Non Euclidean Lp spheres

prusaprinters

<p>Originally published here: <a href="https://www.thingiverse.com/thing:4877814">Non Euclidean Lp spheres by tato_713 - Thingiverse</a></p><p>Of course, "non-Euclidean space" is any space that doesn't follows the Euclidean metrics, but I was bored and wondered how a sphere should look like using a different <strong>Lp</strong> distance. I made a simple script on MATLAB R2020a to graph them on 10cm "spheres". The "normal" distance we know in a Cartesian coordinates system is the <strong>L2</strong>, the "real" distance, that, in <strong>R2</strong> dimension, it follows the Pythagoras Theorem. A sphere is a surface in <strong>R3</strong>, in which each point has the same distance to its centroid. Having that in mind, I chance that "distance", from norm 2 to other norm indexes:</p><ul><li><strong>L0.5</strong>: Every <i>p</i> index lower than 1 (<i>0 &lt; p &lt; 1</i>), actually does not define a norm, or at least a norm with homogeneous function.&nbsp;</li><li><strong>L0.75</strong>&nbsp;</li><li><strong>L1</strong>: The infamous Manhattan distance, or Taxicab distance. Is the distance result of the direct sum of the coordinates, as if it were driving in a square grid of streets. The resulting sphere is a regular octahedron.&nbsp;</li><li><strong>L1.5</strong></li><li><strong>L2</strong>: The Euclidean distance, or Pythagorean distance, the metric we normally use. The resulting sphere is a sphere (duh).&nbsp;</li><li><strong>L2.5</strong>&nbsp;</li><li><strong>L3</strong>&nbsp;</li><li><strong>L4</strong>&nbsp;</li><li><strong>L5</strong>&nbsp;</li><li><strong>L10</strong>: As you may notice from the sequence to this point, while bigger the index, more cube alike the sphere turns. Of course, an <strong>L∞</strong> should be a plain cube, but, beyond this point, my computer takes days generating these geometries running my script.&nbsp;</li></ul><p>My knowledge about these maths is limited, if you want to learn more about these metrics, go and look at other sources. Wikipedia is good enough for this.</p><h4>References</h4><ul><li><a href="https://www.mathworks.com/matlabcentral/fileexchange/4512-surf2stl">Surf to STL function for MATLAB</a></li></ul><p><br>I spend part of my time making these models, letting them available and free for everyone. If you want to support my work, you can contribute with me by <a href="https://www.paypal.com/donate?token=Vs2In7MOO1j3F3ADS858-hZWS3wOL6ebfWXIdzdoOOEZ1cUxa2oRlM9H9kgUT_dfml4aLaLfL32ZgvTv">donating to this PayPal account</a>.</p>