In this post I present an algorithm for splitting an arbitrary, *simple polygon* by a line. My implementation is based on ideas from the article *Spatial Partitioning of a Polygon by a Plane* by George Vanecek, published in the book *Graphics Gems V*. The intention behind this post is covering all the details which the original article omitted. All polygons are assumed to be oriented counter clockwise. You can find the full source code of my implementation in this github repository.

# Introduction

*Splitting* a polygon by a line is very closely related to *clipping* a polygon against a clip region, like a rectangle or another polygon. Splitting a polygon is easier and at the same time harder than clipping a polygon. It’s easier because you only clip against a single line and it’s harder because the resulting polygons cannot simply be discarded. It turns out, that splitting *convex* polygons is a lot easier than splitting *concave* polygons. The reason is that splitting convex polygons results in at most two new polygons, whereas splitting concave polygons can result in arbitrary many new polygons. Look at the following figure for an illustration.

Coming up with a correct implementation for concave polygons is tricky, because there are many corner cases that must be considered. In the following, first, an algorithm for splitting convex polygons is outlined. Afterwards, the algorithm is extended to concave polygons.

# Convex polygons

Splitting convex polygons is pretty simple. Essentially, one iterates over the polygon’s vertices, checks on which side of the split line they lie and adds them accordingly either to the left or the right result polygon. When an edge crosses the split line, additionally, the intersection point must be added to both result polygons. Special care must be taken when multiple successive vertices lie on the split line. In that case, due to the convexity property, the entire polygon lies either on the left-hand or on the right-hand side of the polygon. When for the second time a polygon edge crosses the split line, we know that the first half of the polygon is already completely split off.

# Concave polygons

In contrast to splitting convex polygons it requires some more algorithmic effort to split concave polygons. When the split line crosses a concave polygon for the second time, it doesn’t necessarily mean that the first half of the polygon is now completely split off. On the contrary, it might be that another new polygon to be split off just *opened*. To account for that, first, I tried to iterate over the polygon edges while carrying along a stack of open polygons. Whenever, I came across a vertex which belonged to the open polygon on the top of the stack I knew it just closed. This approach worked out well until I tried accounting for all the corner cases. Finally, I decided to use the method from the article mentioned in the introduction.

## Idea of the algorithm

The intersection points between the split line and the polygon are key to the splitting algorithm. By pairing up 2-tuples of intersection points along the split line, the edges closing the result polygons can be determined. Conceptually, every two intersection points along the split line form a pair of edges, each closing one side of two neighboring result polygons. In a perfect world, the intersection points could be easily paired up after sorting them along the split line, relative to the split line’s starting point (cf. the left figure below). Unfortunately, in the real world special care must be taken when the split line doesn’t cross a polygon edge in middle, but only touches the start and/or end vertex. In such cases the pairing up of *source* points (red) and *destination* points (green) along the split line gets trickier (cf. the figure in the middle blow). As you can see, only certain intersection points qualify as sources and destinations and some intersection points even qualify as both (yellow, cf. the right figure below). Luckily, it’s possible to account for all of these cases as we’ll see later.

## Computing the intersection points

To ease the splitting process it’s helpful to represent the polygon as a doubly linked list. We compute the intersection points between the split line and the polygon by iterating over each polygon edge `e`

and determining on which side of the split line the edge’s start vertex `e.Start`

and end vertex `e.End`

lies. If `e.Start`

lies on the split line `e`

is added to the array `EdgesOnLine`

, which contains all edges starting on the split line. If `e.Start`

and `e.End`

lie on opposite sides of the split line, a new edge starting at the intersection point is inserted into the polygon and added to `EdgesOnLine`

. The edges in this array are of interest, because they start at the intersection points with the split line. The algorithm is outlined below.

for (auto &e : poly) { const LineSide sideStart = GetLineSide(SplitLine, e.Start); const LineSide sideEnd = GetLineSide(SplitLine, e.End); if (sideStart == LineSide::On) { EdgesOnLine.push_back(e); } else if (sideStart != sideEnd && sideEnd != LineSide::On) { Point2 ip = IntersectLines(SplitLine, e); Edge *newEdge = poly.InsertEdge(e, ip); EdgesOnLine.push_back(newEdge); } }

## Sorting the intersection points

After all polygon edges have been intersected with the split line, the edges in the `EdgesOnLine`

array are sorted along the split line by the distance between their starting points and the starting point of the split line. Sorting the `EdgesOnLine`

array is crucial for pairing up the source and the destination edges by simply iterating over the array as we see in the following section.

std::sort(EdgesOnLine.begin(), EdgesOnLine.end(), [&](PolyEdge *e0, PolyEdge *e1) { return (CalcSignedDistance(line, e0->StartPos) < CalcSignedDistance(line, e1->StartPos)); });

