OpenSCAD

Math

This tutorial will show you how OpenSCAD can use math.

How do I make this?

  1. If you do not have OpenSCAD installed, download the application from openscad.org

  2. Download 3dplot.scad and place it in the same directory as your OpenSCAD file that will use it.

  3. Open 3dplot.scad and comment out all the examples. Save the file.

  4. Create a new OpenSCAD document. Save it as 2dgraphing.scad. This code is derived from user Brad Pitcher: 
    pi = 3.1415926536;
e = 2.718281828;

/*************************
* Cartesian equations 
*************************/
/* Sine wave */

module sineWave(){
    linear_extrude(height=5)
    2dgraph([-50, 50], 3, steps=50);
}

module  parabola (){
    linear_extrude(height=5)
    2dgraph([-50, 50], 3, steps=40);
}

/* Ellipsoid - use a cartesion equation for a half ellipse, 
then rotate extrude it */

module ellipsoid(){

    rotate_extrude(convexity=10, $fn=100) 
    rotate([0, 0, -90]) 
    2dgraph([-50, 50], 3, steps=100);
}

/*************************
* Polar equations 
*************************/
/* Rose curve */

module rose(){
    scale([20, 20, 20]) linear_extrude(height=0.15)
    2dgraph([0, 720], 0.1, steps=160, polar=true);
}

/* Archimedes spiral */
module archimedesSpiral(){
    scale([0.02, 0.02, 0.02]) linear_extrude(height=150)
    2dgraph([0, 360*3], 50, steps=100, polar=true);
}

/* Golden spiral */
module goldenSpiral(){
    linear_extrude(height=50)
    2dgraph([0, 7*180], 1, steps=300, polar=true);
}

/**************************
* Parametric equations 
*************************/
/* 9-pointed star */
module ninePointedStar(s,h){
    scale(s) linear_extrude(height=h)
    2dgraph([10, 1450], 0.1, steps=9, parametric=true);
}

/*************************/
// function to convert degrees to radians
function d2r(theta) = theta*360/(2*pi);

// These functions are here to help get the slope of each segment, and use that to find points for a correctly oriented polygon
function diffx(x1, y1, x2, y2, th) = cos(atan((y2-y1)/(x2-x1)) + 90)*(th/2);
function diffy(x1, y1, x2, y2, th) = sin(atan((y2-y1)/(x2-x1)) + 90)*(th/2);
function point1(x1, y1, x2, y2, th) = [x1-diffx(x1, y1, x2, y2, th), y1-diffy(x1, y1, x2, y2, th)];
function point2(x1, y1, x2, y2, th) = [x2-diffx(x1, y1, x2, y2, th), y2-diffy(x1, y1, x2, y2, th)];
function point3(x1, y1, x2, y2, th) = [x2+diffx(x1, y1, x2, y2, th), y2+diffy(x1, y1, x2, y2, th)];
function point4(x1, y1, x2, y2, th) = [x1+diffx(x1, y1, x2, y2, th), y1+diffy(x1, y1, x2, y2, th)];
function polarX(theta) = cos(theta)*r(theta);
function polarY(theta) = sin(theta)*r(theta);

module nextPolygon(x1, y1, x2, y2, x3, y3, th) {
    if((x2 > x1 && x2-diffx(x2, y2, x3, y3, th) < x2-diffx(x1, y1, x2, y2, th) || (x2 <= x1 && x2-diffx(x2, y2, x3, y3, th) > x2-diffx(x1, y1, x2, y2, th)))) {
        polygon(
            points = [
                point1(x1, y1, x2, y2, th),
                point2(x1, y1, x2, y2, th),
                // This point connects this segment to the next
                point4(x2, y2, x3, y3, th),
                point3(x1, y1, x2, y2, th),
                point4(x1, y1, x2, y2, th)
            ],
            paths = [[0,1,2,3,4]]
        );
    }
    else if((x2 > x1 && x2-diffx(x2, y2, x3, y3, th) > x2-diffx(x1, y1, x2, y2, th) || (x2 <= x1 && x2-diffx(x2, y2, x3, y3, th) < x2-diffx(x1, y1, x2, y2, th)))) {
        polygon(
            points = [
                point1(x1, y1, x2, y2, th),
                point2(x1, y1, x2, y2, th),
                // This point connects this segment to the next
                point1(x2, y2, x3, y3, th),
                point3(x1, y1, x2, y2, th),
                point4(x1, y1, x2, y2, th)
            ],
            paths = [[0,1,2,3,4]]
        );
    }
    else {
        polygon(
            points = [
                point1(x1, y1, x2, y2, th),
                point2(x1, y1, x2, y2, th),
                point3(x1, y1, x2, y2, th),
                point4(x1, y1, x2, y2, th)
            ],
            paths = [[0,1,2,3]]
        );
    }
}

