Spline - tangents/tangent weighting to polynomial?

[kbero]

Hello,

Hope there's a spline guru out there who can help me with this. I have found it very convenient to sketch splines and define them using tangents and tangent weighting (SW2014), but for my numerical analyses I need to have the representative polynomials. I can manually brute-force this using points measured in SW and interpolating, but if the two sets of parameters define the same curve, there should be an underlying translation formula which I can use? (I'm not familiar with exactly what "tangent weighting" is in a mathematical sense, and I've unfortunately been able to find but little on the subject.) Example below.

Many thanks in advance.

 

[rstupplebeen]

Years ago I went down this same path and never did find the underlying equations that SWx uses. I ended up deriving my own equations and using equation driven curves to get them into my CAD model.

I hope this helps.

Thank you.

Rob Stupplebeen
OptimalDevice.com
My Personal WP

 

[kbero]

OP here.
Thanks Rob.

What I think I've found is that my major error was assuming that a single polynomial would represent one of these SW spline functions. I've read up more on Bezier curves in the meantime and discovered that they are usually represented in parametric form with different polynomials for x(t) and y(t).

After some investigation, I've found that the tangent weighting seems to be built on a length base of 100/3, at least for the end handles. So in the example pictured, the tangent handle weighted at 40 is equivalent to assigning the control polygon leg a length of (100/3)*0.40=13.333... Plugging the (Cartesian) coordinates for the end points and resulting control points into Bezier polynomials will indeed give you the x(t) and y(t) on a dime -- put them into an equation-driven curve and it will overlap perfectly with the spline. For my application requiring a polar equation, I still then need to interpolate to a higher degree function, but that process can now be automated.

It is still unclear to me what the tangent weighting length base is for intermediate handles, it seems to vary and change with direction etc. So for the moment, I think I will switch to style splines and parameterize the control polygon directly.

Best

Jim/KBE

 

[rstupplebeen]

That's great info, thanks for sharing!

Thank you.

Rob Stupplebeen
OptimalDevice.com
My Personal WP