As Glenn Burkhardt pointed out in a comment, the Euclidean distance cannot be used as distance metric because it’s unsigned. The Euclidean distance can only be used as long as the split line’s starting point lies outside the polygon. In that case all points are on the same side of the split line’s starting point and the Euclidean distance provides a strict order along the split line. However, if the split line’s starting point is inside the polygon, points on two different sides of the split line’s starting point will both have positive distances, which can result in wrong sort orders. A formula for computing the signed Euclidean distance between two points can be derived from the formula for scalar projections. In case of co-linear lines (angle between the two vectors is `|1|`

) the result equals the signed distance along the split line.

static double CalcSignedDistance(const QLineF &line, const QPointF &p) { return (p.x()-line.p1().x())*(line.p2().x()-line.p1().x())+ (p.y()-line.p1().y())*(line.p2().y()-line.p1().y()); }

## Pairing up edges

As we’ve seen before, all edges in the `EdgesOnLine`

array are either source edges, destination edges or both. The edges must be classified accordingly, in order to be able to determine between which pairs of edges a bridge must be created to split the polygon. The source/destination classification is the most important part of the algorithm and the hardest to get right due to plenty of corner cases.

The starting point `e.Start`

of any edge `e`

in the `EdgesOnLine`

array lies on the split line. Each edge `e`

has a predecessor edge `e.Prev`

and a successor edge `e.Next`

. They can lie either on the *left*-hand side (`L`

), on the *right*-hand side (`R`

) or *on* the split line (`O`

). Hence, there’s a total of nine configuration possibilities that must be evaluated in order to find out if an edge is a source, a destination or both. All nine configurations are depicted in the figure below. The vertex order is shown in the caption of each configuration.

I simply examined examples to classify each configuration as source, destination or both. The configurations’ winding orders are key for the classification. Imagine to move along the split line from its starting point to its endpoint. The winding order at the current intersection point indicates if we’re currently inside the polygon or not. Let’s take exemplary the `LOR`

and `ROL`

cases. The `LOR`

case is only encountered when transitioning from the outside to the inside of a polygon. Hence, it’s a source configuration. In contrast, the `ROL`

case is only encountered when transitioning from the inside of the polygon to the outside. Hence, it’s a destination configuration. Below are sample polygons depicted for the classification of each source and each destination configuration.

It follows that `LOR`

, `ROR`

, `LOL`

, `OOR`

and `LOO`

are source configurations and `ROL`

, `ROR`

, `LOL`

, `ROO`

and `OOL`

are destination configurations. Note, that these two sets are not disjoint: `ROR`

and `LOL`

are source and destination configurations. However, these two configurations can only be sources if they have served as destinations before!

Another corner case occurs when two successive edge starting points lie on the split line. Look at the example depicted in the figure below. There are two source edges (red) of which the second one must be used as source, even though it comes second in the `EdgesOnLine`

array.

The `EdgesOnLine`

array contains three edges: two source edges (red) and one destination edge (green). When splitting the polygon it’s crucial to pair up the the second and the third edge, not the first and the third. So how do we figure out which source edge to select when iterating over the `EdgesOnLine`

array? It turns out that the distance between the starting points of the two successive source edges in question and the starting point of the split line can be used. An `OOR`

and `LOO`

source edge is only selected if the following candidate edge isn’t further away from the start of the split line and therefore closer to the next destination.

## Rewiring the polygon

After we’ve figured out how the classification of source and destination edges works, we can finally formulate the algorithm to split concave polygons. All we need to do is to iterate over the `EdgesOnLine`

array while pairing up source and destination edges. Each pair denotes one side of the polygon that has to be be split off. At this point it pays off that we use a linked list representation for the polygon. The current polygon side is split off by inserting two new edges between the source and the destination edge. Therefore, we iterate over the `EdgesOnLine`

array, obtain the next source and the next destination edge and insert two new, oppositely oriented edges between them. This *bridging* operation is depicted in the following figure.

The main loop of the rewiring process is depicted below. For reasons of brevity only the overall structure is shown. For more details look into the full implementation in the github repository.

for (size_t i=0; i<EdgesOnLine.size(); i++) { // find source edge PolyEdge *srcEdge = useSrc; for (; !srcEdge && i<EdgesOnLine.size(); i++) { // test conditions if current edge is a source } // find destination edge PolyEdge *dstEdge = nullptr; for (; !dstEdge && i<EdgesOnLine.size(); ) { // test conditions if current edge is a destination } // bridge source and destination edges if (srcEdge && dstEdge) CreateBridge(srcEdge, dstEdge); }

## Collecting the result polygons

The final step, after all bridge edges have been inserted, is collecting the resulting polygons. For that purpose I use the `Visited`

attribute in the edge data structure to indicate if an edge has been visited already and therefore, has been assigned already to one of the result polygons.

January 20, 2016 at 6:19 pm

I’ve been playing with your polygon splitter. I believe I’ve found a data set for which it fails. Try the polygon:

and the split lines:

The split line is the same line defined by different line segments, and each test yields a different result. Only the first one is correct.

