Generating matrix of distances between points in cylindrical coordinates

Question

I am working on an analytical model for the sound intensity generated from a vibrating cylinder. Part of this work requires me to split the cylinder up into a number of elements, with the more elements increasing the accuracy of my results.

I am fairly new to Mathematica and would like to generate a function that will have the following inputs and output.

Input: Number of vertical partitions (nn), number of circumferential partitions (mm), and Cylinder height (L) and, outer radius (a).

Output: Matrix where each element contains distance between the centroid's of two of the elements.

For example, for a cylinder with height L = 2 m and outer radius a = 1 m and we split the cylinder into 4 elements we would have nn = 1 vertical partition and mm = 2 circumferential partitions. The function would then output a 4 by 4 matrix containing the distance between element 1 and elements 1, 2, 3 and 4, element 2 and elements 1, 2, 3 and 4, etc.

So far I have managed to produce a table containing the coordinates of the centroid of each element. Using,

CentroidList = Table[{tr = a, t\[Theta] = (i*\[Pi])/mm, tz = (j*L)/(2*nn)}, {i,Range[1, 2*mm, 2]}, {j, j = 1, j = 2*nn, j = j + 1}]

Now I am trying to use this Table to generate the matrix which will contain the distance between each of the centroids. What is the best way to do this?

So far I have only been able to produce the distance between two centroids by using the following (where I must manually input the indexes for Part[ ] to get the correct result);

Example, distance between centroid of element 1 and centroid of element 2:

r12 = Sqrt[(Part[CentroidList, 1, 1, 1]*
   Cos[Part[CentroidList, 1, 1, 2]] - 
  Part[CentroidList, 1, 2, 1]*
   Cos[Part[CentroidList, 1, 2, 2]])^2 + (Part[CentroidList, 1, 1,
     1]*Sin[Part[CentroidList, 1, 1, 2]] - 
  Part[CentroidList, 1, 2, 1]*
   Sin[Part[CentroidList, 1, 2, 2]])^2 + (Part[CentroidList, 1, 1,
    3] - Part[CentroidList, 1, 2, 3])^2]

Is is possible to repeat this calculation (automatically changing the indexing in Part[ ]) and assigning the result to a particular location in a matrix?

Answer 1

Here is a slight rewriting of your centroidList for the specific case you mention.

a = 1; nn = 1; mm = 2; L = 2; 
centroidList = Flatten[Table[{a, (i*\[Pi])/mm, (j*L)/(2*nn)}, 
       {i, 1, 2*mm, 2}, {j, 1, 2*nn}], 1]

{{1, \[Pi]/2, 1}, {1, \[Pi]/2, 2}, {1, (3 \[Pi])/2, 1}, {1, (3 \[Pi])/ 2, 2}}

To calculate the distance matrix, you can use the Norm function.

dist = Table[Norm[centroidList[[i]] - centroidList[[j]]], 
         {i, 1, Length[centroidList]}, {j, 1, Length[centroidList]}]

dist//MatrixForm

Of course, this implements the Euclidean norm, and yours is different. Here is a function that calculates the distance d[ ]. Then this can be placed in the Table straightforwardly:

d[{a1_, a2_, a3_}, {b1_, b2_, b3_}] := 
   Sqrt[(a1*Cos[a2] - b1*Cos[b2])^2 + (a1*Sin[a2] - b1*Sin[b2])^2 + (a3 - b3)^2];

Table[d[centroidList[[i]], centroidList[[j]]], {i, 1, 
   Length[centroidList]}, {j, 1, Length[centroidList]}] // MatrixForm

Or, equivalently, you can use the DistanceMatrix command (as suggested by J.M.) with this modified metric:

DistanceMatrix[centroidList, DistanceFunction -> d]

or the Outer product (as suggested by Quantum_Oli)

Outer[d, centroidList, centroidList, 1]

to get the same result.