In an affine space there's the concept of affine combination which states that any point in space can be represented as a affine combination in the form

$a + \sum_{i \in I} \lambda_i \mathbf{aa_i} \quad \quad \text{if \sum_{i \in I} \lambda_i = 1}$

We can add an additional restriction on the values of $$\lambda_i$$ to define a triangle built out of three points, if $$\lambda_1 = \beta, \lambda_2 = \gamma$$, $$\beta + \gamma = 1$$ and $$\beta, \gamma \in [0,1]$$ then a triangle is defined as the affine combination

$a + \beta \mathbf{ab} + \gamma \mathbf{ac}$

One geometric property of the scalar values is that they're the signed scaled distance from the lines that pass through the triangle sides, to compute the scalar values $$\beta$$ and $$\gamma$$ we can use the fact that when the implicit equation of the line that pass through a side is evaluated with points that don't lie on the line the result is equal to

$f(x,y) = d_{(x,y)} \cdot \sqrt{A^2 + B^2}$

Where $$d_{(x,y)}$$ is the distance from the point $$(x,y)$$ to the line, $$A$$ and $$B$$ are the coefficients of $$x$$ and $$y$$ of the general equation of the line that passes through $$a$$ and $$c$$

$Ax + Bx + C = 0$

To find the value of $$\beta$$ we can use the value of the implicit equation of the line to map the distance between any point to the line in the range $$[f_{ac}(x_a, y_a), f_{ac}(x_b, y_b)] = [0, f_{ac}(x_b, y_b)]$$, we can use a simple division to find the value of $$\beta$$

$\beta = \frac{f_{ac}(x,y)}{f_{ac}(x_b, y_b)} = \frac{d_{(x,y)}}{d_{(x_b, y_b)}}$

In a similar fashion the value of $$\gamma$$ is

$\gamma = \frac{f_{ab}(x,y)}{f_{ab}(x_c, y_c)} = \frac{d_{(x,y)}}{d_{(x_c, y_c)}}$

References
• Shirley, P. and Ashikhmin, M. (2005). Fundamentals of computer graphics. Wellesley, Mass.: AK Peters.
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