Calculate “Volume” and “Surface Area” of a voxel-based sphere












5












$begingroup$


By "voxel-based sphere" I mean a sphere made up of cubes. Sorry if that is not the correct terminology. Imagine a sphere made out of legos. Except each voxel is a cube (unlike most legos). Determining the voxel distribution to make the sphere I suppose would involve calculating the x/y/z of a position on the sphere and then 'snapping' it to the nearest multiple of the voxel width/height.



Here's a calculator that can generate one:
http://neil.fraser.name/news/2006/11/17/



And here is an image of one (cross sectioned):



enter image description here



Given the diameter of this voxel sphere (in number of voxels, e.g. '20 voxels in diameter'), how can I calculate:




  1. The number of voxels in the sphere if it were hollow (kind of a 'surface area')

  2. The number of voxels in the sphere if it were solid (kind of a 'volume')


Is there a formula possible here? :)






geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Decompose it into cross sections; each segment should allow an explicit summation involving square roots and floor/ceiling functions.
    $endgroup$
    – anon
    Nov 10 '11 at 3:03
















5












$begingroup$


By "voxel-based sphere" I mean a sphere made up of cubes. Sorry if that is not the correct terminology. Imagine a sphere made out of legos. Except each voxel is a cube (unlike most legos). Determining the voxel distribution to make the sphere I suppose would involve calculating the x/y/z of a position on the sphere and then 'snapping' it to the nearest multiple of the voxel width/height.



Here's a calculator that can generate one:
http://neil.fraser.name/news/2006/11/17/



And here is an image of one (cross sectioned):



enter image description here



Given the diameter of this voxel sphere (in number of voxels, e.g. '20 voxels in diameter'), how can I calculate:




  1. The number of voxels in the sphere if it were hollow (kind of a 'surface area')

  2. The number of voxels in the sphere if it were solid (kind of a 'volume')


Is there a formula possible here? :)






geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Decompose it into cross sections; each segment should allow an explicit summation involving square roots and floor/ceiling functions.
    $endgroup$
    – anon
    Nov 10 '11 at 3:03














5












5








5





$begingroup$


By "voxel-based sphere" I mean a sphere made up of cubes. Sorry if that is not the correct terminology. Imagine a sphere made out of legos. Except each voxel is a cube (unlike most legos). Determining the voxel distribution to make the sphere I suppose would involve calculating the x/y/z of a position on the sphere and then 'snapping' it to the nearest multiple of the voxel width/height.



Here's a calculator that can generate one:
http://neil.fraser.name/news/2006/11/17/



And here is an image of one (cross sectioned):



enter image description here



Given the diameter of this voxel sphere (in number of voxels, e.g. '20 voxels in diameter'), how can I calculate:




  1. The number of voxels in the sphere if it were hollow (kind of a 'surface area')

  2. The number of voxels in the sphere if it were solid (kind of a 'volume')


Is there a formula possible here? :)






geometry







share|cite|improve this question









$endgroup$




By "voxel-based sphere" I mean a sphere made up of cubes. Sorry if that is not the correct terminology. Imagine a sphere made out of legos. Except each voxel is a cube (unlike most legos). Determining the voxel distribution to make the sphere I suppose would involve calculating the x/y/z of a position on the sphere and then 'snapping' it to the nearest multiple of the voxel width/height.



Here's a calculator that can generate one:
http://neil.fraser.name/news/2006/11/17/



And here is an image of one (cross sectioned):



enter image description here



Given the diameter of this voxel sphere (in number of voxels, e.g. '20 voxels in diameter'), how can I calculate:




  1. The number of voxels in the sphere if it were hollow (kind of a 'surface area')

  2. The number of voxels in the sphere if it were solid (kind of a 'volume')


Is there a formula possible here? :)






geometry




geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 10 '11 at 2:56









InfinitiesLoopInfinitiesLoop

12614




12614








  • 1




    $begingroup$
    Decompose it into cross sections; each segment should allow an explicit summation involving square roots and floor/ceiling functions.
    $endgroup$
    – anon
    Nov 10 '11 at 3:03














  • 1




    $begingroup$
    Decompose it into cross sections; each segment should allow an explicit summation involving square roots and floor/ceiling functions.
    $endgroup$
    – anon
    Nov 10 '11 at 3:03








1




1




$begingroup$
Decompose it into cross sections; each segment should allow an explicit summation involving square roots and floor/ceiling functions.
$endgroup$
– anon
Nov 10 '11 at 3:03




$begingroup$
Decompose it into cross sections; each segment should allow an explicit summation involving square roots and floor/ceiling functions.
$endgroup$
– anon
Nov 10 '11 at 3:03










1 Answer
1






active

oldest

votes


















0












$begingroup$

That's very easy, it's a function of an XYZ loop....



