Andrea Peri 2007 wrote:
In the 6.1.5 paragraph of 06-103r3 is reported:
For MultiPoints:
..
A MultiPoint is simple if no two Points in the MultiPoint are equal
(have identical coordinate values in X and Y).
Every MultiPoint is spatially equal under the definition in Clause
6.1.15.3 to a simple Multipoint.
..
Yes, this is clear. MultiPoints with duplicate Point values are non-simple.
For curves:
..
A Curve is simple if it does not pass through the same Point twice
with the possible exception of the two end
points (Reference [1], section 3.12.7.3):
∀ c ∈ Curve, [a, b] = c.Domain, c =: f :[a, b] → ℜ n
c.IsSimple ⇔ ∀ x1, x2 ∈ [a, b]: [ f(x1)=f(x2) ∧ x1<x2] ⇒ [x1=a ∧ x2=b]
..
So a curve with 2 consecutive point like this
linestring (10 10, 20 20, 20 20, 30 30) is violating this definition
I disagree. In the formal definition of Curve above, note the condition
that x1 < x2. In any continuous parameterization f of LINESTRING(10 10,
20 20, 20 20, 30 30), if f(x1) = pt[1] (20 20) and f(x2 = pt[2] (20 20),
then x1 must = x2.
This is why I say that repeated points are topologically irrelevant -
they are topologically indistinguishable under any continuous
parameterization function.
But even this line is invalid for this definition:
linestring (10 10, 20 20, 30 30, 40 40, 50 50, 30 30, 60 60)
(where the repeated points are not consecutive).
because the definition reporting in document:
∀ x1, x2 ∈ [a, b]: [ f(x1)=f(x2) ∧ x1<x2] ⇒ [x1=a ∧ x2=b]
Yes, there's no issue here. Under the continuous parameterization
definition non-consecutive identical points clearly make the linestring
non-simple.
--
Martin Davis
Senior Technical Architect
Refractions Research, Inc.
(250) 383-3022
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