Convex Hull vs Convex Polygon and how to calculate it?

Question

I was given some points to calculate the convex hull. So when I try to find that on internet I also saw that a question which has asked what is the difference of convex hull and convex polygon. According to my knowledge convex hull is the set of points in convex set and polygon is the representation of them in 2D plane. (PLEASE CORRECT ME IF IM WRONG)

However my given points are below and how I tried do calculate convex hull is given below.

Points

$P1=(1,1) , P2=(2,2)$ , $P3=(1.5,3)$ , $P4=(2.5,5)$ , $P5=(3,4)$, $P6=(3,2)$ , $P7=(5,4)$ , $P8=(6,2)$ , $P9=(4,1)$

Tried way

By plotting them in 2D space and going to each point, tried to draw lines to each remaining points and tried to find the internal angle less than 180 while being the largest one among other angles.

Problem

I'm not 100% sure that my method is correct. So if someone can please tell me how to find the convex hull by hand (not with programs) and tell me the difference between convex hull and polygon (if my interpretation is wrong).

Thank you

Answer 1

The convex hull of a subset $S$ of the plane is the smallest convex set that contains all of them. If $S$ is finite, this is a convex polygon whose vertices form some subset of $S$.

In your case, the points look like

Start from, say, the highest point $P_4$, which must be one of the vertices of the convex hull (it wouldn't be in the convex hull of lower points). Think of a line through $P_4$ that starts out horizontal and pivots clockwise. The first point it hits will be $P_7$, so that's the next point of your polygon.

Draw the line segment $P_4, P_7$. This will be an edge of your polygon. Now pivot the line, still clockwise, around $P_7$. The next point it hits will be $P_8$.

Draw the segment $P_7, P_8$, and continue in this way until you come back to $P_4$. And there's your convex hull.

Answer 2

Start with a line parallel to the $x$ axis and far enough up so that all the points are below it. Then imagine that line dropping until it touches one (or maybe more) of the points. That will be the "highest" point $P=P4$. Now pivot that line around $P$ until it meets another point, say $Q$. Then $PQ$ is one of the edges of the polygon that is the boundary of the convex hull. Now pivot around $Q$ to find the next point $R$, and so on until you get back to $P$.

If I had time I'd draw the picture. Maybe someone will edit this and to that. In any case you should.