Convex Hull of a 2D Point Set
Computing a convex hull (or just "hull") is one of the first sophisticated geometry algorithms, and there are many variations of it. The most common form of this algorithm involves determining the smallest convex set (called the "convex hull") containing a discrete set of points. This algorithm also applies to a polygon, or just any set of line segments, whose hull is the same as the hull of its vertex point set. There are numerous applications for convex hulls: collision avoidance, hidden object determination, and shape analysis to name a few. The hull is a minimal linear bounding container, although it is not as easy to use as other containers.
The most popular hull algorithms are the "Graham scan" algorithm [Graham, 1972] and the "divideandconquer" algorithm [Preparata & Hong, 1977]. Implementations of both these algorithms are readily available (see [O'Rourke, 1998]). Both are time algorithms, but the Graham has a low runtime constant in 2D and runs very fast there. However, the Graham algorithm does not generalize to 3D and higher dimensions whereas the divideandconquer algorithm has a natural extension. We do not consider 3D algorithms here (see [O'Rourke, 1998] for more information).
Here is a list of some wellknown 2D hull algorithms. Let n = # points in the input set, and h = # vertices on the output hull. Note that , so . The list is ordered by date of first publication.
Algorithm

Speed

Discovered By

Brute Force


[Anon, the dark ages]

Gift Wrapping


[Chand & Kapur, 1970]

Graham Scan


[Graham, 1972]

Jarvis March


[Jarvis, 1973]

QuickHull


[Eddy, 1977], [Bykat, 1978]

DivideandConquer


[Preparata & Hong, 1977]

Monotone Chain


[Andrew, 1979]

Incremental


[Kallay, 1984]

MarriagebeforeConquest


[Kirkpatrick & Seidel, 1986]


