### Carlo Sequin, Professor

**(1) Modeling Geometrical Sculptures.**

*Applications for Fall 2017 are now closed for this project. *

famous artists such as Max Bill, Charles Perry, or Eva Hild.

We will use an emerging new CAD tool (NOME == Non-Orientable Manifold Editor)

to capture the essential geometrical features of such sculptures,

and then fine-tune and optimize them to make variations of the originals shape,

or new sculptures that &amp;amp;quot;belong into the same family.&amp;amp;quot;

Successful geometries will by made into sculptural maquettes by using some rapid prototyping machine.

TASKS:

Choose a sculpture of moderate complexity by Perry or Hild, and try to express this in a CAD tool of your choice and later in the NOME program.

Qualifications: URAP students should have a love for geometry and should be comfortable enough with some CAD tool (Rhino, Solidworks, Autodesk, Blender, Berkeley SLIDE) so that they can model a smooth Moebius band or a Klein-bottle. [Please bring some visual evidence of such a modeling effort to your interview.]

Weekly Hours: 6-9 hrs

Related website: https://people.eecs.berkeley.edu/~sequin/PAPERS/2015_Bridges_2manifolds.pdf

Related website: https://www2.eecs.berkeley.edu/Pubs/TechRpts/2016/EECS-2016-65.html

**(4) Combined Probabilistic and Greedy Search in the Context of Modular Assemblies**

*Applications for Fall 2017 are now closed for this project. *

http://www.cs.berkeley.edu/~sequin/TALKS/2014_Bridges-Keynote_LEGO-Knots

an intriguing geometrical / mathematical / programming problem has arisen:

How to assemble a chosen knot appearing in the Mathematical Knot Tables:

http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table

from a given modular, tubular component that bends through a fixed angle (e.g., 90, 135, 150 degrees).

A promising component has been defined: See Section 11 in:

http://people.eecs.berkeley.edu/~sequin/PAPERS/2015_CAD-A_SnapSculpt.pdf

For each case the right number of modules has to be chosen, and then all the azimuthal (torsion) angles between subsequent modules have to be selected, so that overall the knotted loop closes smoothly in all six degrees of freedom (x, y, x, two tangent directions, and possibly the azimuthal rotation angle, if the tube has the cross section of a regular n-gon).

If there are more than about a half dozen individual torsion angles that need to be specified, then manual search becomes tedious, and exhaustive search becomes impractical.

The goal is to develop a computer program that will find an acceptably good solution in a reasonably short time.

In addition to the values for all the torsion angles, there are some discrete parameters defining the solution space: the number of modules used, as well as some analog parameters: the placement and orientation of copies of one unique segment of the knot-curve, so that overall a knot with the desired symmetry is created, and all the gaps between all the instances of the identical curve segments close within acceptable tolerances.

TASKS:

In 2016/2017 we have developed greedy gradient-descent programs that were able to approximate a given knot (closed 3D space curve) with a given angular tube component bending through 150 degrees.

We started with a closed-loop poly-line with all vertices on the desired space curve. These points represent the centers of the components we plan to use. Then we moved those points in 3D space, trying to place neighboring points at a constant distance L, and force the angles between adjacent segments at each vertex to be as close to 150 degrees as possible. This approach solves the problem, if the tube has a round. circular cross section.

Now we want to tackle the harder problem that arises when the tube cross section is a regular 16-sided polygon. In this discrete solution space, gradient descent is no longer appropriate. We only have 16 discrete azimuthal angles at each joint. Each change by an angle of 360/16 degrees, may have a huge effect that will drive the two ends of this chain of tube elements far apart.

For small knots (e.g. trefoil) with a high degree of symmetry (e.g. 6-fold D3), we may have only 5 or six unique angles to adjust, and thus an exhaustive search is just possible. We use this to see how good a solution may exist, and then experiment with simulated annealing to see how we can find this solution most efficiently.

Qualifications: -- REQUIRED: Students should be familiar with the basics of geometric modeling, in particular: cumulative rigid-body transformations in 3D Euclidean space. -- REQUIRED: Students must be familiar with the basics of computer-based optimization based on gradient descent and must have written such a program. -- DESIRABLE: Students should have some understanding of searching in discrete spaces, such as simulated annealing or genetic algorithms.

Weekly Hours: 6-9 hrs

Related website: http://people.eecs.berkeley.edu/~sequin/PAPERS/2015_CAD-A_SnapSculpt.pdf

Related website: http://www.cs.berkeley.edu/~sequin/SCULPTS/CHS_miniSculpts/Modular_Knots/Modular%20Knot%207_4%20.JPG