Grasshopper

generative modeling for Rhino

# circular sections on 'arbitrary' surface (villarceau circles)

hi all.

i have a bit of a geometry problem and am wondering if this is something for GH or galapagos or something.

is it possible in some way to get circular sections on an 'arbitrary' surface (i.e. one that is not a torus but that /does/ allow for circular sections with some kind of ability to place the "start point" of the sections and to enter a user desired spacing/

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

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Hi Jonathan,

it might be possible using Galapagos. Basically you need to come up with a metric that defines how well a certain section describes a circle. In the attached file I used a very simple definition, namely the output of the Circle Fit component that represents the largest deviation between the point-set and the circle.

It doesn't work perfectly, it has a slight tendency to 'find' good slicing planes near the ends of the brep, where the slice can yield a few nearly co-linear points. But you can limit the domain of the searcher by picking a shorter rail line segment (explained in the file).

--

David Rutten

david@mcneel.com

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hi david.

can i ask you or one of the other really intelligent folks on the list if this will work on a surface such as attached? i tried creating a line but as a GH newbie this one is not happening for me at the moment.

i know there are obviously two trivial examples of circles on this surface (as principal curvatures I believe) and I have seen /diagrams/ that have shown one set/pair that appears to run diagonally along the intersections of the principal curves but i can't seem to get there from here.

is this possible using the script if i locate the line more correctly in some way? presumably there are two "sets" (mirror images of each other) which would be spaced apart by some user defined amount (?) but i am not following what i need to do for the script to test this.

also, i am sort of curious as to whether there is a dumb way to get this with some kind of circle command.

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There isn't any slice that would yield a single circle on that shape*. It will always yield either two circles, two ellipsish curves or one ellipsish curve.

There's no reason why the rail line couldn't be any other curve you can think of btw. You could also use a circle or some nurbs curve.

Given the surface you posted, what is the answer you are looking for? Can you roughly draw it for us?

* Well ok, there are two. If you slice the shape with a plane that is coplanar with the seam and then similarly at the bottom. But these are trivial solutions for which you don't need Galapagos.

--

David Rutten

david@mcneel.com

hi david. trying to nail this down for awhile over here. pdf showing versions on torus and on cyclides. if there is a trivial solution to the /angled/ ones this is what we have been shooting for. regards.

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argh. sorry. thanks for the heads up!

should be three png's attached

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god you are a true champ.

thanks david.

hi all, anyone able to understand this bit of mathematics enough to help us with an illustration using the figure attached earlier or otherwise able to guide us in GH? I think we only need it for the limited case such as the file attached but we are still struggling a bit with this one...

They appear as the intersections of the cyclide with spheres that are tangent to the cyclide in two points and which all pass through one fixed point. In the case that the ring cyclide is a ring torus, these doubly tangent spheres become doubly tangent planes. (I obtain the description for general ring cyclides from understanding the doubly tangent planes as spheres which all pass through the distinguished point at infinity).

here is that file. i guess there are some spheres that i am supposed to situate within this object that are tangent at two points to the cyclide?

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