## Point in a polygon

August 4, 2010

An important, yet basic problem in computational geometry is the following task: given a polygon (a sequence of points) and a point, determine if the point is inside, or outside the polygon.

To a human eye, this problem is trivial. After all, it seems obvious that point $A$ is outside the polygon, whereas point $B$ is inside it.

Computers, on the other hand, lack this simple intuition. It is not entirely obvious how to go about creating an algorithm that correctly determines this.

### Triangles containing the origin

Let us first focus on an easier version of the problem. Instead of any arbitrary polygon, let our polygon be limited to three points, that is, a triangle. Instead of any point in space, let us determine if the triangle contains the origin.

We observe that we can draw a ray from our point outwards. Any direction will do. If the point is inside the triangle, then the ray will eventually hit a side of it.

On the other hand, if the point is outside the triangle, the ray may hit two sides of the triangle, or it may hit none. However, it will never hit exactly one side:

Aha; this is the approach. Now we can transpose our diagram to a Cartesian plane. For simplicity, let our ray be a line directly upwards from the origin, that is, the entire y-axis above the origin.

Our algorithm now takes the three line segments of the triangle, and counts how many of them intersect the y-axis above the origin.

There are three cases. In the first case, the triangle doesn’t intersect the y-intercept at all above the origin, thus the triangle doesn’t contain the origin:

In the second scenario, the triangle intersects the y-axis twice above the origin, in which case the triangle still does not contain the origin:

In the last case, the triangle intersects the y-axis once above the origin, in which case the origin is in the triangle:

So the algorithm is, for every line segment (pair of coordinates) in the triangle:

1. Calculate the y-intercept, say, $b$
2. If $b<0$ this line segment doesn’t cross the y-axis above the origin so throw it away. Otherwise, continue.
3. Make sure that the line segment actually crosses the y-axis at all. Or, check that the two points are on opposite sides of the y-axis.

Here’s my implementation of the algorithm in C++:


// Slope of a line
double slope(double x1, double y1, double x2, double y2){
return (y1-y2) / (x1-x2);
}

// Y intercept given slope and point
double yint(double px, double py, double m){
return py - m*px;
}

// Determine if a line crosses the y axis
bool cyax(double ax, double bx){
if(ax<0 && bx>0) return true;
if(bx<0 && ax>0) return true;
return false;
}

// Contains origin, or not?
// Input is (A_x, A_y, B_x, B_y, C_x, C_y).
bool cto(double ax, double ay, double bx, double by, double cx, double cy){

// Find slopes
double mAB = slope(ax,ay,bx,by);
double mBC = slope(bx,by,cx,cy);
double mAC = slope(ax,ay,cx,cy);

// Find y intercepts
double yAB = yint(ax,ay,mAB);
double yBC = yint(bx,by,mBC);
double yAC = yint(cx,cy,mAC);

// How many times it crosses the y intercept above 0
int c = 0;

if(yAB>0 && cyax(ax,bx)) c++;
if(yBC>0 && cyax(bx,cx)) c++;
if(yAC>0 && cyax(ax,cx)) c++;

return c==1;
}


### Generalizing the algorithm

We can proceed to generalize the algorithm to not only triangles containing the origin. First, for the problem of determining if an arbitrary point is in a triangle, we can create a bijection:

Translate all the points so that the test point lies on the origin. The position of this point relative to the triangle is exactly the same, so we can proceed with the y-intercept algorithm.

For a polygon with more than 3 points, the same algorithm applies. Sort of. With more complicated polygons, we may encounter this edge case:

Here the ray crosses the polygon three times.

There’s an easy fix to this problem, however. It turns out that if the ray crosses the polygon an odd number of times, then the point is inside the polygon.

## Investigations on star polygons with a star polygon / stellar number generator

June 30, 2010

Note 9/3/2010: Redone all of the diagrams

If you are just looking for a download link to the program, here it is. To run, save as Stellar.jar (Java is required)

As school has been over for more than a week now, I’ve had time to (among other things) study geometry. I’m just spending an hour or two every day working through various exercises in Geometry Revisited.

This blog post is long overdue; it had been sitting in my drafts section with a few sentences written since the middle of April. Maybe it’s still useful now, maybe not. Anyways.

I myself am not in IB, because of my hatred of being forced to do copious and tedious amounts of homework. Many of my friends, however, take IB math. One famous project in IB math (not only in my school, but also in other schools) is a large project known as the math portfolio. From what I’ve heard, this consists of a dozen or so pages of writeup on some extended problem.

I first heard about the existence of such a project when a few of my friends asked me for help on it. The project was related to ‘stellar numbers'; a picture of the assignment can be found here. The problem itself seems rather trivial, thus I don’t feel it necessary to discuss it in this blog post.

