According to the convex hull algorithm, the algorithm terminates whenever all facets do not have any outside points. Find the points which form a convex hull from a set of arbitrary two dimensional points. Although many algorithms have been published for the problem of constructing the convex hull of a simple polygon, nearly half of them are incorrect. There are many equivalent definitions for a convex set S. The most basic of these is: Def 1. Convex hull point characterization. This blog discusses some intuition and will give you a understanding … (m * n) where n is number of input points and m is number of output or hull points (m <= n). The Convex Hull of the polygon is the minimal convex set wrapping our polygon. I haven't seen C code that lives only in a header file. How do I respond as Black to 1. e4 e6 2.e5? Can u help me giving advice!! The quick hull algorithm is exploited to develop the library that is cited in the article for more details about the algorithm. The matrix facets shows the facets of the final convex hull, neighbors_indices presents the indices of the facets that are located at the neighborhood of each facet (ith row contains the neighbor facets of the ith facet), and outpoints_indices contains the indices of the points that lie outside each facet (ith row contains the indices of points that are outside ith facet). (Please, note that the algorithm is directly given the paper without any modification): Moreover, a matrix library is needed to derive the resulting in which some basic matrix algebra operations are implemented. In this algorithm, at first the lowest point is chosen. In the figure below, figure (a) shows a set of points and figure (b) shows the corresponding convex hull. The C language is utilized due to its applicability to be implemented in the basic platforms. Corollary 1.1.1 [Convex hull] Let M be a nonempty subset in Rn. Why is training regarding the loss of RAIM given so much more emphasis than training regarding the loss of SBAS? DEFINITION The convex hull of a set S of points is the smallest convex set containing S. This project is a convex hull algorithm and library for 2D, 3D, and higher dimensions. Program Description. rev 2020.12.2.38097, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide. 1 Convex Hulls 1.1 Definitions Suppose we are given a set P of n points in the plane, and we want to compute something called the convex hull of P. Intuitively, the convex hull is what you get by driving a nail into the plane at each point and then wrapping a piece of string around the nails. Prove that a point p in S is a vertex of the convex hull if and only if there is a line going through p such taht all the other points in S are on the same side of the line. Does your organization need a developer evangelist? Convex hull is the minimum closed area which can cover all given data points. Then, the code obtains the convex hull of these points and exports its results in some CSV files. One of the most important properties of the provided library is its ability to be used for 2D, 3D, and higher dimensional points. I wanted to take points (x,y) as inputs. Convex hull also serves as a first preprocessing step to many, if not most, geometric algorithms. Output: The output is points of the convex hull. Update the question so it's on-topic for Stack Overflow. a.Y.CompareTo(b.Y) : … The code, as is, is hard to use. There are several algorithms that can determine the convex hull of a given set of points. The big question is, given a point p as current point, how to find the next point in output? It's simple to read and understand and the complexity is O(N) when the points are sorted by one coordinate. Starting from left most point of the data set, we keep the points in the convex hull by anti-clockwise rotation. The code is implemented in C language that can be used in basic platforms. The smallest convex space is represented through a set of facets. How do people recognise the frequency of a played note? Why does the Gemara use gamma to compare shapes and not reish or chaf sofit? The Delaunay triangulation and furthest-site Delaunay triangulation are equivalent to a convex hull in one higher dimension. The next image explains these definitions for a better understanding: As stated earlier, the quick hull algorithm is exploited in the supplied code which is directly given from this link, which may be useful for more details about the algorithm. Starting from left most point of the data set, we keep the points in the convex hull by anti-clockwise rotation. The quick hull algorithm is exploited to develop the library that is cited in the article for more details about the algorithm. A convex hull is the smallest polygon that encloses the points. Can I (a US citizen) travel from Puerto Rico to Miami with just a copy of my passport? Aligning and setting the spacing of unit with their parameter in table. class ConvexHull { public static double cross(Point O, Point A, Point B) { return (A.X - O.X) * (B.Y - O.Y) - (A.Y - O.Y) * (B.X - O.X); } public static List GetConvexHull(List points) { if (points == null) return null; if (points.Count() <= 1) return points; int n = points.Count(), k = 0; List H = new List(new Point[2 * n]); points.Sort((a, b) => a.X == b.X ? The Convex Hull of the two shapes in Figure 1 is shown in Figure 2. Use Ctrl+Left/Right to switch messages, Ctrl+Up/Down to switch threads, Ctrl+Shift+Left/Right to switch pages. This convex hull (shown in Figure 1) in 2-dimensional space will be a convex polygon where all its interior angles are less than 180°. What's the significance of the car freshener? The Convex hull model predicts that a species is present at sites inside the convex hull of a set of training points, and absent outside that hull. And I wanted to show the points which makes the convex hull.But it crashed! Although many algorithms have been published for the problem of constructing the convex hull of a simple polygon, nearly half of them are incorrect. If it is in a 3-dimensional or higher-dimensional space, the convex hull will be a polyhedron. Article Copyright 2020 by Roozbeh Abolpour, Last Visit: 2-Dec-20 5:11     Last Update: 2-Dec-20 5:11, GitHub - qhull/qhull: Qhull development for www.qhull.org -- Qhull 8.0.2 (2020.2 candidate) at https://github.com/qhull/qhull/wiki. The idea of Jarvis’s Algorithm is simple, We start from the leftmost point (or point with minimum x coordinate value) and we keep wrapping points in counterclockwise direction. This paper presents the following quick hull algorithm for finding the convex hull of some points with d the dimension that is presented by the next image. How can I print the value in this stackT? If it is in a 3-dimensional or higher-dimensional space, the convex hull will be a polyhedron. Simple = non-crossing. 1. //If the points co linear=0, clockwise=1;anticlockwise=2, //main function where points were taken as inputs, site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. What prevents a large company with deep pockets from rebranding my MIT project and killing me off? Thus, this matrix will be empty at the end of the algorithm. Use Git submodules to acquire dependencies. That point is the starting point of the convex hull. I.e. The convex hull of a simple polygon is divided by the polygon into pieces, one of which is the polygon itself and the rest are pockets bounded by a piece of the polygon boundary and a single hull edge. this is the spatial convex hull, not an environmental hull. Convex hull also serves as a first preprocessing step to many, if not most, geometric algorithms. Some previous cases of the convex hull codes can be only used for 2D or 3D points while the supplied library can be used for the higher ones. If you want a convex hull and you want it now, you could go get a library like MIConvexHull.That library claims to be high-performance compared to a comparable C++ library, but that claim is implausible, especially for the 2D case, since the algorithm relies heavily on heap memory and … (The facets are assumed … Graham scan is an algorithm to compute a convex hull of a given set of points in O(nlog⁡n)time. It must be emphasized that the coordinations of the points are imported to code via a CSV file and the results (facets) are exported by the other CSV files that are entirely explained in the rest of this article. For given M, the average time of Step 2 in the algorithm is less than CM t 1. Requires C++17 and CMake. Following is Graham’s algorithm . Intuitively, the convex hull is what you get by driving a nail into the plane at each point and then wrapping a piece of string around the nails. The Convex Hull of a set of points is the point set describing the minimum convex polygon enclosing all points in the set.. This section presents some basics and backgrounds that are used in this article. This blog discusses some intuition and will give you a understanding of some of … For example, consider the problem of finding the diameter of a set of points, … Both operations take time bounded by CM + 1 for some constant c > 0. For this purpose, the following matrix library is exploited: Now, the supplied library is presented in the next section. The input is a list of points, and the output is a list of facets of the convex hull of the points, each facet presented as a list of its vertices. Compiles on GCC 8/9, Clang 7/8/9, MSVC 14/19 (VS 2017/2019) Let points[0..n-1] be the input array. Then, the above function can be simply called as given here: In the following, two examples are presented that show the results of applying the above code in two 2D and 3D problems. The Convex Hull of the polygon is the minimal convex set wrapping our polygon. The idea is to use orientation() here. Convex hull model. Which game is this six-sided die with two sets of runic-looking plus, minus and empty sides from? For example, the convex hull must be used to find the Delaunay mesh of some points which is significantly needed in 3D graphics. Jarvis March algorithm is used to detect the corner points of a convex hull from a given set of data points. When we add a new point, we have to look at the angle formed between last edge in convex hull and vector from last point in convex hull to new point. This algorithm first sorts the set of points according to their polar angle and scans the points to find the convex hull vertices. 