The Affince Combination Notation

Most often we talk about a certain affine combination in the domain space, and then map the affine combination into the image space. For the sake of simple math formulation, we introduce the notation,

$$\mathfrak{A}(t_1, t, t_2),$$

to mean that $t_1, t_2$ , and $t$ are 3 points in $A^1$ space, and $t$ is an affine combination of $t_1$ and $t_2$ , i.e.,

$$\mathfrak{A}(t_1, t, t_2) = t = \frac{t_2 - t}{t_2 - t_1} t_1 + \frac{t - t_1}{t_2 - t_1} t_2.$$

and the notation

$$\underset{t_1, t, t_2}{\mathfrak{A}}(P_1, P_2)$$

to denote the image $P$ of $t$ under the considered affine mapping from $A^1$ to $A^n$ for some $n$ (usually $n = 3$ ), where $P_1$ and $P_2$ are the images of $t_1$ and $t_2$ , i.e.,

$$\underset{t_1, t, t_2}{\mathfrak{A}}(P_1, P_2) = \frac{t_2 - t}{t_2 - t_1} P_1 + \frac{t - t_1}{t_2 - t_1} P_2.$$