The easiest way is to use Python functions rather than symbolic ones;
define a function that is 1 outside the unit disk, and implicitly plot it.
sage: def f_uv(u,v):
....: if u^2+v^2>=1:
....: return 1
....: else:
....: x=u*sqrt(9/(1-u^2-v^2))
....: y=v*sqrt(9/(1-u^2-v^2))
....: return y^2-x^3+x
....: implicit_plot(f_uv,(u,-1,1),(v,-1,1))
On Wednesday, March 4, 2020 at 12:45:19 AM UTC, Dima Pasechnik wrote:
>
> On Wed, Mar 4, 2020 at 12:20 AM Fernando Gouvea <fqgou...@colby.edu>
> wrote:
> >
> > But no, it doesn't work, since it gives a rectangular plot instead of
> one in polar coordinates. But maybe we are closer.
>
> I looked at the labels on the axes, and they do match the ranges of r
> and phi, so I don't udnerstand
> how it's possible.
>
> >
> > I still think implicit_plot should be smarter about values that do not
> make sense.
> >
> > Fernando
> >
> > On 3/3/2020 6:26 PM, Dima Pasechnik wrote:
> >
> > even better:
> >
> > sage: var('x y u v r phi')
> > ....: u=r*cos(phi)
> > ....: v=r*sin(phi)
> > ....: x=u*sqrt(9/(1-r^2))
> > ....: y=v*sqrt(9/(1-r^2))
> > ....: implicit_plot(y^2-x^3+x==0,(r,0,999/1000),(phi,-pi,pi))
> >
> > On Tue, Mar 3, 2020 at 10:28 PM Dima Pasechnik <dimp...@gmail.com>
> wrote:
> >
> > On Tue, Mar 3, 2020 at 10:10 PM Fernando Gouvea <fqgou...@colby.edu>
> wrote:
> >
> > The whole point of this is to show the behavior of the curve near
> infinity, so changing the limits is not an option.
> >
> > just paste together a number of rectangles where (u,v) stay inside the
> > unit circle.
> > (yes, this would need writing a loop, ideally)
> >
> > Fernando
> >
> > On 3/3/2020 4:15 PM, Dima Pasechnik wrote:
> >
> > On Tue, Mar 3, 2020 at 8:20 PM Fernando Gouvea <fqgou...@colby.edu>
> wrote:
> >
> > Here's what I ended up trying, with r=3:
> >
> > var('x y u v')
> > x=u*sqrt(9/(1-u^2-v^2))
> > y=v*sqrt(9/(1-u^2-v^2))
> > implicit_plot(y^2-x^3+x==0,(u,-1,1),(v,-1,1))
> >
> > That gives an error:
> >
> >
> /opt/sagemath-8.9/local/lib/python2.7/site-packages/sage/ext/interpreters/wrapper_rdf.pyx
>
> in sage.ext.interpreters.wrapper_rdf.Wrapper_rdf.__call__
> (build/cythonized/sage/ext/interpreters/wrapper_rdf.c:2237)()
> > 74 for i from 0 <= i < len(args):
> > 75 self._args[i] = args[i]
> > ---> 76 return self._domain(interp_rdf(c_args
> > 77 , self._constants
> > 78 , self._py_constants
> >
> > ValueError: negative number to a fractional power not real
> >
> > Is there some way to tell implicit_plot to stay inside u^2+v^2\leq 1? Or
> to ignore complex values?
> >
> > I'd just change the limits of u and v to make the rectangle of the
> > values you plot in, anyway,
> > to well stay inside the unit circle.
> >
> > The equivalent code seems to give the correct graph in Mathematica.
> >
> > Fernando
> >
> > On 2/29/2020 5:29 PM, Fernando Gouvea wrote:
> >
> > Some years ago in a book review, David Roberts had the idea of plotting
> an algebraic curve using the transformation (u,v) = (x,y)/(r2 + x2 +
> y2)1/2, which transforms the plane into a circle and makes it easy to
> visualize the projective completion of the curve. You can see some of his
> plots at
> https://www.maa.org/press/maa-reviews/rational-algebraic-curves-a-computer-algebra-approach
>
> >
> > I’d love to do this kind of plot for my students. Can anyone offer help
> on how to do it with Sage? (Of course the dream scenario would be to add
> this option to the plot method for curves...)
> >
> > I’ve been using implicit_plot for most of my examples, which seems to be
> equivalent of using C.plot() when C is a curve.
> >
> > Thanks,
> >
> > Fernando
> >
> >
> >
> >
> >
> > --
> > ==================================================================
> > Fernando Q. Gouvea Editor, MAA
> Reviews
> > Dept of Mathematics and Statistics
> http://www.colby.edu/~fqgouvea
> > Colby College
> http://www.maa.org/press/maa-reviews
> > Mayflower Hill 5836
> > Waterville, ME 04901
> >
> > A training in mathematics is a prerequisite today for work in almost
> > any scientific field, but even for those who are not going to become
> > scientists, it is essential because, if it is only through speech that
> > we can understand what freedom means, only through mathematics
> > can we understand what necessity means.
> > -- W. H. Auden
> >
> >
> > --
> > =============================================================
> > Fernando Q. Gouvea http://www.colby.edu/~fqgouvea
> > Carter Professor of Mathematics
> > Dept. of Mathematics and Statistics
> > Colby College
> > 5836 Mayflower Hill
> > Waterville, ME 04901
> >
> > If little else, the brain is an educational toy.
> > -- Tom Robbins
> >
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>
> >
> > --
> > =============================================================
> > Fernando Q. Gouvea http://www.colby.edu/~fqgouvea
> > Carter Professor of Mathematics
> > Dept. of Mathematics and Statistics
> > Colby College
> > 5836 Mayflower Hill
> > Waterville, ME 04901
> >
> > If little else, the brain is an educational toy.
> > -- Tom Robbins
> >
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>
> >
> > --
> > =============================================================
> > Fernando Q. Gouvea http://www.colby.edu/~fqgouvea
> > Carter Professor of Mathematics
> > Dept. of Mathematics and Statistics
> > Colby College
> > 5836 Mayflower Hill
> > Waterville, ME 04901
> >
> > If little else, the brain is an educational toy.
> > -- Tom Robbins
> >
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