Convex Hulls
The convex hull of a geometric object (such as a point set or a polygon) is the smallest convex set containing that object. There are many equivalent definitions for a convex set S. The most basic of these is:
Def 1. A set S is convex if whenever two points P and Q are inside S, then the whole line segment PQ is also in S.
But this definition does not readily lead to algorithms for constructing convex sets. A more useful definition states:
Def 2. A set S is convex if it is exactly equal to the intersection of all the half planes containing it.
It can be shown that these two definitions are equivalent. However, the second one gives us a better computational handle, especially when the set S is the intersection of a finite number of half planes. In this case, the boundary of S is polygon in 2D, and polyhedron in 3D, with which it can be identified.
Def 3. The convex hull of a finite point set S = {P} is the smallest 2D convex polygon (or polyhedron in 3D) that contains S. That is, there is no other convex polygon (or polyhedron) with .
Also, this convex hull has the smallest area and the smallest perimeter of all convex polygons that contain S.
2D Hull Algorithms
For this algorithm we will cover two similar fast 2D hull algorithms: the Graham scan, and Andrew's Monotone Chain scan. They both use a similar idea, and are implemented as a stack. In practice, they are both very fast, but Andrew's algorithm will execute slightly faster since its sort comparisons and rejection tests are more efficient. An implementation of Andrew's algorithm is given below in our chainHull_2D() routine.
The "Graham Scan" Algorithm
The Graham scan algorithm [Graham, 1972] is often cited ([Preparata & Shamos, 1985], [O'Rourke, 1998]) as the first real "computational geometry" algorithm. As the size of the geometric problem (namely, n = the number of points in the set) increases, it achieves the optimal asymptotic efficiency of time. This algorithm and its implementation has been covered in great detail by [O'Rourke, 1998, Sect 3.5, 7286] with downloadable C code available from his web site. We do not repeat that level of detail here, and only give a conceptual overview of the algorithm. The time for the Graham scan is spent doing an initial radial sort of the input set points. After that, the algorithm employs a stackbased method which runs in just time. In fact, the method performs at most 2n simple stack push and pop operations. Thus, it executes very rapidly, bounded only by the speed of sorting.
Let S = {P} be a finite set of points. The algorithm starts by picking a point in S known to be a vertex of the convex hull. This can be done in time by selecting the rightmost lowest point in the set; that is, a point with first a minimum (lowest) y coordinate, and second a maximum (rightmost) x coordinate. Call this base point P_{0}. Then, the algorithm sorts the other points P in S radially by the increasing counterclockwise (ccw) angle the line segment P_{0}P makes with the xaxis. If there is a tie and two points have the same angle, discard the one that is closest to P_{0}. For efficiency, it is important to note that the sort comparison between two points P_{1} and P_{2} can be made without actually computing their angles. In fact, computing angles would use slow inaccurate trigonometry functions, and doing these computations would be a bad mistake. Instead, one just observes that P_{2} would make a greater angle than P_{1} if (and only if) P_{2} lies on the left side of the directed line segment P_{0}P_{1} as shown in the following diagram. This condition can be tested by a fast accurate computation that uses only 5 additions and 2 multiplications.
After sorting, let the ccwradiallyordered point set be . It looks like a fan with a pivot at the point P_{0}.
We next loop through the points of S onebyone testing for convex hull vertices. The algorithm is an inductive incremental procedure using a stack of points. At each stage, we save (on the stack) the vertex points for the convex hull of all points already processed. This is the induction condition. We start with P_{0} and P_{1} on the stack. Then at the kth stage, we add the next point P_{k}, and compute how it alters the prior convex hull. Because of the way S was sorted, P_{k} is outside the hull of the prior points P_{i} with i < k, and it must be added as a new hull vertex on the stack. But it's addition may cause previous stack points to no longer be a hull vertices. If this happens, the previous points must be popped off the stack and discarded. One tests for this by checking if the new point P_{k} is to the left or the right of the line joining the top two points of the stack. Again, we use the routine isLeft() to quickly make this test. If P_{k} is on the left of the top segment, then prior hull vertices remain intact, and P_{k} gets pushed onto the stack. But, if it is on the right side of the top segment, then the prior point at the stack top will get absorbed inside the new hull, and that prior point must be popped off the stack. This test against the line segment at the stack top continues until either P_{k} is left of that line or the stack is reduced to the single base point P_{0}. In either case, P_{k} gets pushed onto the stack, and the algorithm proceeds to the next point P_{k+1} in the set. The different possibilities involved are illustrated in the following diagram.
