I have a triangle (equilateral triangle ) 3 vertices representing probability that a point would belong to the category. If the centroid of it is at origin and I have a point (x,y,z) representing the probability that they belong to one of the cluster ie., close to the vertex. How do I calculate where would the point lie on the 2d graph.
Expected Output
Trivial Example:- ( if l is the length of the side then )
(1,0,0) = (0, $ l \% \sqrt3\ $)
(0,1,0) = (-l/2, $ l \% (2*\sqrt3\ ) $)
(0,0,1) = (l/2, $ l \% (2*\sqrt3\ ) $)
(0.3,0.3,0.3) = (0,0)
Wondering how do I convert 3 points into 2d systematically
Apply this matrix.
$\begin{bmatrix} \frac 1{\sqrt 2} & - \frac 1{\sqrt 2} & 0 \\ -\frac 1{\sqrt 6}& -\frac 1{\sqrt 6} & \frac {2}{\sqrt 6}\\ \end {bmatrix} \begin {bmatrix} x_2\\x_2\\x_3\end{bmatrix}$
You need two otho-normal vectors, that in turn are orthogonal to the vector from the origin to the centroid, $(\frac 13, \frac 13, \frac 13)$