Mathematics Of 3D Printing
3D Printing For Mathematics
Alexander Hulpke
Department of Mathematics
Colorado State University
Fort Collins, CO, 80523, USA
http://www.math.colostate.edu/~hulpke
Feb/6/15
Thank You
College of Natural Sciences
What Is 3D Printing
•
Additive (not subtractive)
manufacturing.
•
A hot glue gun on a robot
arm.
•
Build object in thin slices
(layers).
•
Material: thermoplastic
(PLA, ABS, PET).
Temperature 215-270°C
What Is 3D Printing
•
Additive (not subtractive)
manufacturing.
•
A hot glue gun on a robot
arm.
•
Build object in thin slices
(layers).
•
Material: thermoplastic
(PLA, ABS, PET).
Temperature 215-270°C
What Is 3D Printing
•
Additive (not subtractive)
manufacturing.
•
A hot glue gun on a robot
arm.
•
Build object in thin slices
(layers).
•
Material: thermoplastic
(PLA, ABS, PET).
Temperature 215-270°C
What Is 3D Printing
•
Additive (not subtractive)
manufacturing.
•
A hot glue gun on a robot
arm.
•
Build object in thin slices
(layers).
•
Material: thermoplastic
(PLA, ABS, PET).
Temperature 215-270°C
Other Technologies
•
Other meltable material.
•
Sliced Paper.
•
Sintering (Sugar, Plastic,
Metal Powder)
•
Photosensitive liquid
(Stereolithography).
Other Technologies
•
Other meltable material.
•
Sliced Paper.
•
Sintering (Sugar, Plastic,
Metal Powder)
•
Photosensitive liquid
(Stereolithography).
Other Technologies
•
Other meltable material.
•
Sliced Paper.
•
Sintering (Sugar, Plastic,
Metal Powder)
•
Photosensitive liquid
(Stereolithography).
Other Technologies
•
Other meltable material.
•
Sliced Paper.
•
Sintering (Sugar, Plastic,
Metal Powder)
•
Photosensitive liquid
(Stereolithography).
Other Technologies
•
Other meltable material.
•
Sliced Paper.
•
Sintering (Sugar, Plastic,
Metal Powder)
•
Photosensitive liquid
(Stereolithography).
New Features
•
Individualized, adaptable
dimensions.
•
Replacement parts.
•
Complicated craft
structures.
•
Impossible structures.
•
Replace mechanical skills
with mathematics.
New Features
•
Individualized, adaptable
dimensions.
•
Replacement parts.
•
Complicated craft
structures.
•
Impossible structures.
•
Replace mechanical skills
with mathematics.
New Features
•
Individualized, adaptable
dimensions.
•
Replacement parts.
•
Complicated craft
structures.
•
Impossible structures.
•
Replace mechanical skills
with mathematics.
New Features
•
Individualized, adaptable
dimensions.
•
Replacement parts.
•
Complicated craft
structures.
•
Impossible structures.
•
Replace mechanical skills
with mathematics.
New Features
•
Individualized, adaptable
dimensions.
•
Replacement parts.
•
Complicated craft
structures.
•
Impossible structures.
•
Replace mechanical skills
with mathematics.
Object Features
•
Outer shell
•
Honeycomb structure
inside, save on material
•
Complete Rubik's Cube
would be ~100g.
•
Sharing Websites
(thingiverse.com)
Extruder
Some Use
New
Extruder, Back And Inside
Back
Inside
(old model)
Cost
•
Printer: $500-$5000 (Fuse Deposit Modeling)
•
$5000-$30000 (Stereolithography)
•
$100000 and +++ (Metal Sintering)
•
Plastic: $20- $50 per kg, special material (mixed
with carbon fiber, metal, wood) more.
•
Replacement Extruder: $180
Use For Mathematics
•
Models of structures in R3, both research and
teaching
•
Being able to manipulate with hands is genuine
benefit.
•
Easier than perspective drawings etc.
•
Sufficiently cheap and stable for class use.
Simplices
For k+1 points
n
p0,…,pk⊆R ,
affinely independent (that
is p1-p0,p2-p0,…,pk-p0 lin. independent), define the
simplex as the set {Σi≧0αipi | αi ≧0, Σi≧0αi =1}.
