Is out there scripts for determing convex sub-meshes?


(Charlls) #1

Hi folks

I would like to know if there is a way (API or a already done script) to determine the convex (i.e: simply connected ) sub-meshes in a mesh

Regards,


(S68) #2

‘Convex’ and ‘simply connected’ are not synonimous

Stefano

P.S. the answer to your question is ‘I dont know’


(Charlls) #3

convex is not synonymous of simply-connected…

do you understand what a convex and a simply-connected mesh are?

convex is that curvature doesn’t changes his sign along the surface (Riemaniann curvature), but im pretty sure you understand what Riemaniann curvature is

simply-connected is a set in which two arbitrary points are connected by a straight line (psst, do you know what a straight line is, right?)

in N dimensional cartesian space, a simply-connected set is ALWAYS bounded by a convex surface. the converse is true as long as the surface itself is weakly-connected i.e:two arbitrary point are connected by a arbitrary curve

in computer graphics, real-time rendering of complex sites depend on the ordering of the drawables (the meshes, which btw are models of sets and surfaces, hence the relation) in convex sets in a tree-structure, but even if this is the essence of my question, of course this goes outside the scope of your ‘doubt’

I hope that clears you up about the relation between the terms and why they are used interchangeably

Regards,


(S68) #4

I LOVE people who goes so deep in mathematics :slight_smile:

For the engeenieristic point of view it is NOT

Here a surface (N-1D set in ND space) is convex if the radius of curvature
does not change sign and is simply connected if you can go from any point on the surface to any other point in the surface with a curved line belonging to the surface :smiley:

For me a sphere is a convex, simply connected mesh.

Two non-intersecating non-one-inside-the-other spheres non are two convex simply connected mesh.

If I Join the meshes (Select both CTRL+J in Blender) I ge a Single mesh which is absolutely not simply connected, and of which I can say that is made by two convex meshes.

If I take the same sphere and punch a vertex towards the center in mesh editing I get a single simply connected mesh which is not convex.

All this with good peace of

Georg Friedrich Bernhard Riemann
Born: 17 Sept 1826 in Breselenz, Hanover (now Germany)
Died: 20 July 1866 in Selasca, Italy

     1.  R Dedekind, Biography of Riemann, in H Weber and R Dedekind (eds.), The Collected Works of Riemann (New York, 1953).
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     4. M Monastyrsky, Rieman, Topology and Physics (Boston-Basel, 1987).
     5. G Schulz, Riemann, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).
     6. W L Gonczarow, On the scientific papers of Riemann (Polish), Wiadom Mat. (2) 2 (1959), 155-196.
     7. H Grauert, Bernhard Riemann and his ideas in philosophy of nature, in Analysis, geometry and groups: a Riemann legacy volume (Palm Harbor, FL, 1993), 124-132.
     8. Y K Hon, Georg Friedrich Bernhard Riemann, Bull. Malaysian Math. Soc. 6 (2) (1975), 1-6.
     9. F Klein, Riemann und seine Bedeutung für die Entwicklung der modernen Mathematik, Ges. Math. Abh. 3 (1923), 482-497.
     10. S Kulczycki, On Riemann's habilitational address (Polish), Wiadom. Mat. (2) 1 (1955/1956), 180-193.
     11. E Portnoy, Riemann's contribution to differential geometry, Historia Math. 9 (1) (1982), 1-18.
     12. E Scholz, Riemann's vision of a new approach to geometry, in 1830-1930: a century of geometry (Berlin, 1992), 22-34.
     13. F G Tricomi, Bernhard Riemann e l'Italia, Univ. e Politec. Torino Rend. Sem. Mat. 25 (1965/1966), 57-72.
     14. A Weil, Riemann, Betti and the birth of topology, Arch. Hist. Exact Sci. 20 (2) (1979), 91-96.
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(Charlls) #5

Two non-intersecating non-one-inside-the-other spheres non are two convex simply connected mesh.

If I Join the meshes (Select both CTRL+J in Blender) I ge a Single mesh which is absolutely not simply connected, and of which I can say that is made by two convex meshes.

That is why the requirement for the convex surface needs to be also weakly connected is important to assure that the bounded region is simply connected

i think you’re confusing some terms:

weakly connected is that you can join every pair of points with a continuous curve (not necessarily a line)

simply connected is that you can join every pair of points with a line (a line is a straight curve)

is important to understand the difference between a straight line and a continuous curve:

every line IS a continuous curve
BUT
every continuous curve IS NOT necessarily a line

Hope that helps

Regards,

btw: nice bibliography, have you read any of them?


(theeth) #6

yes, he probably did. (www.selleri.org, under Stefano, you can look at his CV if you understand italian a bit).

Martin