Interactive Symmetry Generator

A CS285 course project by Michael Wittman

A chain of tori produced by the symmetry generator.

Goals

The goal of this project was to provide an interactive facility to allow users to experiment with with 3D symmetry. A second, more specific goal was to give users the ability to interactively create and prototype 3D symmetric sculptures. The basic problems addressed by the project were the addition of symmetry operations and interactive controls to the existing sculpture generator.

Accomplishments

To support the underlying symmetry operations, classes were developed to be used within the sculpture generator. These provide the support for displays of symmetry such as glide or rotational symmetry, for arbitrary objects within the sculpture generator. These classes are meant to provide symmetry operations on an abitrary class, regardless of whether it is interactively controlled.

The two available symmetry operations are defined as follows. The glide symmetry class gives the user control of 9 parameters. These are three triples of rotation, vector, and extent. The symmetric object is then generated by taking the original object, replicating it extent times in the vector direction, adding a rotation at each replication. This is done for all three triples to produce a parallelepiped-like region (assuming the vectors are linearly independent).

A fairly simple application of the glide symmetry class. This was the result of setting the vectors to (9,0,0), (0,9,0), (0,0,9), the rotations to 90, 0, 0, and the extents to 4, 3, 4. (The torus has inner radius 8, outer radius 10.)

A not-so-simple application, but this object was produced with solely the glide symmetry operations. The extents were 3, 3, 3, the vectors were in the x, y, and z directions and had lengths less than the torus outer radius, and I used an arbitrary rotation for each vector to make it look interesting.

The rotational symmetry class allows the user to specify a 3D rotational symmetry group, and symmetry operations are performed based on that group. Here are a few examples, starting with our original object in C1 symmetry.

D1H

S21

The other part of the project provides interactive controls for the above classes. This is accomplished by using user-interface classes (one per symmetry class) and short Tcl files for the actual UI elements. This gives the user a uniform window for each symmetry operation, which is available through the "Root" button in the standard UI. Also available through both symmetry user interfaces is a button which will pop-up the original object's user interface.

Implementation

All of the symmetry operations in the symmetry classes are performed at the level of OpenGL, making the operations transparent to the replicated object. This works by drawing the object multiple times for each call to the symmetry object's Draw function, once for each replication. Each replication (via symmetry operation) is done on the OpenGL matrix stack, followed by a call to the object's Draw function to draw the object at the replication location.

Conclusion

With the exception of the C++/Tcl interface necessary for a user interface class, the implementation of the project was fairly straightforward. The existing object-oriented class structure of the viewer was essentially what allowed the symmetry object to be included in a relatively elegant manner. In the design phase of this project, the main stumbling block was distinguishing that the glide symmetry and rotational symmetry are two separate operations, not one big symmetry operation. With that observation, it was relatively easy to code each of the two simpler operations.

Further Info

Symmetry classes user's manual
Symmetry classes source code
A tarfile of this directory (for Carlo)

Notes

The code available is in runnable condition, however it does have some rough edges and a few things which are not fully implemented. I will try to correct the most egregious of these and update it sometime soon.

wittman@cs.berkeley.edu