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Creating logarithmic spirals using a conic and a helix. Need some math help!

I've been trying to create logarithmic spirals in Onshape, and have found a few special features that other users have kindly developed, but none of these met my needs. You can project a regular (Archimedean) spiral on to a sketch easily enough using a conical helix: 

But I wanted a logarithmic spiral (i.e. one where the distance between the turnings increase in geometric progression) and unfortunately you can't project a helix on to anything other than a cone or a cylinder. So instead, I used the loft feature to create a surface from a helix to its axis. I then created a horn shaped object by rotating a sketch with a conic about its axis. I then subtracted (Boolean) the horn from the surface. I could then 'use' (i.e. project) the inside edge of the surface on to a sketch in a perpendicular plane:


In this way the spiral shape is determined by only three variables: The radius of the horn's base:

the number of turns (rotations) the helix makes:

and the Rho of the conic used to sketch the horn's cross section.

The height of the horn has no influence.

What I would like to do is understand the relationship between these variables and the spiral's logarithmic ratio (the geometric progression of the distances between the turnings of the spiral), so that I can (a) figure out what variables to use to create a golden spiral (i.e. with a ratio equal to the golden ratio) and then perhaps at some future point once I've learned how, (b) incorporate these variables into a feature script that draws logarithmic spirals using just these three variables.

So, can anyone point me in the right direction?

Best Answer

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    mahirmahir Member, Developers Posts: 1,292 ✭✭✭✭✭
    edited January 24 Answer ✓
    The equation for a logarithmic spiral is r = ae^(bθ). This can either be implemented via custom FS, or you can use my Parametric Curve FS to generate it. Here's an example.


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