A simplicial complex is a set of simplices, closed under
faces (simplices defined by subsets of corners) with
any two simplices intersection being a face of both.
As a subset of Rn it is also called a polyhedron.
Geometric Structures
•
Given only the outside faces, we can reconstruct
an equivalent simplicial complex.
•
File format: STL (surface triangulization
language). Describe the outer triangles (as tuples
of vertices). (There are other file formats.)
•
Floating point coordinates of vertices, rounding
issues can produce holes that require fixing.
It Slices, It Dices
The print builds up the
object in many layers.
For each layer the slicer
determines the polygon of
object intersecting the plane:
•
Edges intersecting plane (special case if triangle in plane)
•
Edges on same triangle connect
•
Determine inside by having triangles oriented (clockwise,
when viewed from outside).
•
Many heuristics in optimizing travel time.
Print Process
Print Process
Practical Issues
•
Gravity might imply need
for nonstandard orientation.
•
Support overhangs (>45°)
with extra material.
•
Software can add support,
not well.
•
Object must sit firmly on
build-surface — add raft.
(Flat bottom advantageous.)
Practical Issues
•
Gravity might imply need
for nonstandard orientation.
•
Support overhangs (>45°)
with extra material.
•
Software can add support,
not well.
•
Object must sit firmly on
build-surface — add raft.
(Flat bottom advantageous.)
Practical Issues
•
Gravity might imply need
for nonstandard orientation.
•
Support overhangs (>45°)
with extra material.
•
Software can add support,
not well.
•
Object must sit firmly on
build-surface — add raft.
(Flat bottom advantageous.)
Practical Issues
•
Gravity might imply need
for nonstandard orientation.
•
Support overhangs (>45°)
with extra material.
•
Software can add support,
not well.
•
Object must sit firmly on
build-surface — add raft.
(Flat bottom advantageous.)
Less Than Perfect
•
Object might not hold
well on build platform
(and move/tip over
while building).
•
Stringing.
•
Removing support can
be difficult.
Less Than Perfect
•
Object might not hold
well on build platform
(and move/tip over
while building).
•
Stringing.
•
Removing support can
be difficult.
Less Than Perfect
•
Object might not hold
well on build platform
(and move/tip over
while building).
•
Stringing.
•
Removing support can
be difficult.
Real-World Aspects
•
Extruder may clog. (Oil helps)
•
Material may not stick (or stick too much) on
build surface (blue painter tape).
•
Brittle material can break when feeding.
•
Vertical resolution .1, .2 (default), or .3 mm.
Printing time @default is about 45 minutes for
matchbox-sized object.
•
Infill can be increased for stability.
•
Post-Processing: Sanding, Painting, Casting
Mathematical Problems
•
How to orient an object for optimal printing
(firm hold on platform, little overhangs).
•
If not, decomposition? Good support?
•
Thin connections, Singularities, are hard
Software
Construction: Maple
Use plot3d to form parametric
surfaces.
Grid needs to be chosen
sufficiently fine.
Export in Collada (.dae) format —
Need conversion.
Output often is badly formed
(orientation, holes).
Intersecting surfaces problematic
(Surface vs. volume.)
Conversion, Fixing
Convert to .STL format: MeshLab (Open Source)
Add thickness: Blender (Open Source)
Fix holes, orientation: netfabb (Commercial, free
use), online service (requires signup) at
netfabb.azurewebsites.net
Additional Fixes, Add Support: meshmixer
(Commercial, free use)
Construction: OpenSCAD
Programming language
(limited). Open-Source.
Geometric primitives
(spheres, boxes, cylinders/
cones).
Affine transformations,
Boolean operations.
Limited calculation
functionality — produce code
with Maple program etc.
Construction: 123D Design
Freemium Software
(Autodesk).
Mouse-oriented.
Geometric Primitives,
Boolean Operations.
STL input and output.
Geometric Calculations
•
Linear algebra for coordinate change, affine
transformations.
•
Parameterization of surfaces.
•
Need to solve polynomial equations (e.g. circle through
three points): Gröbner-basis, Resultant, Numerical
Alg.Geom. techniques
•
Symmetric structures: Map by coset representatives for
stabilizer.