Convex hulls for a finite amount of points

Question

I'm trying to understand what a convex hull intuitively is, and say given for a set of points $(x,y)\in\mathbb{R}^2$ how is it generated from these points? I tried reading the wikipedia article and some other sources but they seem very inaccessible. I figure that there must be some simpler explanation as to how a convex hull can be generated from a finite number of points in $\mathbb{R}^2$

Answer 1

Start with a line $L$ such that all the points lie on one side of $L$. Move $L$ parallel to itself towards the points, until one of them, say $p_1$, first hits $L$.

Rotate $L$ about that point slowly until the first moment that another point in your set, say $p_2$, hits $L$. Stop and call that line $L_1$, which contains $p_1$ and $p_2$ with has the points on one side of $L_1$ or on $L_1$ itself.

Now rotate $L_1$ slowly about $p_2$ until another point, say $p_3$, hits it. Stop and call that line $L_2$, which contains $p_2$ and $p_3$ with all the points on one side of $L_2$ or on $L_2$ itself.

Et cetera.

Eventually you get a line $L_{k-1}$ which contains $p_{k-1}$ and $p_{k}$ with all the points on one side of $L_{k-1}$ or on $L_{k-1}$ itself, and when you rotate $L_{k-1}$ slowly about $p_k$ you will get a line $L_{k+1}$ that contains $p_k$ and $p_1$.

The line segments connecting up the cyclic sequence $p_1,p_2,\ldots,p_{k+1},p_1$ forms a convex polygon which is the convex hull of your given set of points.