You would be actually doing a marching cubes kind of logic, and an ISOsurface logic for a spherical volume.



The first result is this programming query (in metacode):



var cubecount:
ForLoop(x,y,z){
If space(x,y,z) corresponds to DistanceFunction(x,y,z) == 1;
Then cubecount +=1:



question 2 is the same using >=1



For a sphere on zero of radius 1. (distancefunction is xposvectormagnitude, same thing if the sphere is origin is 0; otherwise the sphere pos equation is for the xyz loop is:



This code makes a 3d printable off axis voxel ball if you want:



num = 45.9874;



for (x =[-num-5:num+5]) 

{

    for (y =[-num-5:num+5]) 

    {    

        for (z =[-num-5:num+5]) 

        {    

              if ((x+1.24)*(x+1.24)+(y+1.66)*(y+1.66)+(z+1.88)*(z+1.88)<=num)  

                translate([x,y,z])    

                cube(1);





        }





    }

}


https://www.thingiverse.com/thing:3075040






share|cite|improve this answer









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f80726%2fcalculate-volume-and-surface-area-of-a-voxel-based-sphere%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    That's very easy, it's a function of an XYZ loop....



    You would be actually doing a marching cubes kind of logic, and an ISOsurface logic for a spherical volume.



    The first result is this programming query (in metacode):



    var cubecount:
    ForLoop(x,y,z){
    If space(x,y,z) corresponds to DistanceFunction(x,y,z) == 1;
    Then cubecount +=1:



    question 2 is the same using >=1



    For a sphere on zero of radius 1. (distancefunction is xposvectormagnitude, same thing if the sphere is origin is 0; otherwise the sphere pos equation is for the xyz loop is:



    This code makes a 3d printable off axis voxel ball if you want:



    num = 45.9874;
    

    
for (x =[-num-5:num+5]) 
    
{
    
    for (y =[-num-5:num+5]) 
    
    {    
    
        for (z =[-num-5:num+5]) 
    
        {    
    
              if ((x+1.24)*(x+1.24)+(y+1.66)*(y+1.66)+(z+1.88)*(z+1.88)<=num)  
    
                translate([x,y,z])    
    
                cube(1);
    

    

    
        }
    

    

    
    }
    
}


    https://www.thingiverse.com/thing:3075040






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      That's very easy, it's a function of an XYZ loop....



      You would be actually doing a marching cubes kind of logic, and an ISOsurface logic for a spherical volume.



      The first result is this programming query (in metacode):



      var cubecount:
      ForLoop(x,y,z){
      If space(x,y,z) corresponds to DistanceFunction(x,y,z) == 1;
      Then cubecount +=1:



      question 2 is the same using >=1



      For a sphere on zero of radius 1. (distancefunction is xposvectormagnitude, same thing if the sphere is origin is 0; otherwise the sphere pos equation is for the xyz loop is:



      This code makes a 3d printable off axis voxel ball if you want:



      num = 45.9874;
      

      
for (x =[-num-5:num+5]) 
      
{
      
    for (y =[-num-5:num+5]) 
      
    {    
      
        for (z =[-num-5:num+5]) 
      
        {    
      
              if ((x+1.24)*(x+1.24)+(y+1.66)*(y+1.66)+(z+1.88)*(z+1.88)<=num)  
      
                translate([x,y,z])    
      
                cube(1);
      

      

      
        }
      

      

      
    }
      
}


      https://www.thingiverse.com/thing:3075040






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        That's very easy, it's a function of an XYZ loop....



        You would be actually doing a marching cubes kind of logic, and an ISOsurface logic for a spherical volume.



