`implicit_plot` complex function in SageMath

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I'd like to plot the circle described by the equation |z - 1| = 1 for complex z. I expected the following to work:

f(x, y) = abs(x +  i * y -1)

implicit_plot(f(x, y) - 1, xrange=(-2, 2), yrange=(-2, 2))

but it fails with:

TypeError: unable to coerce to a real number

which seems strange, since abs does't return a complex number. The following fails with the same error:

f(x, y) = abs(x +  i * y -1)

implicit_plot((f(x, y) - 1).real_part(), xrange=(-2, 2), yrange=(-2, 2))

What's the right way to plot this function?

complex-numbers sage plot python
1 Answers

The command defining f defines it as a symbolic function:

sage: f(x, y) = abs(x +  i * y - 1)

sage: f
(x, y) |--> abs(x + I*y - 1)
sage: f(x, y)
abs(x + I*y - 1)

sage: parent(f)
Callable function ring with arguments (x, y)
sage: parent(f(x, y))
Symbolic Ring

The plotting commands to plot f(x, y) - 1 as in the question trigger internal code to convert the required expression into a "fast-callable function", which fails.

A workaround, valid for many plotting problems in Sage, is to use a lambda function instead.

For this, just introduce lambda x, y: before the expression in x and y to be plotted:

sage: implicit_plot(lambda x, y: f(x, y) - 1, xrange=(-2, 2), yrange=(-2, 2))
Launched png viewer for Graphics object consisting of 1 graphics primitive

Implicit plot of a circle using its complex numbers definition

user_38275750
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