It is easy to understand why this works by viewing it as an incremental algorithm. The old stack , with P_{k–1} at the top, is the convex hull of all points P_{i} with i < k. The next point P_{k} is outside this hull since it is left of the line P_{0}P_{k–1} which is an edge of the S_{k–1} hull. To incrementally extend S_{k–1} to include P_{k}, we need to find the two tangents from P_{k} to S_{k–1}. One tangent is clearly the line P_{k}P_{0}. The other is a line P_{k}P_{t} such that P_{k} is left of the segment in S_{k–1} preceding P_{t} and is right of the segment following P_{t} (when it exists). This uniquely characterizes the second tangent since S_{k–1} is a convex polygon. The way to find P_{t}_{ }is simply to search from the top of the stack down until the point with the property is found. The points above P_{t} in S_{k–1} are easily seen to be contained inside the triangle , and are thus no longer on the hull extended to include P_{k}. So, they can be discarded by popping them off the stack during the search for P_{t}. Then, the kth convex hull is the new stack .
At the end, when k = n1, the points remaining on the stack are precisely the ordered vertices of the convex hull's polygon boundary. Note that for each point of S there is one push and at most one pop operation, giving at most 2n stack operations for the whole algorithm.
This procedure is summarized by the following pseudocode.
PseudoCode: Graham Scan Algorithm
Andrew's Monotone Chain Algorithm
[Andrew, 1979] discovered an alternative to the Graham scan that uses a linear lexographic sort of the point set by the x and ycoordinates. This is an advantage if this ordering is already known for a set, which is sometimes the case. But even if sorting is required, this is a faster sort than the angular Grahamscan sort with its more complicated comparison function. The "Monotone Chain" algorithm computes the upper and lower hulls of a monotone chain of points, which is why we refer to it as the "Monotone Chain" algorithm. Like the Graham scan, it runs in time due to the sort time. After that, it only takes time to compute the hull. This algorithm also uses a stack in a manner very similar to Graham's algorithm.
First the algorithm sorts the point set by increasing x and then y coordinate values. Let the minimum and maximum xcoordinates be x_{min} and x_{max}. Clearly, , but there may be other points with this minimum xcoordinate. Let be a point in S with first and then min y among all those points. Also, let be the point with first and then max y second. Note that when there is a unique xminimum point. Similarly define and as the points with first, and then y min or max second. Again note that when there is a unique xmaximum point. Next, join the lower two points, and to define a lower line . Also, join the upper two points, and to define an upper line . These points and lines are shown in the following example diagram.
The algorithm now proceeds to construct a lower convex vertex chain below and joining the two lower points and ; and also an upper convex vertex chain above and joining the two upper points and . Then the convex hull of S is constructed by joining and together.
The lower or upper convex chain is constructed using a stack algorithm almost identical to the one used for the Graham scan. For the lower chain, start with on the stack. Then process the points of S in sequence. For , only consider points strictly below the lower line . Suppose that at any stage, the points on the stack are the convex hull of points below that have already been processed. Now consider the next point P_{k} that is below . If the stack contains only the one point then put P_{k} onto the stack and proceed to the next stage. Otherwise, determine whether P_{k} is strictly left of the line between the top two points on the stack. If it is, put P_{k} onto the stack and proceed. If it is not, pop the top point off the stack, and test P_{k} against the stack again. Continue until P_{k} gets pushed onto the stack. After this stage, the stack again contains the vertices of the lower hull for the points already considered. The geometric rationale is exactly the same as for the Graham scan. After all points have been processed, push onto the stack to complete the lower convex chain.
The upper convex chain is constructed in an analogous manner. But, process S in decreasing order , starting at , and only considering points above . Once the two hull chains have been found, it is easy to join them together (but be careful to avoid duplicating the endpoints).
PseudoCode: Andrew's Monotone Chain Algorithm

