Clothoids in GH/Rhino

I am desperately trying to find a way to connect two lines with a curve with the following properties:

1. The curve is tangent to the endpoints of the two lines

2. The curve has zero curvature where it connects to the lines and varying curvature between the ends.

A clothoid has these properties, but i do not know how to create it.

The image attached shows an arc connecting the two lines in the desired way - but the curvature must be zero at endpoints.

I hope somebody can help me with this!

Best, Jacob

Replies to This Discussion

Permalink Reply by David on March 15, 2012 at 10:05am

Hi Jacob, did you know how to do it? I am also very interested. I have found a lot of information (about equations) but it's too difficult to me understand it.

I can only recommed this person: http://www.grasshopper3d.com/profile/MNettelbladt I have watch his work and seems to have a lot of knowledge about Clothoids.

(Sorry for my English!)

Permalink Reply by Mateusz Zwierzycki on March 15, 2012 at 10:28am

just following this discussion...

as I can google, clothoid is most used in roller coasters designs

Permalink Reply by taz on March 15, 2012 at 1:47pm

And railroad tracks!

http://en.wikipedia.org/wiki/Track_transition_curve

Permalink Reply by Mateusz Zwierzycki on March 15, 2012 at 9:34pm

As I flied over some text about clothoid, this is one of most elegant curve Ive seen.

And to foreigners - were you teached in school about it ? (I mean up to high school).

Permalink Reply by Jacob Drachmann on March 16, 2012 at 7:48am

They are really nice yes, i want to use them as railcurves for a sweep, which will hopefully result in a very nice surface too:)

Haven´t been taught about it in scool no, but road and railroad engineers definetely do. They can also be created using AutoCAD Civil, but for my purpose it is not suitable.

Permalink Reply by Ola Jaensson on March 16, 2012 at 3:00am

Are the lines allowed to be shortened or elongated?

Do you want the curves to be symmetric around the bisector?

Do you only need U-curves or S-curves as well?

Try searching for Euler spirals and Cornu spirals too, but you probably knew that.

Some inspiration could probably be found here: http://www.lems.brown.edu/vision/publications/Kimia's_Publication/J...

And of course at Mårten's blog: http://thegeometryofbending.blogspot.com/

Permalink Reply by Jacob Drachmann on March 16, 2012 at 7:45am

Hi Ola

Thank you for the reply. I need symmetric u-curves as the one the figure i have now posted. Thank you for the article.

Best, Jacob

Permalink Reply by Jacob Drachmann on March 16, 2012 at 7:44am

Thank you for replies, here is an update:

In an article by Walton and Meek from St. Pauls College "A controlled clothoid spline" (see figure) , i have found a method to create exactly the curve i need using Fresnel integrals. But i do not know how to create a parametrized curves in GH. I can create the curves in a math software, import the curve as a polyline and then create an interpolated splinethrough the vertices. However, this is not exact enough for my purpose, at the ends of the curve there is not zero curvature because of the interpolation (figure). So my problem now is actually to be able to create a smooth curve in GH/Rhino, which has exactly the properties it is supposed to have. I hope this makes sense, i am bit on thin ice :)

Permalink Reply by Ola Jaensson on March 16, 2012 at 8:49am

Microstation has clothoids as curve primitives. If no polyline/spline approximations are exact enough for your purpose, maybe you have to use Generative Components in Microstation.

Permalink Reply by Scott Davidson on March 16, 2012 at 9:44am

In Rhino 5 is the adjustable blend curve command. It makes the curve you are looking for:

Unfortunately this function is not in the SDK yet. You could make a G2 or G3 blend curve by projecting lines, dividing with point and feeding a degree 5 curve. increasing degree is important to this type of curve. Here are the results:

With some refinement I think the control point curve version might work. I did this by hand in Rhino, I do not have a GH definition for it.

Permalink Reply by Jacob Drachmann on April 1, 2012 at 9:45am

Hi Scott

This is great - suits my need perfectly!

Thanks a lot.

Jacob

Permalink Reply by Hannes Löschke on March 16, 2012 at 10:12am

My understanding of BSplines is: For a degree3 BSpline, cuvature is 0 somewhere within the range of 3 control points, that form a linear row. If the controlpoints form an end, the 0 crossing will be at the Splines end.

So if you have the tangent vector of the line ends, you need to create two more points tangential to the endpoint. Using these control points you create a spline that starts and ends with 0 curvature. the spacing of the controlpoints along the tangent vector controls the curvature of the blend curve.