I'm dealing with a problem involving a linear combination on a bulk of lines.
A standard line equation is:
$$Ax+By+C=0$$
I define a "bulk" of lines as a linear combination of two boundary lines, represented as:
$$\alpha(A_1x+B_1x+C_1)+(1-\alpha)(A_2x+B_2x+C_2) = 0$$
Here, $\alpha$ is a parameter that ranges from $0$ to $1$. It represents a kind of "transition" from the first boundary line to the last one:
The lines share the same $x$, $y$ coordinates, and it is known which point corresponds to which line in the "bulk".
By holding $\alpha$ constant, the bulk of lines simplifies to a single line:
$$\left(\left(1-\alpha \right) \mathit{A_2} +\mathit{A_1} \alpha \right) x +\left(\left(1-\alpha \right) \mathit{B_2} +\mathit{B_1} \alpha \right) y +\left(1-\alpha \right) \mathit{C_2} +\mathit{C_1} \alpha = 0$$
This can be restated as:
$$Ux+Vy+M = 0$$
Where $U$, $V$, and $M$ are dependent on $\alpha$ and defined as follows: $$U = \left(1-\alpha \right) \mathit{A_2} +\mathit{A_1} \alpha$$ $$V = \left(1-\alpha \right) \mathit{B_2} +\mathit{B_1} \alpha$$ $$M = \left(1-\alpha \right) \mathit{C_2} +\mathit{C_1} \alpha$$
As $\alpha$ varies, the slopes of the lines in the bulk change uniformly, maintaining a structural coherence within the bulk.
My aim is to calculate the line parameters $A_1,B_1,C_1,A_2,B_2,C_2$ (or alternatively $U,V,M$), subject to certain slope restrictions.
Question: Given the structural similarities between this problem and Linear Regression, is it possible to apply Linear Regression or similar statistical techniques to compute these line parameters $A_1,B_1,C_1,A_2,B_2,C_2$ (or $U,V,M$)? I would appreciate any advice or insights.