Planarization, Smoothing, Equilateralization etc.

Hope you are all enjoying the new release.

Has anyone tried out the planarization or smoothing functions yet ?

I'm interested to hear how you are getting along if you are, or if you have any questions about them.

It's a direction for Kangaroo I very much intend to continue developing - and I am still getting to grips with the possibilities and experimenting with how different optimization and fairing forces work in combination with one another, so I would value your input and experience.

For those interested in some background reading material -

[1] http://www.cs.caltech.edu/~mmeyer/Research/FairMesh/implicitFairing...

[2] http://mesh.brown.edu/taubin/pdfs/taubin-eg00star.pdf

[3] http://www.pmp-book.org/download/slides/Smoothing.pdf

[4] http://graphics.stanford.edu/courses/cs468-05-fall/slides/daniel_wi...

[5] http://www.evolute.at/technology/scientific-publications.html

[6] http://www.math.tu-berlin.de/~bobenko/recentpapers.html

[7] http://spacesymmetrystructure.wordpress.com/2011/05/18/pseudo-physi...

[8] http://www.evolute.at/technology/scientific-publications/34.html

[9] http://www.evolute.at/software/forum/topic.html?id=18

At the moment the Laplacian smoothing is uniformly weighted, which tends to even out the edge lengths as well as smoothing the form, which is sometimes desirable, and sometimes not. It also tends to significantly shrink meshes when the edges are not fixed.

I plan to try some of the other weighting possibilities, such as Fujiwara or cotangent weighting (see [1] and [3]), as well as other fairing approaches, such as Taubin smoothing [2], Willmore flow[4], and so on. This also has applications in the simulation of bending of thin shells.

Planar quad panels are often desirable, but I'm finding that planarization forces alone are sometimes unstable, or cause undesirable crumpling, so need to be combined with some sort of fairing/smoothing, but the different types have quite different effects, and the balance is sometimes tricky.

There's also the whole issue of meshes which are circular (I posted a demo of circularization on the examples page), or conical (this one still isn't working quite right yet), and their relationship with principal curvature grids and placement of irregular vertices, all of which is rather different when the whole form is up for change, rather than having a fixed target surface [7].

I'm also trying to get to grips with ways of making surfaces of planar hexagons, which need to become concave in regions of negative Gaussian curvature (see this discussion)

and I hope to release soon a component for calculating CP meshes, as described in [8], which I think could have many exciting construction implications.

While there are a number of well developed smoothing algorithms, their main area of application so far seems to be in processing and improving 3D scan data, so using them in design in this way is somewhat new territory. There can be structural, fabrication or performance reasons for certain types of smoothness, but of course the aesthetic reasons are also often important, and I think there are some interesting discussions to be had here about the aesthetics of smoothness.

Anyway, that's enough rambling from me, hopefully something there triggers some discussion - I'm really keen to hear about how all of you envision these tools might be used and developed.