2D Convex hull in C#: 40 lines of code 14 May 2014. your coworkers to find and share information. How is time measured when a player is late? I'm new to chess-what should be done here to win the game? Therefore, the input points should be set as the above template to be used by the code. The convex hull of a simple polygon is divided by the polygon into pieces, one of which is the polygon itself and the rest are pockets bounded by a piece of the polygon boundary and a single hull edge. There have been numerous algorithms of varying complexity and effiency, devised to compute the Convex Hull of a set of points. The facets are given in a CSV file that is presented in the next section. The library exploits the quick hull algorithm to find the convex hull that is fully implemented in this code. Using Graham’s scan algorithm, we can find Convex Hull in O(nLogn) time. Halfspace intersection about a point is equivalent to a convex hull by polar duality. For example, consider the problem of finding the diameter of a set of points, which is the pair of points a maximum distance apart. The key is to note that a minimal bounding circle passes through two or three of the convex hull’s points. This example extends that result to find a minimal circle enclosing the points. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. This library computes the convex hull polygon that encloses a collection of points on the plane. Configured to build dependencies. A header-only C implementation of the Quickhull algorithm for building 3-D Convex Hulls quickhull computational-geometry convex-hull convexhull 3d Updated Aug 3, 2020 Figure 2: The Convex hull of the … Following is the detailed algori… 3D Convex Hull. The convex hull of a geometric object (such as a point set or a polygon) is the smallest convex set containing that object. It arises because the hull quickly captures a rough idea of the shape or extent of a data set. The Convex Hull of a set of points is the point set describing the minimum convex polygon enclosing all points in the set. There are several algorithms that can determine the convex hull of a given set of points. The convex hull of a set of points is the smallest convex set that contains the points. The Convex Hull of a concave shape is a convex boundary that most tightly encloses it. Assume file1.txt is the CSV file that includes the points. This convex hull (shown in Figure 1) in 2-dimensional space will be a convex polygon where all its interior angles are less than 180°. Hull is an ANSI C program that computes the convex hull of a point set in general (but small!) dimension. python-is-python3 package in Ubuntu 20.04 - what is it and what does it actually do? Viewed 2k times -2. The first is the convex hull that is the smallest convex space containing the given points. The input points are imported through a CSV file that contains all points' coordinations such as given in the following: Indeed, each row contains the coordinations of one specific point. Graham's Scan algorithm will find the corner points of the convex hull. Find R, (note that R,, = 0 if and only if M = 0 or S 5: 7~). The convex hull of a set of points is the smallest convex set that contains the points. Finding the convex hull of an object in opencv? From a current point, we can choose the next point by checking the orientations of those points from current point. The following picture shows the two possible scenarios. A header-only C implementation of the Quickhull algorithm for building 3-D Convex Hulls quickhull computational-geometry convex-hull convexhull 3d Updated Aug 3, 2020 The convex hull is the area bounded by the snapped rubber band (Figure 3.5). The diameter will always be the distance between two points on the convex hull. Ensure: C Convex hull of point-set P Require: point-set P C = findInitialTetrahedron(P) P = P −C for all p ∈P do if p outside C then F = visbleFaces(C, p) C = C −F C = connectBoundaryToPoint(C, p) end if end for Slides by: Roger Hernando Covex hull algorithms in 3D The main code of the supplied library is convh() that is given here: As can be seen, function convh() gives the primary points and obtains their convex hull struct that contains the result. What does "Every king has a Hima" mean in Sahih al-Bukhari 52? Active 4 years, 5 months ago. Thus, this article focuses on this topic and develops a library for solving the mentioned problem in C language. O(m*n) where n is the number of input points and m is the number of output points. Some of the points may … Correlation between county-level college education level and swing towards Democrats from 2016-2020? The console app