        The first result is this programming query (in metacode):



        var cubecount:
        ForLoop(x,y,z){
        If space(x,y,z) corresponds to DistanceFunction(x,y,z) == 1;
        Then cubecount +=1:



        question 2 is the same using >=1



        For a sphere on zero of radius 1. (distancefunction is xposvectormagnitude, same thing if the sphere is origin is 0; otherwise the sphere pos equation is for the xyz loop is:



        This code makes a 3d printable off axis voxel ball if you want:



        num = 45.9874;
        

        
for (x =[-num-5:num+5]) 
        
{
        
    for (y =[-num-5:num+5]) 
        
    {    
        
        for (z =[-num-5:num+5]) 
        
        {    
        
              if ((x+1.24)*(x+1.24)+(y+1.66)*(y+1.66)+(z+1.88)*(z+1.88)<=num)  
        
                translate([x,y,z])    
        
                cube(1);
        

        

        
        }
        

        

        
    }
        
}


        https://www.thingiverse.com/thing:3075040






        share|cite|improve this answer









        $endgroup$



        That's very easy, it's a function of an XYZ loop....



        You would be actually doing a marching cubes kind of logic, and an ISOsurface logic for a spherical volume.



        The first result is this programming query (in metacode):



        var cubecount:
        ForLoop(x,y,z){
        If space(x,y,z) corresponds to DistanceFunction(x,y,z) == 1;
        Then cubecount +=1:



        question 2 is the same using >=1



        For a sphere on zero of radius 1. (distancefunction is xposvectormagnitude, same thing if the sphere is origin is 0; otherwise the sphere pos equation is for the xyz loop is:



        This code makes a 3d printable off axis voxel ball if you want:



        num = 45.9874;
        

        
for (x =[-num-5:num+5]) 
        
{
        
    for (y =[-num-5:num+5]) 
        
    {    
        
        for (z =[-num-5:num+5]) 
        
        {    
        
              if ((x+1.24)*(x+1.24)+(y+1.66)*(y+1.66)+(z+1.88)*(z+1.88)<=num)  
        
                translate([x,y,z])    
        
                cube(1);
        

        

        
        }
        

        

        
    }
        
}


        https://www.thingiverse.com/thing:3075040







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Oct 5 '18 at 15:40









        com.prehensiblecom.prehensible

        18511




        18511






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f80726%2fcalculate-volume-and-surface-area-of-a-voxel-based-sphere%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Biblatex bibliography style without URLs when DOI exists (in Overleaf with Zotero bibliography)

            Image
            2 2 I have a Zotero bibliography in Overleaf. I use biblatex . usepackage[backend=biber,citestyle=authoryear]{biblatex} addbibresource{zoteroALH.bib} Bibliography entries look like: @book{de_saint-gervais_uniformization_2016, title = {Uniformization of {Riemann} {Surfaces}}, isbn = {978-3-03719-145-3}, url = {https://www.ems-ph.org/books/book.php?proj_nr=198&srch=series%7Chem}, urldate = {2019-02-03}, author = {de Saint-Gervais, Paul Henri}, year = {2016}, doi = {10.4171/145} } In the bibliography down the paper the reference shows like this: There is the DOI, great, but also the URL which is unnecessary when the DOI is there. How can I remove the URL when the DOI is present? biblatex bibliographies overleaf zote
            Read more

            ComboBox Display Member on multiple fields

            Image
            2 I'm trying to set the .DisplayMember property of a ComboBox in C#, but I want to bind it to multiple columns in the .DataSouce . My SQL looks like this: SELECT PersNbr, PersFirstName, PersMiddleName, PersLastName FROM Pers WHERE PersNbr = :persNbr; I'm saving this query in a DataTable so each column selected has it's own column in the Datatable . I want to make the .DisplayMember a combination of PersFirstName + PersMiddleName + PersLastName so their full name appears like this: comboBox.DisplayMemeber = "PersFirstName" + "PersMiddleName" + "PersLastName" I know I can just to this on the query: SELECT PersNbr, (PersFirstName || PersMiddleName || PersLastName) PersName and then just do this: comboBox.DisplayMember = "PersName"; but I
            Read more

            Is it possible to collect Nectar points via Trainline?

            Image
            2 Since I am not eligible for any Railcard, the only available discount/bonus scheme is Nectar Card for me. GWR, SWR, LNER and Virgin Trains allow to collect Nectar points and I have already linked a card to my GWR account. A friend suggested me to use Trainline App, because it can search for bus tickets either. I had a look at the application, but could not see any option for Nectar. Is there a way to collect Nectar points when I buy tickets via Trainline? uk trains loyalty-programs trainline share | improve this question asked Dec 2 '18 at 15:50 ahmedus 3,224 5 19 49
            Read more