{"id":7328,"date":"2023-10-31T06:38:08","date_gmt":"2023-10-31T06:38:08","guid":{"rendered":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/?p=7328"},"modified":"2023-11-02T02:51:52","modified_gmt":"2023-11-02T02:51:52","slug":"polar-curve-and-parametric-curve-for-the-g-code","status":"publish","type":"post","link":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/2023\/10\/31\/polar-curve-and-parametric-curve-for-the-g-code\/","title":{"rendered":"polar curve and parametric curve for the g-code"},"content":{"rendered":"\n

When I was in high school, polar curve is one of my favorite math subject. So I decided to use some concepts from the polar curve to finish this project. I will do the project based on three polar curves: spiral curve, rose curve, and heart shape curve. The major philosophy behind this is to define R and theta, then find the x y value based on them. In order to make things more interesting, some of the z values are the function of x and y.<\/p>\n\n\n\n

The first curve I tried is the spiral curve, r = theta; this one is easy:<\/p>\n\n\n\n

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To add some flavors to this simple shape, I make the z value as a function of x and y; to be specifically z = Zmax*(1-(x^2+y^2)\/(x_max^2+y_max^2)); this can add a weight to the z value, when x and y values are small, z will be larger\/taller. <\/p>\n\n\n\n

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This is harder to print out, I tried different parameters such as the extruder width and speed. But the result is still not ideal. I will try this again using broader width. <\/p>\n\n\n\n

The next curve I tried is the heart curve, but I tried the polar curve of r = a+bcos(theta); but no matter how I tune the a and b values, the shape is not exactly what we need. <\/p>\n\n\n\n

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So I turned into parametric equations to draw a better heart shape curve. Like this one:<\/p>\n\n\n\n

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What I tried is to use the neural network method to try different combinations of the trigonometric function order, phase, and magnitude. It will take a long time if we use brutal force approach. I manually developed a neural network for it. The learning rate of the neural network is set to 1 since I want all of these parameters to be integers. The neural network method can stop within 1 seconds by setting the mean square error tolerance threshold to 0.01. The fit I find is x = 16sin(theta)^3; y=13cos(theta) – 5cos(2theta)-2cos(3theta)-cos(4theta); and here is the result:<\/p>\n\n\n\n

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Last curve is rose curve: r = 35cos(theta); the center is the most tricky part since the nozzle will path the center point for multiple times. This will cause a hump forming in the center point. So what I do is adding a if statement logic, such that when x^2+y^2 is small(near center point), I will set z value aliitle bit smaller. In this way the hump will be gone. <\/p>\n\n\n\n

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The code I used can be find here:
https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-content\/uploads\/2023\/10\/p3.gh<\/a><\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

When I was in high school, polar curve is one of my favorite math subject. So I decided to use some concepts from the polar curve to finish this project. I will do the project based on three polar curves: spiral curve, rose curve, and heart shape curve. The major philosophy behind this is to define R and theta, then […]<\/p>\n","protected":false},"author":29,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[40],"tags":[],"class_list":["post-7328","post","type-post","status-publish","format-standard","hentry","category-gcode_23"],"_links":{"self":[{"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/posts\/7328","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/users\/29"}],"replies":[{"embeddable":true,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/comments?post=7328"}],"version-history":[{"count":9,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/posts\/7328\/revisions"}],"predecessor-version":[{"id":7546,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/posts\/7328\/revisions\/7546"}],"wp:attachment":[{"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/media?parent=7328"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/categories?post=7328"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/tags?post=7328"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}