Stereographic projection

by henryseg, published

Stereographic projection by henryseg Dec 13, 2013

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221132Views 40103Downloads Found in Math Art


This model is featured in figure 3.12 of Visualizing Mathematics with 3D Printing.

Stereographic projection is a map from the sphere to the plane: Imagine straight lines going from the north pole of the sphere down onto the plane. Each line hits the sphere in one point and the plane in one point. Stereographic projection maps that point on the sphere to the point on the plane.
If you put a light source at the north pole of this sculpture, the rays of light effectively do stereographic projection! The curves on the sphere cast shadows, mapping them to a straight line grid on the plane.
Stereographic projection is also available at Shapeways.

Update: added a version of the file with a wider base for easier printing.

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It's an amazing effect! I love it. But I want to break it in 4 pieces (to do a sphere puzzle) in 3ds Max, but the .stl file isn't quadrify, so it becomes an impossible task to do. Can you upload an .obj file or .fbx? someone which is quadrify to modify it and work with easyly. Besides that, it's an amazing work to make maths physicaly represented!
Thank you very much!

Hi Diana, I'm not familiar with quadrify - does it work on any obj? I could convert my stl to obj, but it's the exact same data - if 3ds Max can't quadrify the stl, I wouldn't think that it would be able to do it to the obj. For that matter, 3ds Max should be able to convert an stl file to an obj file itself?

Which 4 pieces are you thinking of breaking it into? Four quarters? Or with less regular, "jigsaw" style pieces? I built the original design in Rhino, using NURBS surfaces. It might make more sense to do the modifications at the NURBS level rather than the mesh level.

oh! some news. I've tried to make it from the begining by my own, and after all the afternoon I've found the way to model it by myself!! (I'm sure it's a quite simple task to achieve, but I'm pretty happy about it), so now that's a more simple polygon shape, It will be easier to make the puzzle from it, so I don't need the original modeling anymore. But THANK YOU VERY MUCH for your help! And so fast :)
Even if I'm repeating myself: your work it's AMAZING

Glad to hear it worked out!

Mar 23, 2017 - Modified Mar 23, 2017
DianaLacruz - in reply to henryseg

Max can transform an .stl to .obj, and quadrify too, but since the model is made of thousands of triangles, it doesn't do it very well. The problem is not the file itself, it's 3ds max that doesn't work very well with triangles. If the original model is made like that, there's no quick solution in my case, but I kow that exporting to .stl convert the model to triangles (always), while .obj and .fbx keeps the original structure.

I want to break the model in 4 irregular (like "jigsaw", yes) pieces (like the spheres puzzles like the earth ones that you can buy in puzzle stores, like a game), but I've already modify a bit the original covering some holes to project a shadow that shows a "path", so I need to "make the puzzle" from this version. I cover them as well as i could, but not really well anyways.

Want to make this print look even cooler ?!? Check out this sweet remote controlled LED light for this 3D print!


Feb 8, 2017 - Modified Feb 8, 2017

Printed very well on my Prusa v1 printer. Thank you!

Filament Material: ABS
Colour: Transparent Green
Filament Diameter: 2.82mm
Layer Height: 200 um
Hotend: J-Head MK IV

Great model, but unprintable ;-)

@Kolous: See my print parameters above.

Thanks, but I wouldn't consider it parmeters ;-) (Speeds, temperatures, fan, infil, Z lift...)
As I see you got the overhangs burn. ABS is not best material for this because you can't cool it by fan.

Comments deleted.
May 5, 2016 - Modified May 5, 2016

AWESOME!!!!!!!!!! I just can't print it though :(

Be cool if you made one to go over a light bulb in a room, maybe a little too extreme? but would be very nice. Make it with transparent color filament and the coloured shadows would be super.

The problem with standard light bulbs is that the light comes from all over the bulb, rather than from a single point. So the shadows wouldn't be sharp.

You must be older than I am. We have LEDs now. Even a single diode is as bright as incandescent bulbs once were, with virtually no heat.

Do you know of a make of lightbulbs that fit a standard screw mount and have a single small LED light?

There you go with that word "Bulb" again. The 'D' in LED stands for diode. You don't need bulbs anymore, nor a screw socket. I see 9 diodes in this $5 flashlight I have here. Any one of them would work perfectly, and they're only 5mm in diameter. It even has a switch on it & a battery inside.