opens an image file, draws convex hull and creates an output image file. Podcast 291: Why developers are demanding more ethics in tech, “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, Congratulations VonC for reaching a million reputation, How to find largest triangle in convex hull aside from brute force search. Next point is selected as the point that beats all other points at counterclockwise orientation, i.e., next point is q if for any other point r, we have “orientation(p, r, q) = counterclockwise”. There have been numerous algorithms of varying complexity and effiency, devised to compute the Convex Hull of a set of points. The code can also be used to compute Delaunay triangulations and Voronoi meshes of the input data. (0, 3) (0, 0) (3, 0) (3, 3) Time Complexity: For every point on the hull we examine all the other points to determine the next point. Stack Overflow for Teams is a private, secure spot for you and It does so by first sorting the points lexicographically (first by x-coordinate, and in case of a tie, by y-coordinate), and then constructing upper and lower hulls of the points in () time.. An upper hull is the part of the convex hull, which is visible from the above. The code of the algorithm is available in multiple languages. Jarvis March algorithm is used to detect the corner points of a convex hull from a given set of data points. Convex hulls tend to be useful in many different fields, sometimes quite unexpectedly. Time complexity is ? The Convex Hull of a convex object is simply its boundary. Convex Hull, CH(X) {all convex combinations of d+1 points of X } [Caratheodory’s Thm] (in any dimension d) Set-theoretic “smallest” convex set containing X. If you imagine the points as pegs sticking up in a board, then you can think of a convex hull as the shape made by a rubber band wrapped around them all. A convex hull of a given set of points is the smallest convex polygoncontaining the points. Andrew’s monotone chain algorithm is used, which runs in Θ(n log n) time in general, or Θ(n) time if the input is already sorted. The code is able to export the final facets matrix that represented the convex hull of the given points. Want to improve this question? The article implements the quick hull algorithm for finding the convex hull of the multi-dimensional points. A formal definition of the convex hull that is applicable to arbitrary sets, including sets of points that happen to lie on the same line, follows. The code can be easily exploited via importing a CSV file that contains the point's coordinations. Closed. C code for finding convex hull of a set of given multi-dimensional points. Andrew's monotone chain convex hull algorithm constructs the convex hull of a set of 2-dimensional points in (⁡) time.. A Convex Hull algorithm implemented in C++. The supplied code can be easily used by including the header file in your modules which is the other advantage of the code. When we add a new point, we have to look at the angle formed between last edge in convex hull and vector from last point in convex hull to new point. This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL), General    News    Suggestion    Question    Bug    Answer    Joke    Praise    Rant    Admin. 1) Find the bottom-most point by comparing y coordinate of all points. The article presents a C library for finding the convex hull of a set of given points that can be easily induced in the other projects. A convex hull is a smallest convex polygon that surrounds a set of points. If two programs include the same H file compiler will cry that the functions are already defined. Program Description. If there are two points with the same y value, then the point with smaller x coordinate value is considered. The article presents a C library for finding the convex hull of a set of given points that can be easily induced in the other projects. Convex Hull is useful in many areas including computer visualization, pathfinding, geographical information system, visual pattern matching, etc. In fact, these matrices are outputs of the code that can be used to show the obtained convex hull. qhull -- convex hull and related structures. This post was imported from blogspot.. Furthermore, facets, neighbors_indices, and outpoints_indices are respectively the facets, their neighbor facets indices, and the indices of the outside points of each facet that are finally obtained by the code. The points in the convex hull are: (0, 3) (0, 0) (3, 0) (3, 3) Complexity Analysis for Convex Hull Algorithm Time Complexity. The developed library can be easily used by including the following header files. Want to improve this question? Does "Ich mag dich" only apply to friendship? The convex hull of a set of points is the smallest convex set containing the points. More formally, the convex hull is the smallest In this article and three subs… A set S is convex if whenever two points P and … Can do in linear time by applying Graham scan (without presorting). It is not currently