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

john_concannon

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:


https://cad.onshape.com/documents/8f65c1667813235a037348da/w/e6d6a886e68d914a56993e10/e/b02f51d28e59bd8b08e851b9

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?

mahir

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.

MichaelPascoe

What exactly are you having trouble with? If you can make it in the part studio, it can be either instantiated or recreated via FS.

john_concannon

I can make it in the part studio, and in time I’ll learn how to implement it in feature script, but what I want to do is understand the relationship between the independent variables (number of turns, and Rho) and the dependent variable, the ratio of the spirals’ increase in distance from the origin per turn.  Because ultimately, I want to be able to draw a golden spiral (i.e. one with a ratio equal to the golden ratio).  

MichaelPascoe

Not sure I can help you with understanding the variables. But I can help with FeatureScript. 

Here is an example of how to create a spiral from scratch within FS using solid modeling approaches: 
https://cad.onshape.com/documents/abb998532f54a132a42bea8b/w/b0d311a0342c2b1e722bdaf1/e/79973a...

And here is an example of how to model it in the part studio, and simply instantiate it using your custom feature:
https://cad.onshape.com/documents/bf218127a4290c050e205e75/w/42342e313f737ad5d1388f67/e/dd21cb39f...

mahir

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.

MichaelPascoe

Boom! @mahir knows his spirals!

john_concannon

Thank you Mahir. It certainly looks like I can use the Parametric Curve FS to create a golden spiral, a more direct method than I was aiming for, so I'll close this question as answered.

And thank you Michael, I'll return to your examples when I start learning to write feature scripts.

robert_melascaglia

Thank you Mahir.  I used your fine script and pushed the logarithmic curve into the 3rd dimension using "1 * t" as the last variable.  Would you know how to allow just one revolution?  Regards, Robert.

john_concannon

Hi Robert. You can set upper and lower bounds for the angle of t 

Here's an example - I'm using Radians so t is bounded between -pi and pi (2 pi radians in a single rotation)



The document is here: Single rotation parametric curve

John

mahir

robert_melascaglia said:

Thank you Mahir.  I used your fine script and pushed the logarithmic curve into the 3rd dimension using "1 * t" as the last variable.  Would you know how to allow just one revolution?  Regards, Robert.

You can use @john_concannon 's method, but I personally prefer to leave #t bounded between 0 to 1 and instead vary the formula for #theta by replacing #t with #t*360*deg. That way the other formulas aren't impacted.

kendrick_feller

https://forum.onshape.com/discussion/comment/98986#Comment_98986

Just a heads up the formula for radius in this example is missing a factor of 2 it should be:

Radius = #a*exp(#b*2*PI*#c*#t)*mm

to match the theta formula

Theta = #c*#t*2*PI*rad

I thought my spiral was unwinding a bit too slow.