Just a friendly reach-out to let you know that your comments make you appear to be a bit of a dick. I understand this may just be your way of speaking, but I thought I'd make sure you're aware.

Right, you can make one, but are there any commercially available to fit into a standard lightbulb socket? It's a lot of hassle to get the light source in the right place unless it's a mass produced light and fitting that I can design the print for.

In a word ... Genius!
Thanks for sharing this superb design.

Hey, I was thinking it would be great if you could do it 8x8 field, so it could be used as a chess table.

Wow! It is so cool! Who knows what the app I can create something like this with?

Very cool! I was just thinking it would be awesome if someone designed an led stand so it can act as a projection lamp.

The issue is that different LEDs will need different kinds of mount attachments.

Could you make it 8x8 fields? So that you can make a chess field projection?

That is f***ing awesome :D!

Hi all!
I'm printing with M3D (Micro 3D printer) with ABS filament from M3D.
Sadly print failed for me.
And here's the video of the failed process so that you can see where it failed.
As it can be seen in the video the problem are that there's no support on these for ends so they starts moving around...
https://www.youtube.com/playlist?list=PLgF8LJ1a_jc3Ux9Pv1gZSMrsKccXp6fWB - whole playlist, at the last video you can see what happened :D looks "cool" :P

I've printed at medium quality and hollow thick walls, without support, adding a raft and spraying the bed with Plastik spray for better adhesion.
Bad adhesion didn't caused this problem. Only problem was bad support for the whole construction.
Maybe adding something on these 4 sides would help?

Also I opened a discussion on M3D forums, available only for M3D users:

Thank you, and I'll try to print again when I get more information, or fix the 3D object to have better support.

I am not sure why M3D provide abs filament. They are not suitable for unheated plate which m3D doesn't has.
You might have better luck with pla filamen

Comments deleted.

Congrats! I thought you model was one of the best choices for a 3D Printed Christmas gift! I made a link from here: http://3dprintingninja.blogspot.com/2014/12/top-20-free-3d-printed-christmas-gifts.html

Any chance you can widen the base supports where it touches the plate? I've tried to print this a few times now only to have it come loose. I realize there are a few things I could do the make it stick better during a print, but it would make the design much more easily printable with more contact area. Thanks.

Maybe try printing with a raft...

How much bigger would be good? Double thickness?

Thanks for the quick reply! Looking at the printed part I have, I'd say you could go triple the width without detracting from the awesome aesthetic of the design.

Ok, wider base version is up (also with subtle hs logo!)

Excellent! Solved the adhesion problems I was having. Much easier to print now. No raft required. Thanks!


Great idea, like it!

Comments deleted.

hey, I made a customizable version so anyone can make stereographic projections without bothering with the math: http://www.thingiverse.com/thing:294899http://www.thingiverse.com/thi...

Customizable stereographic projection

Any idea if you could make one of these to be cut from CNC or Laser cutter rather than 3d printed? I'm guessing it would be quite hard to get the shape right?

Actually, I am planning on doing just that! Stay tuned!

No idea sorry, I don't have any experience with CNC or laser cutting.

what settings did you print at? Infill etc?

nice, like it!

would be cool as a candle holder - maybe one of those led simulated ones. wonder how that would work?

Hi, when I try to import this .stl into slic3r, Slic3r quits. Anyone got a gcode for a printrbot jr. V2 or a tip?

Hi, How did you make this, mathematically that is.

Mathematically, this is the inverse stereographic projection of the square grid to the sphere. To design it, I drew the square grid, coned it to the north pole of the sphere, then used the intersection of that cone surface with the sphere to cut out windows in the sphere. There is also a little more messing around beveling to make sure that none of the edges of the model are too sharp.

What program did you use? I'm looking for a good program to make more mathematical style prints and such. Thanks!

I use Rhinoceros. It has python scripting, and it can be used in a "ruler and compass" construction style, so it works well for mathematical things.

="thingiverse-021123d53714a8c7bbbf9015a755dc76:disqus" href="http://disqus.com/thingiverse-021123d53714a8c7bbbf9015a755dc76/henryseg what was this printed with? Thanks!

I don't have my own printer - I just use Shapeways. So my copy was printed on an EOS SLS machine.

I too have a Printrbot Jr. V2 and as far as size goes it will fit, but it is a very difficult print for a FDM machine. One must be very keen with settings and know how your printer reacts to things like thin walled prints, flimsy prints, small bed surface area (adhesion problems).

Need support when printing?

I printed with no supports, in PLA, and with a cooling fan. Some of the overhangs needs a tiny bit of cleanup, but it looks kinda cool eighter way.


Aside from the math, it makes a cool lamp shade?

I have slic3r this and it says it will take 1 hour and 31 minutes. What I am trying to ask is, Can this be printed on a Printrbot jr?

which do you have? the V2 or the V1? You can check to make sure it fits on your print bed by uploading it to tinkercad, then
changing the measurements to inches. I think you will find this a useful tool.

I don't know what version it is. Where can I find the version out? But it does fit my print bed. It won't even slice if it doesn't fit.

Do you have a heated bed? if so, it's the V2. if not, V1. When you say it broke off, how so? Did it fall off? Not extrude? stop printing?

I don't have a heated bed... but on the back top near the gears I found a 5. And by broke off I meant the extruder caught the piece and it came off the print bed.

That happens on my printrbot jr sometimes. try adjusting infill, supports, and scale. Sometimes adjusting the z offset might help.

I have a Printrbot Jr V2 as well - ="thingiverse-e8493eb7b5ef1095328dc3186bb20c4e:disqus" href="http://disqus.com/thingiverse-e8493eb7b5ef1095328dc3186bb20c4e/Shantorian3d what settings have you found to work? I've tried adjusting just about everything I can think of.

I haven't printed this design yet, but with designs similar to this, i find the trick is to adjust the z-offset to make sure the bottom sticks well. If the base is too small, make sure you have supports enabled in slic3r. A tip for the z-offset, adjust manually. Make sure the extruder is cool, first. Then place a normal sheet of (2d) printer paper under the extruder. Then home on the Z axis. if it is positioned well, you should be able to pull the paper out with only a slight resistance.

Nice, besides it being a very interesting object to show to others because it's some sort of 'dynamic' it's also a nice test for my self-built printer ;)

The overhang is too much on some areas making this a very hard one, but it's coming along just fine, if you disregard the loops hanging on the bottom edges of some of the areas inside the object.

The Grid. A digital frontier.

this is awesome! ^^

That's pretty cool. Can you do other designs? Like say, the hylian crest from zelda?(my picture)

Copyright issues aside, yes, any connected design with not too thin lines should be possible. Although things get difficult as parts of the design get near the top of the sphere - the plastic has to get very thin up there.

I know you explained the method you used to design this in another comment, but would you be willing to post a more detailed guide? This floored me and I'd really like to create my own model using the same principle.

I'm hesitant, because I think that any significantly more detailed explanation would depend to a large extent on the specific software being used.

That's too bad. I found methods to create something similar using different versions of AutoDesk, but it didn't have the same look as your model; the models produced all were wrapped perfectly to a sphere, rather than having the intersection at the top of the model.

I'm not sure I understand you - what do you mean by "wrapped to a sphere"? And what do you mean by "intersection"? Is the issue to do with how you are thickening up the sphere for printing?

Here's the thread I was thinking of when I said that. http://forums.autodesk.com/t5/Modeling/Wrap-mesh-or-lines-around-sphere/td-p/4132220/page/2http://forums.autodesk.com/t5/...

The light source for you is at the top of the sphere, for the model that would be created based on the thread I linked all points of the image/pattern would intersect at the middle of the sphere.

Ok, now I see what you mean - the extrusion in that model is radial: towards the center of the sphere, rather than towards the north pole. I'm not familiar with Autodesk, nor the commands being used in that example. It looks like they are wrapping a cylindrical projection onto the sphere, but with the map having some thickness that is also wrapped. If that is what's happening, I don't think it will be easy to get the thickness to point towards the north pole instead of the center.

till you not sell it, there should be no copyright issues..
but it should not be that difficult to make, simply project splines of the picture to the light-point and cut holes at the crossing with the sphere..or any other object..

i dont know what a spline is, but i am now going to learn

I'm definitely not smart enough to do it myself lol

but it should not be that difficult to make, simply project splines of the picture to the light-point and cut holes at the crossing with the sphere..or any other object..