I’d be happy to continue this conversation by email: gbburkhardt@gmail.com. Thanks.

February 11, 2016 at 12:20 pm

Thanks for letting me know about this bug. I’ll look into it. Did you give it a try already? Do you have already any idea what the problem could be?

February 17, 2016 at 1:00 am

I believe I’ve found the problem, which is with the comparison function used to sort the

`EdgesOnLine`

list. The key is in the description George Vanecek gives of what the sort is supposed to do:Note that Dr. Vanecek used the dot product of

`cutDir`

with a vector with origin`refP`

to the edge source point. The`cutDir`

is the cross product of the unit normal vectors of the cut plane unit and the polygon`plane`

.So, in the

`PolySplitter`

code, the comparison function doesn’t give a signed value. Thus, the order of the on edge points with respect to cut direction is lost.I think the two points that define the cut line need to be treated like a vector, and each source vertex used with one of those two points to form a second vector, and the dot product (or scalar product) to get the value on which to sort. I think that one can safely leave out the division by the cut line vector magnitude (saving some cycles) for the purposes of your code.

I also pulled the original code by Dr. Venecek, and found several errors. I got the code here: http://www.realtimerendering.com/resources/GraphicsGems/ and have submitted the patches to Eric Haines at the ACM, who is listed as the maintainer. Probably no one has been using that code since it was written 20 years ago, or they never bothered to submit a patch.

I was really interesting to read the original implementation in 3D, and your 2D implementation in modern C++. I haven’t done a lot work with 3D analytical geometry, and it certainly adds another (more difficult) dimension.

I was trying to use his constructor to generate a plane from three points, and discovered that there’s an issue of “orientation” of the cut plane. For his unit test case, he had explicitly defined the plane using a unit vector and distance (

`Plane(Vector(0,1,0),-3)`

). I wanted to try his code using 2 points to define cut line, so I defined three points to define the cut plane using the constructor he provided:One would think that changing

`p3`

to`Point p3(p1.x(), p1.y(), 10)`

that one would get the same plane, but for the purposes of the algorithm, one doesn’t. He’s using the Hessian normal form to represent the plane, and here’s where my understanding of the math gets fuzzy. My first guess is that these two representations should be the same plane:but clearly for the purposes of the algorithm they are not. The code fails to correctly split the polygon when the second definition of the plane is used. If you can shed any light on this issue, I’d appreciate it. You mention in your blog that your code handles several cases that the original Graphic Gems code did not. I’d be interested in knowing what those are.

In any case, thanks for posting your code and the explanation in your blog.

February 18, 2016 at 1:30 pm

Thank you very much for the detailed information. I’ve applied your fix already to the source code on Github. Additionally, I’ve investigated a bit more. Here are my findings.

I agree that the

`SortEdges()`

function is broken.`PointDistance()`

always returns a value >= 0, because it computes the Euclidean distance between the two point arguments. This only works correctly as long as the split line’s starting point,`line.p1()`

, lies outside the polygon. In that case all points are on the same side of the split line’s starting point and the Euclidean distance provides a strict order along the split line. If the split line’s starting point is inside the polygon, points on two different sides of the split line’s starting point will both have positive distances, which can result in wrong sort orders. I didn’t find this issue in my tests and applications, because the split lines I use always start outside the polygon.As you pointed out, we need a signed distance metric to fix this problem. I did some research my-self and eventually came up with the same solution you’ve described: basically, it’s a scalar projection along the split line. In case of co-linear lines (angle between the two vectors is

`|1|`

) the result equals the signed distance along the split line. Starting with the formula from Wikipedia, I could simply derive your solution, except that you’re not normalizing`b^`

. This has no adverse effect on the sort order, as you mentioned in your comment as well already.Taking this equation without the normalization gives:

Which is equal to your

`dotProduct()`

implementation.November 25, 2016 at 5:46 pm

It looks like the algorithm fails when I change the first test polygon (

`TestPolys[0]`

) with it’s reverse:December 13, 2016 at 1:47 am

You can find a good test for CCW polygon orientation on Dan Sunday’s site:

http://geomalgorithms.com/a01-_area.html

Search for

`orientation2D_Polygon`

.Note that those algorithms depend on a polygon with

`n`

vertices having`V[0] == V[n]`

. Not everyone does it that way, but it makes the code cleaner.December 4, 2016 at 4:40 pm

Please see comment in introductory paragraph “All polygons are assumed to be oriented counter clockwise.”.

December 4, 2016 at 5:10 pm

Glenn found it! The polygon must be oriented counter clockwise for the algorithm to work. Otherwise, the checks for determining if an edge is a source/destination edge don’t work anymore. In contrast, the orientation of the split line doesn’t matter, because it’s only used to test if the start and end point of a polygon edge lies on two different sides of the split line.

December 6, 2016 at 8:26 pm

Thanks for pointing this out. I obviously missed that very important property input polygons must satisfy.