Context: trying to draw a 3d geometry using points only.
Looking for ways (efficiency vs precision) to evenly distribute N amount of 3d points over a 3D plane made of 3 vertices - see image below:
So, with given three 3D points (A.xyz, B.xyz, C.xyz), how to generate an array of N 3D points that evenly spread/distribute over the surface.
Put another way, if I had N points to paint a triangle surface, where should I place these points on the plane so that the surface appears fully painted.
Your help would be much appreciated!
For an even spreading, I advocate a barycentrical approch. It suffices to use the set of points:
$$\begin{pmatrix}x\\y\\z\end{pmatrix}=\frac{u}{20}\begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}+\frac{v}{20}\begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}+\frac{20-u-v}{20}\begin{pmatrix}x_c\\y_c\\z_c\end{pmatrix}$$
with a double loop "for $u=0..20$, for v=$0..(20-u)$" (take care of this limit $20-u$).
Explanation: points such as the ones described in (1) are of the symbolic form $M=\alpha a+\beta b+\gamma c$ with $\alpha + \beta + \gamma = 1$ (check it...) ; they are inside the triangle defined by points $a,b,c$, and, in a reciprocal way, any point in this triangle is such that it can be written in this way ($\alpha, \beta, \gamma$ are called the "barycentrical coordinates" of $M$).
If you don't want the sides of the triangle to be "painted", use loops beginning at 1 and ending at $19.$
Of course, $20$ is just there as an example ; it can be replaced by any other integer constant.