module 2dgraph(bounds=[-10,10], th=2, steps=10, polar=false, parametric=false) {

    step = (bounds[1]-bounds[0])/steps;
    union() {
        for(i = [bounds[0]:step:bounds[1]-step]) {
            if(polar) {
                nextPolygon(polarX(i), polarY(i), polarX(i+step), polarY(i+step), polarX(i+2*step), polarY(i+2*step), th);
            }
            else if(parametric) {
                nextPolygon(x(i), y(i), x(i+step), y(i+step), x(i+2*step), y(i+2*step), th);
            }
            else {
                nextPolygon(i, f(i), i+step, f(i+step), i+2*step, f(i+2*step), th);
            }
        }
    }
}
  5. Open another OpenSCAD document.

  6. To display the axes in the model view press Command+2 (MAC) or CTRL+2 (Windows and Linux).


  7. Save the new document as openscadMath.scad in the same place as you saved 2dgraphing.scad.

  8. The first line of the new file should be : 
    include <2dgraph.scad>
include <3dplot.scad>


  9. This is how you would create a sine wave: 
    function _f(x) = 20*sin((1/4)*x);
function f(x) = _f(d2r(x));
sineWave();
    Press F5 to render

  10. Here are some other shapes:
    ShapeCode
    Parabola
    function f(x) = pow(x, 2)/20;
parabola();
    ellipsoid
    function f(x) = sqrt((2500-pow(x, 2))/3);
ellipsoid();
    rose
    function r(theta) = cos(4*theta);
rose();
    Archimedes Spiral
    function r(theta) = theta/2*pi;
archimedesSpiral();
    Golden Spiral
    function r(theta) = pow(e, 0.0053468*theta);
goldenSpiral();
    Nine Pointed Star
    function x(t) = cos(t);
function y(t) = sin(t);
ninePointedStar([10,10,10],0.25);


  11. For the 3d graphing there are four example graphs to play with:
    GraphCode
    Square ripples in a pond
    function z(x,y) = 2*cos(rad2deg(abs(x)+abs(y)));
3dplot([-4*pi,4*pi],[-4*pi,4*pi],[50,50],-2.5);
    A wash board
    function z(x,y) = cos(rad2deg(abs(x)+abs(y)));
3dplot([-4*pi,4*pi],[-4*pi-20,4*pi-20],[50,50],-1.1);
    Uniform bumps and dips
    function z(x,y) = 5*cos(rad2deg(x)) * sin(rad2deg(y));
3dplot([-2*pi,2*pi],[-2*pi,2*pi],[50,50],-6);
    sombrero function
    function z(x,y) = 15*cos(180*sqrt(x*x+y*y)/pi)/sqrt(2+x*x+y*y);
3dplot([-4*pi,+4*pi],[-4*pi,+4*pi],[50,50],-5);
    Select one or two, uncomment them and render the 3D graph (F5)

  12. OpenSCAD also includes syntax for for loops. Here is an example that takes advantage of the cos() function: 
    for(i=[0:36]){
    for(j=[0:10]){
        translate([i*10,j*10,0])cylinder(r=5,h=cos(i*10)*50+60);
    }
}


  13. Here is an example that takes advantage of the sin() function: 
    for(i=[0:36]){
    for(j=[0:10]){
        translate([i*10,j*10,0])cylinder(r=5,h=sin(i*10)*50+60);
    }
}


  14. Create an interesting math form



Why am I doing this?



OpenSCAD allows you to generate interesting forms by taking advantage of math. This tutorial also demonstrated how to reuse modules across files.

Now what?

  1. Upload your math form to Thingiverse
  2. Come back tomorrow for more information and inspiration!