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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> > https://groups.google.com/d/msgid/sage-support/b1b0c01c-7dfd-0e17-a3d4-61012ab66d8b%40colby.edu.
> >
> > --
> > =============================================================
> > 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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