Determine whether a point lies inside the curve or outside a random curve using pencil and scale Say, I am given a point and a closed curve. I don't know anything about the curve (where it is, what it is, its size etc.;say it is hidden somewhere)."I just can't see the curve but I can see the point where it is." I am supplied with a pencil and a ruler with no measurement marks on it, or in other words, I can only draw straight lines with the ruler, nothing else I can do with the ruler.
What is the general procedure that I should follow to determine whether the point lies inside the curve or outside it irrespective of the nature, size, shape or any other parameter of the curve?
 A: What the OP really wants, I think, is a crossing number algorithm for a simple closed continuous curve. Such a curve can be considered, however, as the limiting case of a simple polygon, for its edges becoming infinitesimally small. A relevant reference therefore may be this Wikipedia one:

Point in polygon

and especially the crossing number algorithm.But there is much more to tell about the Inside / Outside Problem, once you take a good look at it.
After some (in vain) attempts to answer the question in a concise manner, and while making too many duplicates of existing material, I've decided to simply redirect to my best shot so far:


Inside / Outside Problem

If the points of the simple closed curve are considered as pixels (i.e. integer coordinates), then we can even devise a never fail algorithm (only assuming that our picture is not too large).

It is seen in the picture below that, for a continuous curve, simple and straightforward application of the crossing number algorithm will be OK for the points $A$ and $B$, but it certainly will go wrong for the points $C$ and $D$.
According to Murphy's law , Anything that can go wrong will go wrong. Meaning that ignoring special cases, especially in a computational geometry environment, sooner or later, will be disastrous. It is noticed that the "wrong" cases have to do with rays that are tangent to the curve.

A: By the order of intersection between curve and a cutting line which produces two points of intersection $P,Q$.
If given point is $O$, the line cuts the closed curve at extreme points $P$ and $Q$, then proceeding in one direction without retracing path,
you are outside if the order is $ O,P, Q $ or $ Q,P,O $  and inside if the order is $ P,O,Q  $ or $ Q,O,P.$
Ignore (all even number of) points of intersection between the extreme points.
