Derivation of equation of the straight line through two points $x_1$ and $x_2$ in $\mathbb{R}^N$ (in convex function definition).

Question

I am getting started on reading convex optimization. One equation that is being used to represent traveling from one point to another in a straight line in a convex set is:

$$y = (1-\theta) x_1 + \theta x_2$$ for two points $ x_1 \neq x_2$ where $x_1, x_2 \in \mathbb{R}^N$.

I think I have an intuitive understanding of how this is a straight line, but I am trying to derive is from the usual equation of straight line going through two points. $y = y_1 + m(x-x_1)$ where $m = \frac{y_1 - y_2}{x_1-x_2}$. Could anyone convince me how the first equation was derived and what the parameter $\theta$ means or represents?

Thanks a lot.

Answer 1

If $x, y\in \mathbb{R}^n$, then $$ L := \{ x + \theta(y-x) \mid \theta \in \mathbb{R} \} $$ represents the line passing through $x$ (when $\theta = 0$) and $y$ (when $\theta = 1$) in the direction of the vector $y-x$.

Notice that $x + \theta(y-x) = (1-\theta)x + \theta y$ and $$ L(x,y) := \{ (1-\theta)x + \theta y \mid \theta \in [0,1] \} $$ is the line-segment containing $x$ and $y$.

This can be visualized in $\mathbb{R}^2$ using the parallelogram law: