Barycentric coordinates

Sometimes it is desired to derive attributes for an arbitrary position based on the attributes of some known points. Imagine for example a triangle where each of the three base points has its own colour and you want to derive the colour of any other position in the grid. In such cases, barycentric coordinates are useful. This coordinate system is built upon some base points (say three for a triangle) and every other point inside or outside of the triangle can be represented by a linear combination of these base points

\begin{equation*} \fvec{x} = \alpha_1 \fvec{x}_1 + \alpha_2 \fvec{x}_2 + \alpha_3 \fvec{x}_3. \end{equation*}

Each point \(\fvec{x}_i\) is weighted into the resulting point by the corresponding weighting factor \(\alpha_i\). Weights can be positive (for points inside the triangle) or negative (for points outside of the triangle). Important is that in total they sum up to

\begin{equation*} \sum_{i=1}^3 \alpha_i = 1. \end{equation*}

As an example consider the representation of the first base point itself

\begin{equation*} \fvec{\alpha} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \Rightarrow 1 \cdot \fvec{x}_1 + 0 \cdot \fvec{x}_2 + 0 \cdot \fvec{x}_3 = \fvec{x}_1. \end{equation*}

The resulting weights can also be used to interpolate between colours associated with the base points. In total, each point in the grid gets its own (unique) weight vector.

Figure 1: Different weight vector \(\fvec{\alpha}\) for each position in the grid. You can change the position of the locator by mouse events. The grey dots denote the possible positions of the locator. At the point of each of the highlighted lines, one weight is fixed to 1, e.g. at all points on the green line the equation \(\alpha_3 = 1\) holds.

The technique can also be expanded to higher dimensions, e.g. four weights and corresponding points forming a tetrahedron.

List of attached files:


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