3D Printing Mathematical Formulas

INTRODUCTION

LESSON PURPOSE

• Understand mathematical equations.
• Be able to use MathMod, Blender and Meshmixer to create and prepare 3D models for 3D printing
• Be able to explain the elements of an algebraic equation.
• Be able to control a surface through manipulating a mathematical equation.

MATERIALS

• 3D printer
• Access to the internet
• Applications: MathMod, Blender and Meshmmixer

RESOURCES

• About Quadratic Surfaces
• Mo Labs
• singsurf.org
• metamagics.wordpress.com
• Implicit Algebraic Surfaces
• paulbourke.net
• www.econym.demon.co.uk:Isosurfaces

MathMod Editor

1. Download and install the appropriate application for your system from here.
2. Open MathMod. On the Mac you need to locate and open the mathmodcollection.js file
3. To start, Toggle Isosurfaces and click on one of the named surfaces. You should see the rendered image in the second window.
4. To change the render, adjust the x, y, and z min and max values then press the (Update|Add|Cut) & Run button
5. To Export a surface click on the Export menu and save the image as a Wavefront.obj file
6. To view the script, click on the script icon
   
7. Deconstructing the script:

   {
   //The two choices for forms are "Iso3D" and "Param3D"
   "Iso3D": 
   //if set to null the CND Equation input will not be displayed
   	{"Cnd": null,
   //name of component or components
   	"Component": ["Octahedron"],
   //Description
   	"Description": ["This is a Description"],
   //formula
   	"Fxyz": ["abs(x)+abs(y)+abs(z)-1"],
   //Name of surface
   	"Name": ["Octahedron"],
   //xmin and max
   	"Xmax": ["2"],
   	"Xmin": ["-1"],
   //ymin and max
   	"Ymax": ["1"],
   	"Ymin": ["-1"],
   //z min and max
   	"Zmax": ["1"],
   	"Zmin": ["-1"]
   	},
   }
   

8. To create your own, CTRL click on a shape on the left and select Add Current Model to MySelection
   
9. Scroll down to MySelection and toggle the view
   
10. Edit the script to reflect your formula.

    {"Iso3D": 
    	{"Cnd": ["x^2+y^2+z^2-1"],
    	"Component": ["Sphere"],
    	"Description": ["A sphere"],
    	"Fxyz": ["x^2+y^2+z^2-4^2"],
    	"Name": ["Sphere"],
    	"Xmax": [" 6"],
    	"Xmin": ["-6"],
    	"Ymax": [" 6"],
    	"Ymin": ["-6"],
    	"Zmax": [" 6"],
    	"Zmin": ["-6"]
    	}
    }
    

11. Press Run to test
    
12. Here is the formula for a squared off sphere:

    x^4 + y^4 + z^4 -1

    It is a generalization of a sphere. Try increasing the powers to get nearer to a cube.
13. Many of the parametric surfaces and some of the IsoSurfaces appear to be solids. When you export them open them up in NetFabb. Chances are you will have to invert them.
    • If you see the exclamation point, CTRL +click on the the form and select Extended>Invert Part
      
      
    • Replace Part
      
    • CTRL_ click on the part again and Export as STL (ASCII)
14. If you are working with a surface, you will have to make the surface a solid before printing.
    
    Here are two options:BlenderMeshMixer

    Blender

    1. Open The latest version of Blender.
    2. Delete the default cube by pressing X and then clicking on the DELETE button.
    3. Select User Preferences and navigate to Addons:
       
    4. Enable Mesh:3D Print Tools.
    5. Save the preferences and close the window.
    6. Select File>Import>wavefront .obj:
    7. Select the object (an light orange outline around the object):
       
    8. Press N to bring up the numbers palette. Change the size to a desirable size:
       
       
    9. Click on the Modifiers (wrench icon):
       
    10. Click on Add Modifier:
        
    11. Select Solidify:
        
    12. Enable Even Thickness and make the thickness between 2-10:
        
    13. Click Apply
        
    14. Open the 3D Printing Tools and click on the Isolated, Distorted and Non-Manifold buttons:
        
    15. Select File>Export>STL or open 3D Printing Tools and specify the path:
        
        And then Export by clicking the Export Button:
        

    MeshMixer

    Meshmixer

    1. Open Meshmixer
    2. Import your model
    3. Click on Analysis and select Units/Dimensions:
       
    4. Adjust the size of your object:
       
    5. Click on Select:
       
    6. Click on your form to select some of it:
       
    7. Press E to extend your selection:
       
    8. From the Edit menu select Extrude:
       
    9. Select Normal:
       
    10. Change the extrusion to 2:
        
    11. Accept the changes:
        
    12. Clear the selection:
        
    13. Click on print to prepare your object for printing (repairing, adding support, etc.):
        
    14. After clicking on Add Support Structure, select Support All Overhangs:
        
    15. When support has been added click on Done:
        
    16. Click on Export:
        

    Blender

Alfred Clebsch

Image from mo-labs.com

Modell der Diagonalfläche von Clebsch -Schilling VII, Oliver Zauzig - universitaetssammlungen.de

Use MathMod to create and print a Clebsch diagonal Cubic surface.

