Segment: also use dot product test for contains

[cbm755]

This is coming from discussion in #13302 .

This currently fails at least one slow test. So just RFC for now. CC @jksuom, @siefkenj, for discussion.

[cbm755]

According to the tests,

p = Point(x,y)
L = Segment2D(Point2D(0, 0), Point2D(4, 4))
p in L

should be False. Whereas I might've thought None in the usual SymPy way. I don't know enough about the geometry module to know if this is deliberate. It happens because rank is used in is_collinear and it behaves that way:

>>> Matrix([[1,0],[1,x]]).rank()
2

If this is how it is supposed to work, I cannot swap the order of the collinearity test and the dot prod tests...

[jksuom]

I would also have expected that is_collinear(p, L.p1, L.p2) return None in this case but that may be hard to arrange. Apparently Point regards p as a generic point, i.e., a point whose coordinates are algebraically independent. Maybe we'll have to accept this.

[cbm755]

I reverted my swap. And I found a new test for the triangle inequality stuff.

So I figure we can either:

(a) cherry-pick the test only to master: 0da2db5

(b) discuss whether adding back the dot prod test is desirable (this PR).

Thoughts?

[siefkenj]

Unfortunately sympy matrices assume symbols are different form other values in a matrix, so Matrix([[1,0],[1,x]]).rank() returns 2, even though it made an assumption that x != 0. The rref routine, with a small modification, could be made to inform the user if any non-zero assumptions needed to be made (_find_reasonable_pivots reports its assumptions).

I knew there was some reason I used the triangle inequality, and it looks like complex symbols were it! The relationship between complex numbers and the geometry module seems weird at the moment. If a point is initialized with symbols, should those symbols be recast with the real=True assumption? Or should we allow complex numbers as entries and make all routines complex-friendly?

[cbm755]

I don't know enough about the geometry module to offer an informed opinion. Numerically stable algorithms with Floats would be nice to have too.

[jksuom]

To say that a point p is in the segment p1p2 means that p = t*p1 + (1 - t)*p2 where t is in the interval [0, 1]. If the points are collinear then that is true for some t which can be explicitly computed from the coordinates (unless p1 == p2). Then there only remains to test whether that t is real and in the interval [0, 1]. That way there would be no need to declare everything real.

[czgdp1807]

@cbm755 @jksuom Please provide updates on this PR.

[czgdp1807]

@cbm755 Any updates on this?

[czgdp1807]

>>> u, v, w, z = symbols('u, v, w, z')
>>> l = Segment(Point(u, w), Point(v, z))
>>> p = Point(2*u/3 + v/3, 2*w/3 + z/3)
>>> l.contains(p)
True

The above depicts that the tests in this PR pass on master. So, closing this for now. However, if there is a need please re-open this. Thanks.