Input: a set S = {P = (P.x,P.y)} of N points
Sort S by increasing x and then ycoordinate. Let P[] be the sorted array of N points.
Get the points with 1st x min or max and 2nd y min or max minmin = index of P with min x first and min y second minmax = index of P with min x first and max y second maxmin = index of P with max x first and min y second maxmax = index of P with max x first and max y second
Compute the lower hull stack as follows: (1) Let L_min be the lower line joining P[minmin] with P[maxmin]. (2) Push P[minmin] onto the stack. (3) for i = minmax+1 to maxmin1 (the points between xmin and xmax) { if (P[i] is above or on L_min) Ignore it and continue. while (there are at least 2 points on the stack) { Let P_{T1} = the top point on the stack. Let P_{T2} = the second point on the stack. if (P[i] is strictly left of the line from P_{T2} to P_{T1}) break out of this while loop. Pop the top point P_{T1} off the stack. } Push P[i] onto the stack. } (4) Push P[maxmin] onto the stack.
Similarly, compute the upper hull stack.
Let = the join of the lower and upper hulls.
Output: = the convex hull of S.


Implementation
Here is a "C++" implementation of the Monotone Chain Hull algorithm.
// Copyright 2001, 2012, 2021 Dan Sunday // This code may be freely used and modified for any purpose // providing that this copyright notice is included with it. // There is no warranty for this code, and the author of it cannot // be held liable for any real or imagined damage from its use. // Users of this code must verify correctness for their application.
// Assume that a class is already given for the object: // Point with coordinates {float x, y;} //===================================================================
// isLeft(): tests if a point is LeftOnRight of an infinite line. // Input: three points P0, P1, and P2 // Return: >0 for P2 left of the line through P0 and P1 // =0 for P2 on the line // <0 for P2 right of the line inline float isLeft( Point P0, Point P1, Point P2 ) { return (P1.x  P0.x)*(P2.y  P0.y)  (P2.x  P0.x)*(P1.y  P0.y); } //===================================================================
// chainHull_2D(): Andrew's monotone chain 2D convex hull algorithm // Input: P[] = an array of 2D points // presorted by increasing x and ycoordinates // n = the number of points in P[] // Output: H[] = an array of the convex hull vertices (max is n) // Return: the number of points in H[] int chainHull_2D( Point* P, int n, Point* H ) { // the output array H[] will be used as the stack int bot=0, top=(1); // indices for bottom and top of the stack int i; // array scan index
// Get the indices of points with min xcoord and minmax ycoord int minmin = 0, minmax; float xmin = P[0].x; for (i=1; i<n; i++) if (P[i].x != xmin) break; minmax = i1; if (minmax == n1) { // degenerate case: all xcoords == xmin H[++top] = P[minmin]; if (P[minmax].y != P[minmin].y) // a nontrivial segment H[++top] = P[minmax]; H[++top] = P[minmin]; // add polygon endpoint return top+1; }
// Get the indices of points with max xcoord and minmax ycoord int maxmin, maxmax = n1; float xmax = P[n1].x; for (i=n2; i>=0; i) if (P[i].x != xmax) break; maxmin = i+1;
// Compute the lower hull on the stack H H[++top] = P[minmin]; // push minmin point onto stack i = minmax; while (++i <= maxmin) { // the lower line joins P[minmin] with P[maxmin] if (isLeft( P[minmin], P[maxmin], P[i]) >= 0 && i < maxmin) continue; // ignore P[i] above or on the lower line
while (top > 0) // there are at least 2 points on the stack { // test if P[i] is left of the line at the stack top if (isLeft( H[top1], H[top], P[i]) > 0) break; // P[i] is a new hull vertex else top; // pop top point off stack } H[++top] = P[i]; // push P[i] onto stack }
// Next, compute the upper hull on the stack H above the bottom hull if (maxmax != maxmin) // if distinct xmax points H[++top] = P[maxmax]; // push maxmax point onto stack bot = top; // the bottom point of the upper hull stack i = maxmin; while (i >= minmax) { // the upper line joins P[maxmax] with P[minmax] if (isLeft( P[maxmax], P[minmax], P[i]) >= 0 && i > minmax) continue; // ignore P[i] below or on the upper line
while (top > bot) // at least 2 points on the upper stack { // test if P[i] is left of the line at the stack top if (isLeft( H[top1], H[top], P[i]) > 0) break; // P[i] is a new hull vertex else top; // pop top point off stack } H[++top] = P[i]; // push P[i] onto stack } if (minmax != minmin) H[++top] = P[minmin]; // push joining endpoint onto stack
return top+1; }
References
S.G. Akl & Godfried Toussaint, "Efficient Convex Hull Algorithms for Pattern Recognition Applications", Proc. 4th Int'l Joint Conf. on Pattern Recognition, Kyoto, Japan, 483487 (1978)
A.M. Andrew, "Another Efficient Algorithm for Convex Hulls in Two Dimensions", Info. Proc. Letters 9, 216219 (1979)
A. Bykat, "Convex Hull of a Finite Set of Points in Two Dimensions", Info. Proc. Letters 7, 296298 (1978)
W. Eddy, "A New Convex Hull Algorithm for Planar Sets", ACM Trans. Math. Software 3(4), 398403 (1977)
Ronald Graham, "An Efficient Algorithm for Determining the Convex Hull of a Finite Point Set", Info. Proc. Letters 1, 132133 (1972)
R.A. Jarvis, "On the Identification of the Convex Hull of of a Finite Set of Points in the Plane", Info. Proc. Letters 2, 1821 (1973)
M. Kallay, "The Complexity of Incremental Convex Hull Algorithms in R^{d}", Info. Proc. Letters 19, 197 (1984)
D.G. Kirkpatrick & R. Seidel, "The Ultimate Planar Convex Hull Algorithm?", SIAM Jour. Comput. 15, 287299 (1986)
Joseph O'Rourke, Computational Geometry in C (2nd Edition), Chap. 3 "Convex Hulls in 2D" (1998), with online code at www.science.smith.edu/~jorourke/books/compgeom.html
Franco Preparata & Michael Shamos, Computational Geometry: An Introduction, Chap. 3 "Convex Hulls: Basic Algorithms" (1985)
Franco Preparata & S.J. Hong, "Convex Hulls of Finite Sets of Points in Two and Three Dimensions", Comm. ACM 20, 8793 (1977)