My investigations here, although inspired by this assignment, is only marginally related to the assignment itself. The end product to be handed in for grades was supposed to include diagrams. However, it isn’t so easy to draw diagrams of stars (see above image) either by hand or on the computer. I saw many of my friends hastily put together their diagrams of stars using Powerpoint or MS Word, using the line and circle tools and a heck lot of copy-and-pasting. The result, unsurprisingly, is messy and inexact. One friend of mine wrote a computer program to output Latex code to draw the stars; the details of how this is done is something I consider more difficult and interesting than the project itself.

Keeping along with my theme of geometry this week, it is useful to figure out the geometry of star polygons before writing programs to generate them.

### Some geometric constructions

Consider the five sider star (fig.1):

We can see that the five corners of the star, ABCDE, are the same distance from the center K of the star. This means the five points lie on a circle; furthermore the distances between the corners are the same.

A similar phenomenon occurs with the five points between the corners, FGHIJ. They, too, lie on a circle centered around K.

Let us call the radius of the larger circle r, and the radius of the smaller circle i. Any star can be built around the framework of two circles. For example, our five pointed star (fig.2):

There is, however, one more characteristic present in star polygons that we neglected in the above diagram. It is that we chose the values of r and i pretty much arbitrarily.

Notice that in (fig.1), the sides EF and GB are parallel. It can be shown geometrically that if parallel, EF and GB are also concurrent. In (fig.2), EF and GB are not parallel nor concurrent.

Although any values of i and r for $i are enough to produce a star, it is necessary to further adjust the ratio between i and r in order to produce a correct star polygon.

Let us denote n to mean the number of points of a star, and d to denote the density of a star.

What do we mean by density? In order to define density, we will take a step back and take a look at alternative ways that star polygons can be constructed.

At present we are looking at star polygons as points on two distinct circles joined together with alternating line segments. Here is a completely different way of presenting it (fig.3):

Points ABCDE are spread equally on a circle. For the five points, every two points are connected by a line. We define density to be this number: the density for this star is 2.

It is possible, perhaps easier, to implement a drawing routine this way. However, by doing this we would have to erase the lines FGHIJ, which seems straightforward but is troublesome in implementation. We are going to stick to the two circles paradigm.

If $d=1$, then our ‘star’ would be degenerate; it would merely be a n-sided polygon. On the other hand, we assume that the inner circle has positive radius, thus,

$d < \frac{n}{2}$

For any integer n with $n \geq 3$, the maximum value of d is given by:

$d \leq \lfloor \frac{n-1}{2} \rfloor$

Next is finding a way to express the ratio between the radii in terms of n and d.

Here we combined the previous two diagrams, and added some new lines. Since our star contains so much symmetry, what follows can be extended to all corners of the star, and also to stars with different densities and number of corners.

Let us call the angle $\angle EKA$ to be x, which is $\frac{1}{n}$ of the circle, or $\frac{2 \pi}{n}$ radians.

Let y be the angle such that

$x:x+y = 1:d$

.. whence here $d=2$, therefore $x=y=72^\circ$.

As $\triangle EKB$ is isosceles, $\angle BEK = \angle EBK$, and $\angle BEK + \angle EBK + x + y = 180^\circ$; therefore

$\angle BEK = \frac{1}{2}\left[ 180-(x+y) \right] = 90-\frac{1}{2}(x+y)$

.. whence here $\angle BEK = 18^\circ$.

Next, $\angle EKL = \frac{1}{2}x$, in our case $36^\circ$, thus the measure of $\angle EFK$ can be found:

$\angle EFK = 180 - \left[ 90-\frac{1}{2}(x+y) \right] - \frac{1}{2}x = 90 + \frac{1}{2}y$

In our case $\angle EFK = 126^\circ$.

By the sine law in $\triangle EKF$,

$\frac{r}{\sin (90 + \frac{1}{2}y)} = \frac{i}{\sin \left[90-\frac{1}{2}(x+y) \right]}$

Or,

$i = \frac{r \sin \left[90-\frac{1}{2}(x+y) \right]}{\sin (90 + \frac{1}{2}y)}$

Simplifying, we get

$i = \frac{r \cos \left[ \frac{1}{2}(x+y) \right]}{\cos (\frac{1}{2}y)}$

We defined that $x = \frac{2 \pi}{n}$ and that $x+y = \frac{2 \pi d}{n}$, thus we write y:

$y=\frac{2 \pi (d-1)}{n}$

And that brings us to the final formula:

$i = \frac{r \cos (\frac{\pi d}{n})}{\cos \left[ \frac{\pi (d-1)}{n} \right]}$

Evaluating it for our star, we find that if $r=1$, then $d \approx 0.38197$.

### Implementation in Java

Now that we have the math done, it is a fairly easy programming exercise to implement this as a java application. Here’s what I came up with after ~2 hours of hacking in Netbeans (animated gif):

This may be useful to future people doing the same assignment (if it ever gets assigned again), or with modifications it may be useful for other purposes too.

To use it, adjust the settings, take a screenshot, crop it, and it may be necessary to further edit it in photoshop.

Without further ado, here’s the application as a jar file: