take a while (minutes? hours?) to complete. Note from 5.9 on this is 64bit only. 32bit for raring 13.04 is available from the sagemath.org binary download section. Regards, Jan On 6 May 2013 13:02, Ed Scheinerman edward.sc...@gmail.com javascript: wrote: I'm using Ubuntu 13.04 and am having trouble upgrading to 5.9. I get this error message: W: Failed to fetch http://ppa.launchpad.net/aims/sagemath/ubuntu/dists/raring/main/binary-amd64/Packages Ideas? Thanks. -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support...@googlegroups.com javascript:. To post to this group, send email to sage-s...@googlegroups.comjavascript: . Visit this group at http://groups.google.com/group/sage-support?hl=en. For more options, visit https://groups.google.com/groups/opt_out. -- .~. /V\ Jan Groenewald /( )\www.aims.ac.za ^^-^^ -- .~. /V\ Jan Groenewald /( )\www.aims.ac.za ^^-^^ -- You received this message because you are subscribed to the Google Groups sage-support group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
[sage-support] Question about scoping
I'm confused by the fact that variables defined inside functions can
leak out and become global variables. Here's what I've noticed.
(1) I create a file called steiner.py for finding the Steiner point
given three points in the plane. Here's my code:
def dist(p,q):
Eulidean distance between two points
n = len(p)
d2 = sum( (p[k]-q[k])**2 for k in range(n) )
return sqrt(d2)
def steiner(a,b,c):
Given three points in the plane, find the point p that minimizes
the sum of the distances to those three points.
x = var('x')
y = var('y')
p = (x,y) # unknown point
# objective function to minimize
obj = dist(p,a) + dist(p,b) + dist(p,c)
# start search at center of mass of the three points
p0 = ( (a[0]+b[0]+c[0])/3., (a[1]+b[1]+c[1])/3. )
print Starting optimiztion at, p0
print obj.subs(x=p0[0], y=p0[1])
return minimize(obj,p0)
(2) I attach this file in a Sage session. If I have global variables x
or y, they are wiped out when I invoke the steiner function. Here's a
transcript.
$ sage
--
| Sage Version 4.7.2, Release Date: 2011-10-29 |
| Type notebook() for the GUI, and license() for information.|
--
sage: attach steiner.py
sage: x = [1,2,3]
sage: y = [4,5,6]
sage: x
[1, 2, 3]
sage: y
[4, 5, 6]
sage: steiner((1,1), (2,3), (5,2))
Starting optimiztion at (2.6665, 2.0)
5.4788343901030965
Optimization terminated successfully.
Current function value: 5.303240
Iterations: 6
Function evaluations: 7
Gradient evaluations: 7
(2.20179756615, 2.46316721759)
sage: x
x
sage: y
y
My apologies if this is something already covered, but it's confusing
and not the behavior I would expect. Other variables defined inside
functions don't leak out like this.
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