Parametric Equation (involute) Curve?

Rhino Rhino for Windows
1

Hi all,
What is the best way to draw an involute curve in Rhino?

I would prefer using a Parametric Equation formula in GH and/or Script, but even so, with regard to my current Rhino skill level, I can think of only plotting points and then draw an Involute curve using _CrvThroughPt, which won’t be very “mathematically correct” or “exact”. (see Bx points in Fig 1 below)

With correct/exact I mean according to Rhino global tolerance.

The “simple” but not so precise next-to-best method would be to draw a number of arcs and “unwrap” them as in the picture below, but exactly that method I’d like to replace with an parametric equation instead. But, is it possible to draw such a trajectory curve directly based on an parametric equation, without first plotting the path with points?

Fig 1. Involute curve (the point based geometry here (Bx) resembles unwrapping a tape or a rope from a cylinder at certain angle intervals):

// Rolf

2

Hi Rolf,

the parametric equations for an involute curve is
x(t) = r * (cos(t) + t * sin(t))
y(t) = r * (sin(t) - t * cos(t))

There is an example of a parametric curve in the script help. You should be able to use these equations.

Alternatively you can use my MathPlugin http://rhino3.de/_develop/__v3_plugins/math/

The involute you’ll find in the library.

Best,
Jess

3

Yes, the equation is well known. But the Involute function in the Math-lib was unknown to me. Thanks!

// Rolf

4

Rolf,have you find the solution? @RIL

5

Not sure this helps. This plug-in makes involute gears: https://www.food4rhino.com/app/gear

And the clusters are open so you can see how they did it.

6

Thank you Scott