How can I use graph object info to find: areas, number of sides, and perimeter lengths of a each connected polygon within a network of polygons?

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Steve 2019-12-9

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编辑： Steve 2019-12-13

graph_object_G.mat

I was told by an MVP in the MathWorks community, "You can save the graph object, G, which contains all the connection information. You should read up on the capabilities of graph objects, as they allow you to do many things!

https://www.mathworks.com/help/matlab/ref/graph.html#d117e525223". However, I was never told how to do this and the link does not provide information on how to do this either. Can someone explain how I could use the attached graph object info to find: areas, number of sides, and perimeter lengths of a each connected polygon within a network of polygons?

Thanks in advance for your help!

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Matt J 2019-12-9

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编辑：Matt J 2019-12-9

在 MATLAB Online 中打开

Using one of my recent FEX postings,

https://www.mathworks.com/matlabcentral/fileexchange/73630-spatialgraph2d

you should be able to convert a TripletGraph object tg to a polyshape array.

function [pshape,pgonNodes] = getPolyshapes(tg)
%
%IN:
%
%  tg: TripletGraph object
%
%OUT:
%
% pshape: polyshape array. Each pshape(i) is a constituent polygon.
% pgonNodes: A cell array such that pgonNodes{i} contain the node indices of the
%            i-th polygon.
  G=tg.CGraph;
  x=tg.C(:,1);
  y=tg.C(:,2);
  [pshape,pgonNodes]=polyshape( spatialgraph2D(G,x,y) );
end

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Matt J 2019-12-9

编辑：Matt J 2019-12-9

Well, first of all you need to come up with better terminology for it than "polygons", because polygons by definition have straight sides. You will invite much confusion otherwise. I will call them piecewise circular cells (PCCs).

For the PCC case, getPolyshapes will still give you useful information. It will output for you all the node numbers that participate in each PCC. Then, it's just a geometry game. I believe you already have the radius and central angle

(in radians) of each of the circular arcs, so each arc length will be,

ArcLength =

*Radius.

Then you just add up the arc lengths of all the sides of the PCC to get the perimeter.

For area calculation, it will be a bit more complicated, but not too bad. You can get the bulk area of the polygon (with straight sides) from the pshape output of getPolyshapes and then add/subtract the area of the bulgy orange part of any given circular arc like in the figure below. The area will need to be added or subtracted, depending on whether the arc bulges out of the polyshape or inward. You can check which way the arc bulges by testing whether one of the end points V1,V2 lies inside the polyshape or not. This can be done with the isinterior() function for polyshapes.

Steve 2019-12-10

Matt,

Here's one way we could get the required info--please let me know what you think and how we could implement this in Matlab, in conjunction with your existing scripts:

We have a list of all edges (IDs 1...2750 or so) with the corresponding

IDs of two central points (obj.C) belonging to each. The goal is to construct

n-tuples of edge IDs that describe PCCs (e.g. maybe the 5 edge IDs

10, 35, 47, 98, 110 are the edges of one of the closed PCCs). One algorithm

that does that is:

Make a second list of the three (at most) central points connected to each other central point.

Start a for loop with the first edge ID, say E1, choose one end's central point (C ID, say C1); make that point the head of a vector pointing there from the other central point (chord vector).

Find the other two central points connected to this head point (from the other list) and the corresponding vectors that point from the head to them (two edges, two chords).

Take cross products of the first chord with each of the other two.

Only one of these will be a positive cross product (counterclockwise turning); retain the ID of that edge and concatenate it to the edge ID we started with; we now have a 2-tuple [E1 E2]; also retain the ID of its central point to make [C1 C2].

Keep doing this iteratively (in a while loop), with E2 taking the role of E1, etc. After every step [E1 ... Ek], check if Ek=E1. If yes, we have gone too far, and BE=[E1 ... E(k-1)] is the polygon edge set, i.e., n=k-1; likewise, you get BV=[C1 ... Cn] as the vertex set. Store BE,BV as new variables and exit the while loop.

Close the for loop by taking the next edge ID, and do it for all of them.

This will generate a lot of duplicate polygon variables BE(), BV(), but that's OK. Discard duplicates.

Once we have the BE, BV vectors of edge IDs, the length of the vectors is the topology of the PCC.

The area is computed in two steps: the polygonal area (straight edges) is obtained by the standard formula using the (x,y) coordinates of the central points belonging to the IDs of BV, or by the method you had described above.

The circle segment areas within the arcs are either added or subtracted from that. Here we may have to go back to the way the theta is determined: we computed inner products of chords and end point directors (right?), but in the end we must have discarded their sign. We should keep the sign, because it tells us whether the arc curves counterclockwise or clockwise with respect to our edges as we go around the PCC. If we consistently concatenate our edges (as described above) by turning ccw at each central point, a ccw turning arc will subtract, a cw arc will add.

If this makes sense, how can we implement this into the existing code? Thanks again for all your help!

Matt J 2019-12-11

编辑：Matt J 2019-12-11

在 MATLAB Online 中打开

Nothing we've been discussing involves any code other than TripletGraph.m (orTripletGraph2.m, but you really should rename it), spatialgraph2D.m, and the above short getPolyshape() function.

tg is not the name of a file. It is just an arbitrary hypothetical name I chose for a variable generated using TripletGraph.m, for example,

load fpep.mat
tg=TripletGraph(fpep);

You can of course choose any names you wish for your variables.

Since I have told you several times that tg is a "TripletGraph object", I am going to assume that you did not understand that terminology. It is important that you do, however. Every variable in Matlab has a type or "class". These include traditional kinds of Matlab variables like doubles, chars, cells, and structs. It is also possible to define our own classes, however. TripletGraph is not just the name of the file that created, tg. It is also the name of its class - a class defined by us using the classdef command. You can see the classes of different variables using either the whos or class commands, for example,

>> a=1:3; b='ABC';
>> class(a)
ans =
    'double'
>> class(b)
ans =
    'char'
>> class(tg)
ans =
    'TripletGraph'
>> whos a b tg
  Name      Size             Bytes  Class           Attributes
  a         1x3                 24  double                    
  b         1x3                  6  char                      
  tg        1x1             191213  TripletGraph              

When I told you that tg is a TripletGraph object, this simply means that it is a variable of type 'TripletGraph'.

Matt J 2019-12-12

Like I said here, I don't think it's too bad.

Steve 2019-12-12

编辑：Steve 2019-12-13

Well, we know the chord-lengths (c) and arc-lengths (s) of each edge. We also know the radii (r), and thus the central angles for the same. So, we can definitely calculate the segment areas via the following equation: circular segment area = (1/2)*(r^2)*((180/Pi)*alpha - sin(alpha)). I am just not sure how to have Matlab to do it (in terms of syntax, etc.). Also, I'm not sure how to set up isinterior() to determine concavity (for +/- areas). Have any suggestions on how to set this up? Thanks again.

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