Preprocessor

Triangulation for non-uniform data can be demanding, and a few algorithm steps may get stuck if the data is not preprocessed properly. It is highly recommended that the user prepares the input data on their own; however, this project provides a few built-in methods.

Preprocessor	Description
None	Default, no effect.
COM	Transforms input into normalized local space, i.e. [-1, 1] box.
PCA	Transforms input into normalized coordinate systems obtained with principal component analysis.

To use one of the following preprocessors, use the corresponding settings:

triangulator.Settings.Preprocessor = Triangulator.Preprocessor.COM;

PCA transformation

This algorithm can help in situations when the Sloan algorithm gets stuck. The transformation can be applied using the following steps:

1. Calculate com: \(\mu = \displaystyle\frac1n\sum_{i=1}^n x_i\).
2. Transform points: \(x_i \to x_i -\mu\).
3. Calculate covariance matrix: \(\text{cov} = \frac1n\sum_i x_i x_i^{\mathsf T}\).
4. Solve eigenproblem for \(\text{cov}\): \(\text{cov}u_i =v_i u_i\).
5. Transform points using matrix \(U = [u_i]\): \(x_i \to U^{\mathsf T} .x_i\).
6. Calculate vector center \(c = \frac12[\max(x_i) + \min(x_i)]\) and vector scale \(s=2/[\max(x_i) - \min(x_i)]\), where \(\min\), \(\max\), and "\(/\)" are component wise operators.
7. Transform points: \(x_i \to s (x_i-c)\), assuming component wise multiplication.

To summarize, the transformation is given by:

\[ \boxed{x_i \to s[U^{\mathsf T}(x_i - \mu) - c]} \]

and the inverse transformation:

\[ \boxed{x_i \to U(x_i / s + c) + \mu}. \]

Note

The PCA transformation does not preserve the RefinementThresholds.Angle used for refinement. As a result, triangles can be classified as bad in the PCA local space.