What must you do to achieve a taller form?

What must you do to achieve a more stout form?

Formulas

1. Create a script in the MySelection and plug in these formulas:
   • Steiner's Roman Surface

     x^2*y^2 + y^2*z^2 + z^2*x^2 -2*x*y*z

   • Boys surface

     64*(1-z)^3*z^3- 48*(1-z)^2*z^2*(3*x^2+3*y^2+2*z^2)+    12*(1-z)*z*(27*(x^2+y^2)^2-24*z^2*(x^2+y^2)+    36*sqrt(2)*y*z*(y^2-3*x^2)+4*z^4)+    (9*x^2+9*y^2-2*z^2)*(-81*(x^2+y^2)^2-72*z^2*(x^2+y^2)+    108*sqrt(2)*x*z*(x^2-3*y^2)+4*z^4)
     

   • Chub's Surface

     x^4 + y^4 + z^4 - x^2 - y^2 - z^2 + 0.5

   • Tetrahedral skeleton

     (x^2 + y^2 + z^2)^2 + 8*x*y*z - 10 (x^2 + y^2 + z^2) + 25

     Use +/- 5 for the bound
   • Hunt's Surface

     4*(x^2 + y^2 + z^2 - 13)*(x^2 + y^2 + z^2 - 13)*(x^2 + y^2 + z^2 - 13) + 27*(3*x^2 + y^2 -4*z^2 - 12)^2 

     Use +/- 5 for the bound
   • Five fold oddity

     z + ((x,y)*(x,y)*(x,y)*(x,y)*(x,y)).(1,0)

     use +/-2 for the domain
   • Sum of Sins

     sin(x) + sin(y) + sin(z)

   • P (Schwarz P)

     cos(x) +  cos(y) +  cos(z)

   • D (Diamond)

     sin(x) sin(y) sin(z) +sin(x) * cos(y) * cos(z) +cos(x) * sin(y) * cos(z) +cos(x) * cos(y) * sin(z)

   • G (Gyroid)

     cos(x) * sin(y) +cos(y) * sin(z) +cos(z) * sin(x)

   • N (Neovius)

     3*(cos(x) + cos(y) + cos(z)) +4* cos(x) * cos(y) * cos(z)

   • W (iWP)

     cos(x) * cos(y) +cos(y) * cos(z) +cos(z) * cos(x) -cos(x) * cos(y) * cos(z)

   • P_W (P W Hybrid)

     4*(cos(x) * cos(y) + cos(y) * cos(z) + cos(z) * cos(x))- 3* cos(x) * cos(y) * cos(z) +2.4

2. Orbis Gravis

   -(cos(x) * sin(y) + cos(y) * sin(z) + cos(z) * sin(x))
   *(cos(x) * sin(y) + cos(y) * sin(z) + cos(z) * sin(x))
   +0.02
   +exp((x*x+y*y+z*z)-32.0)

   use +/-6 for the domain
3. A Param3d:
   HypocycloidFiveSides

   "Fx":sin(u)* (3 + 0.2*cos(u/5+(0.2-1)/0.2*v)+(1-0.2)*cos(u/5+v))"],
   "Fy": ["cos(u)* (3 + 0.2*cos(u/5+(0.2-1)/0.2*v)+(1-0.2)*cos(u/5+v)) "],
   "Fz": ["0.2*sin(u/5+(0.2-1)/0.2*v)+(1-0.2)*sin(u/5+v)]

   use +/-pi"for domain
4. A Param3d:
   HypocycloidTenSides

   Fx": ["sin(u)*(3+0.1*cos(u/10+(0.1-1)/0.1*v)+(1-0.1)*cos(u/10+v))"],
   "Fy": ["cos(u)*(3+0.1*cos(u/10+(0.1-1)/0.1*v)+(1-0.1)*cos(u/10+v))"],
   "Fz": ["0.1*sin(u/10+(0.1-1)/0.1*v)+(1-0.1)*sin(u/10+v)"]

   use +/-pi"for domain
5. Here is a torus:

   x(u,v) =  (a + b cos v) cos u,
   y(u,v) =  (a + b cos v) sin u,
   z(u,v) = b sin v

   Here is how it would appear in MathMod:

   {"Param3D": 
   	{
   	"Component": ["Torus"],
   	"Description": ["Description of the model"],
   	"Fx": ["(1+0.5*cos(v))*cos(u)"],
   	"Fy": ["(1+0.5*cos(v))*sin(u)"],
   	"Fz": ["0.5*sin(v)"],
   	"Name": ["Torus"],
   	"Umax": ["2*pi"],
   	"Umin": ["0"],
   	"Vmax": ["2*pi"],
   	"Vmin": ["0"]
   	}
   }
   

Christopher Olah's Equations

Neumann vs Dirichlet BC demo (wave eq)

Plot of a bijection between [0,1]² and [0,1]

Re(cos(z)) on [-τ,τ]²

Diffusion Eq. with Heavside Function Initial Conditions

Minimal Suface in Minkowski Space

Nomials on the Complex Unit Circle

NYS Learning Standards for Mathematics, Science, and Technology: