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Sunday, January 12, 2014

Day 139 - Stereographic Spheres

Henry Segerman is well-known for the intricate mathematical modeling that he designs and prints at his Shapeways shop.  So to put our Replicator 2 to the test, today we printed henryseg's Stereographic projection model from Thingiverse. It came out beautifully!

Thingiverse link: http://www.thingiverse.com/make:61807

Settings: Printed with some looping on a Replicator 2 with MakerWare .3mm/low (the red one) but very nicely on MakerWare .15mm/high (the purple one). Raft used but no supports.

Technical notes, math flavor: This model enables you to visualize the stereographic projection of a sphere to a plane. From a point light source at the north pole of the sphere, imagine a line extending from the point, through the surface of the sphere, and then down to the plane. Such lines define a mapping of each point on the surface of the sphere to a point on the surface of the plane. In this model the curved lines on the sphere are the preimage of a straight grid on the plane. The solid curved lines in the model block the light source so you can visually see the image of the curved lines as straight lines on the plane. A beautiful idea, as it is in some sense the opposite of what we usually see in a stereographic world map projection, where latitude and longitude lines become curved lines after the projection, as you can see in this image from subblue's Little Planets:

My favorite thing about Segerman's model it is that it really makes clear where the north pole gets mapped under steregraphic projection. Can you see it?

UPDATE: This post caught the eye of evelynjlamb at the American Mathematical Society's Blog on Math Blogs, and subsequently this MakerHome blog was featured there in the post The Revolution will be 3D Printed!

MORE UPDATE: The Blog on Math Blogs post on MakerHome apparently inspired an article written in Greek at Τρισδιάστατη εκτύπωση. So cool to see our pictures alongside text written in another language.  :)