Declaration

TBarycentricGradientSampler = class(TCustomSparsePointGradientSampler)

Description

A sparse point gradient interpolator using a barycentric coordinate system for interpolation.

Based on three (and only three) vertices, with each vertex specified with a certain color, a linear triangle of color is calculated. Outside this triangle the colours extends as before.

Mathematics & Algorithm

Given a triangle with vertices $P_1,P_2,P_3$ and associated colors $C_1,C_2,C_3$, any sampling point $P=(X,Y)$ is expressed in barycentric coordinates $({\lambda }_1,{\lambda }_2,{\lambda }_3)$:

$$P={\lambda }_1{P}_1+{\lambda }_2{P}_2+{\lambda }_3{P}_3,{\lambda }_1+{\lambda }_2+{\lambda }_3=1$$

The weights ${\lambda }_i$ are computed via determinants:

$${\lambda }_1=\frac{(Y_2-Y_3)(X-X_3)+(X_3-X_2)(Y-Y_3)}{(Y_2-Y_3)(X_1-X_3)+(X_3-X_2)(Y_1-Y_3)}$$$${\lambda }_2=\frac{(Y_3-Y_1)(X-X_3)+(X_1-X_3)(Y-Y_3)}{(Y_2-Y_3)(X_1-X_3)+(X_3-X_2)(Y_1-Y_3)}$$$${\lambda }_3=1-{\lambda }_1-{\lambda }_2$$

The interpolated color $C(P)$ is:

$$C(P)={\lambda }_1{C}_1+{\lambda }_2{C}_2+{\lambda }_3{C}_3$$

References

• Barycentric Coordinate System - Wikipedia

See also

• TBarycentricGradientPolygonFiller