# What is convex combination of two points?

I am studying algorithms and i saw a definition like the following:

Given $3$ points $p_1 = (x_1, y_1)$, $p_2 = (x_2, y_2)$ and $p_3 = (x_3, y_3)$, $p_3$ is a convex combination of $p_1$ and $p_2$ iff $\exists 0 \le a \le 1$ such that:

(i) $x_3 = x_1 + (1 −a )x_2$
(ii) $y_3 = y_1 + (1 − a)y_2$

If $p_3$ is a convex combination of $p_1$ and $p_2$, we also write $p_3 = p_1 + (1 −a )p_2$

My question is, i did not understand what this definition means, and what does convex combination means intuitively? I appreciate any help. Thanks

• You mean $p_3 = ap_1 + (1-a)p_2$. And this means geometrically that $p_3$ is on the line between $p_1$ and $p_2$. – Vobo May 27 '13 at 19:17

## 2 Answers

The point $ap_1+(1-a)p_2$ can also be written as $p_2+a(p_1-p_2)$. Perhaps you recognize this as a parametric representation of a line through $p_2$ with direction vector $p_1-p_2$ (with $a$ as the parameter).

When $a=0$, $ap_1+(1-a)p_2=p_2$. And when $a=1$, $ap_1+(1-a)p_2=p_1$. So the points $ap_1+(1-a)p_2$ with $0\le a\le1$ make up the line segment between the two given points.

It just means that you can get the point $p_3$ by simply adding the other points ($p_1$, $p_2$). But you don't want to just add them instead you'd like to give some sort of "weight" to each point.

Simply put, convex combination means weighted sum with the condition that the weights have to sum to 1.

In general, for a vector ($x_1, x_2, ...., x_n$), the convex combination is:

$\alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3+ ... \alpha_n x_n \hspace{2em}$

where $\hspace{0.5em} \alpha_i \ge 0 \hspace{0.5em}$ and $\hspace{0.5em} \alpha_1+ \alpha_2+ \alpha_3+ ... +\alpha_n = 1$