Symbolic CCD

Symbolic Continuous Collision Detection (CCD) using Mathematica.

Contents

• src:
  • roots_ee.wl/roots_vf.wl: Mathematica sources for computing the CCD roots
  • compare_toi.wl: Utility for comparing TOI accuracy
• scripts: Python scripts we used to process our dataset
  • These scripts call the Mathematica code using the wolframclient package
• hpc: Scripts designed for an HPC environment where jobs can be dispatched to process queries

Usage

The source files roots_ee.wl and roots_vf.wl each contains a root function that performs CCD using symbolic mathematics and returns a Boolean answer. Optionally, you can provide an outfile argument to save the symbolic expressions for the roots to a .wxf Mathematica file.

The input data is an $8\times6$ matrix of integers where each row represents a vertex position and every two columns represent the x/y/z coordinates as a rational number (numerator and denominator pair). The order of the rows is:

• Edge-edge queries: ea0_t0, ea1_t0, eb0_t0, eb1_t0, ea0_t1, ea1_t1, eb0_t1, eb1_t1 where a or b indicates which edge, 0 or 1 indicates the end-point of the edge, and t0 or t1 indicates the starting or ending positions of the edge.
• Vertex-face queries: v_t0, f0_t0, f1_t0, f2_t0, v_t1, f0_t1, f1_t1, f2_t1 where v indicates the vertex, f0, f1, or f2 indicates the face vertices, and t0 or t1 indicates the starting or ending positions of the vertex.

Each pair of columns (1/2, 3/4, and 5/6) comprise a single rational number for the x, y, and z coordinates of the vertex, respectively.

Notes

• If you want the TOI and not only the Boolean result, you can modify the roots functions to return them in your desired format.
• We chose to use rational numbers because we saved our data from C++ and found it was the only reliable way to transfer the data unchanged from C++ to Python to Mathematica (esp., cross-platform and for data longevity).
  • You can try changing the input data format to an $8\times3$ matrix of doubles, but this may or may not introduce a round-off error.