Daily Equation #3 – Spiral r = a*theta


This is Day 3 of my collection of printable equations. To see the whole collection, go to:http://www.thingiverse.com/thing:227210

For today's equation, we're moving to polar coordinates! By defining distance based on the angle, you can get beautiful symmetries in curves. The other advantage of polar coordinates is that, since theta and theta+360*n define the same rotational position, it's easy to define a curve with multiple y values for the same x value (vs. cartesian coordinates which require absolute value, square roots, or more complex methods for the same effect).

For the first polar equation, I picked the simple: r = atheta, which is the simplest version of Archimede's spiral (r = atheta + b, http://en.wikipedia.org/wiki/Archimedean_spiral). In this equation a controls the distance between the spirals, while b rotates the whole curve about the central axes. Archimedean spirals are used in a number of real world applications. The one I'm most familiar with is scroll compressors, which use two interleaved spirals that rotate relative to one another to compress a fluid (there's a good gif of this on the Wikipedia page).


  • units are in mm
  • a = 1/18 (to give a diameter of 100 after 5 turns)
  • Range(theta) = 5 to 1800 (starting at zero causes the thickend line to overlap on itself, which Openscad's polygon command can't handle.)
  • stl resized to make x axis = 100 mm

As always, I'm going to continue to post one equation per day until get bored or people lose interest. If there's a particular equation you'd like to see, leave a comment and I'll give it a shot.

Technical Note:

  • The thickness in the y-direction is created by plotting a second curve whose points are a constant distance in the direction perpendicular to the tangent of the first curve. This can create artifacts at regions of rapidly changing slope. However, the lower or inner curve is always correct.

Daily Equation #3 - Spiral r = a*theta by skaye is licensed under the Creative Commons - Attribution - Non-Commercial license.

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