On Thu, Mar 5, 2020 at 2:32 PM Fernando Gouvea <fqgou...@colby.edu> wrote:
>
> This works, in the sense that there's no error. One does get a bunch of
> extraneous points near the boundary of the disk. It's as if plot_points were
> trying to connect the point at (0,1) and the point at (0,-1) along the
> circle, even though f_uv is 1 on the circle.
>
> Strangely, they occur only on the right hand side (i.e., positive u, not
> negative u). I tried setting plot_points to be 500, but the bad points don't
> go away. Changing the curve to y^2-x^3+x-1=0 doesn't make them go away either.
>
the reason is that implicit_plot attempts to approximate the function
it assumes continuous, so if it's negative inside, but near, the
boundary, and positive nearby, but outside, then a fake zero is being
drawn very close to the boundary.
That's why it should be better to create a plot in polar coordinates
and then transform it.
> Fernando
>
> On 3/5/2020 8:22 AM, Dima Pasechnik wrote:
>
> 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 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 http://www.colby.edu/~fqgouvea
> Carter Professor of Mathematics
> Dept. of Mathematics and Statistics
> Colby College
> 5836 Mayflower Hill
> Waterville, ME 04901
>
> We now face a choice between Christ and nothing, because Christ has
> claimed everything so that renouncing him can only be nihilism.
> -- Peter Leithart
>
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