Can SymPy's existing set implementation handle parameterized curves?

[EKW]

For example if I wanted to represent the line segment from (0,0) to (1,1) (f(t) = (t, t) for t in (0, 1))- can this be done with an image set? How? Or a curve in 3d space (for example, f(t) = (t^2, t, 1 - t) for t in (0, 1)).

[Oscar Benjamin]

Yes, you can do that with ImageSet:

In [11]: ImageSet(Lambda(t, (t, t)), Interval(0, 1))
Out[11]: {(t, t) | t ∊ [0, 1]}

In [14]: ImageSet(Lambda(t, (t**2, t, 1-t)), Interval(0, 1))
Out[14]:
⎧⎛ 2 ⎞ ⎫
⎨⎝t , t, 1 - t⎠ | t ∊ [0, 1]⎬
⎩ ⎭

On Tue, 1 Oct 2019 at 22:58, EKW <etk...@gmail.com> wrote:
>
> For example if I wanted to represent the line segment from (0,0) to (1,1) (f(t) = (t, t) for t in (0, 1))- can this be done with an image set? How? Or a curve in 3d space (for example, f(t) = (t^2, t, 1 - t) for t in (0, 1)).
>

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[EKW]

OK, I wasn't sure if that was the intended way, since using tuples quickly leads to errors such as:

>>> a = ImageSet(Lambda(t, (t, t)), Interval(0, 1))

>>> a

ImageSet(Lambda(t, (t, t)), Interval(0, 1))

>>> (0,0) in a

True

>>> b = ImageSet(Lambda(t, (1 - t, t)), Interval(0, 1))

>>> b

ImageSet(Lambda(t, (1 - t, t)), Interval(0, 1))

>>> a.intersect(b)

Intersection(ImageSet(Lambda(t, (t, t)), Interval(0, 1)), ImageSet(Lambda(t, (1 - t, t)), Interval(0, 1)))

>>> simplify(a.intersect(b))

Traceback (most recent call last):

  File "<stdin>", line 1, in <module>

  File "/home/eward/se/sympy/simplify/simplify.py", line 582, in simplify

    return done(expr)

  File "/home/eward/se/sympy/simplify/simplify.py", line 541, in done

    rv = e.doit() if doit else e

  File "/home/eward/se/sympy/core/basic.py", line 1691, in doit

    for term in self.args]

  File "/home/eward/se/sympy/core/basic.py", line 1691, in <listcomp>

    for term in self.args]

  File "/home/eward/se/sympy/sets/fancysets.py", line 460, in doit

    return SetExpr(base_set)._eval_func(f).set

  File "/home/eward/se/sympy/sets/setexpr.py", line 84, in _eval_func

    res = set_function(func, self.set)

  File "/home/eward/se/sympy/sets/sets.py", line 2255, in set_function

    return _set_function(f, x)

  File "/home/eward/se/sympy/multipledispatch/dispatcher.py", line 198, in __call__

    return func(*args, **kwargs)

  File "/home/eward/se/sympy/sets/handlers/functions.py", line 70, in _set_function

    sing = [i for i in singularities(expr, var)

  File "/home/eward/se/sympy/calculus/singularities.py", line 83, in singularities

    if not expression.is_rational_function(symbol):

AttributeError: 'Tuple' object has no attribute 'is_rational_function'

[Oscar Benjamin]

Yeah, a lot of work is needed in sets.

I see that error on master but not on my PR:
https://github.com/sympy/sympy/pull/17593

There it doesn't raise but just returns unevaluated:
In [3]: simplify(a.intersect(b))
Out[3]: {(t, t) | t ∊ [0, 1]} ∩ {(1 - t, t) | t ∊ [0, 1]}

There is no code to handle intersections of this kind of ImageSet yet.

Oscar

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