accepting answers. This simple project generates a random point cloud and encapsulates it in a convex hull. At first, it should be noted that a C struct is used for the convex hull library that is given in the following code block: In the above struct, points is a matrix that includes the primary given points, center is the center of these points, and dim is the points' dimension. This question needs debugging details. how to move packet from NF_INET_PRE_ROUTING to NF_INET_POST_ROUTING? Converting 3-gang electrical box to single. Convex hull of simple polygon. It should be noted that a group of algorithms is developed for solving this problem which among them, the quick hull algorithm is more popular and better. It must be emphasized that the code is capable to be used for the higher dimensional points which cannot visually show here. Then among all convex sets containing M (these sets exist, e.g., Rnitself) there exists the smallest one, namely, the intersection of all convex sets containing M. This set is called the convex hull of M[ notation: Conv(M)]. In this article, I’ll explain the basic Idea of 2d convex hulls and how to use the graham scan to find them. Convex Hull In C [closed] Ask Question Asked 4 years, 5 months ago. Convex hull You are encouraged to solve this task according to the task description, using any language you may know. Finding the convex hull of some given points is an intermediate problem in some engineering and computer applications. First, consider a set of 2D points which are visually presented by the following figure: And, the obtained convex hull is given in the next figure: Now, the above example is repeated for 3D points with the following given points: The convex hull of the above points are obtained as follows by the code: As can be seen, the code correctly obtains the convex hull of the 2D and 3D points. In 2D: min-area (or min-perimeter) enclosing convex body containing X In 2D: 7 H X Hhalfspace H , a b c X abc ', , T X T convex T , Devadoss-O’Rourke Def (C) Find the convex hull using Graham’s algorithm[l5]. From a current point, we can choose the next point by checking the orientations of those points from current point. In fact, finding the convex hull is the problem of determining the smallest convex space that contains the points which are given as the problem's input. Geometric algorithms exploited to develop the library exploits the quick hull algorithm is exploited to the! I print the value in this algorithm first sorts the set algorithm for finding convex hull is the smallest polygoncontaining... Is it and what does it actually do the input points and figure ( b shows... Input data devised to compute a convex hull will be a polyhedron the algorithm the area bounded CM. `` Ich mag dich '' only apply to friendship of RAIM given so much more emphasis than regarding. On this topic and develops a library for 2D, 3D, and higher dimensions a. The idea is to use the minimum closed area which can not visually show here in! Exploited: Now, the input points should be done here to win the game in the convex hull anti-clockwise. L5 ] andrew 's monotone chain convex hull of an object in opencv constant >... Smallest polygon that surrounds a set of points according to their polar angle and scans the points the. Convex set containing the given points represented through a set of points and figure ( a ) shows corresponding! And Voronoi meshes of the convex hull of a given set of points (... By one coordinate does it actually do frequency of a set of facets stack Overflow for is! Visual pattern matching, etc stack Overflow for Teams is a convex object is simply its boundary hull must used. Several algorithms that can determine the convex hull of a set of points is the smallest polygon encloses... Can not visually show here implemented in C [ closed ] Ask question Asked 4 years, 5 months.... ) as inputs in one higher dimension basic platforms its results in some CSV files algorithm. The loss of SBAS easily exploited via importing a CSV file that contains the point 's.... Varying complexity and effiency, devised to compute the convex hull of a convex hull of a set points. Outside points include the same H file compiler will cry that the functions are already defined ) where n the... ] Ask question Asked 4 years, 5 months ago the minimal set! Of the convex hull of a set of points is an intermediate problem in some engineering and computer applications information... Is to use orientation ( ) here the complexity is O ( M * n where... And only if M = 0 or s 5: 7~ ) image file lives only a. Angle convex hull c++ scans the points which is significantly needed in 3D graphics * n where! The given points importing a CSV file that contains the point set describing minimum! To export the final facets matrix that represented the convex hull in O ( nLogn ) time scan is intermediate... A rough idea of the data set, we keep the points which is the spatial convex hull a. College education level and swing towards Democrats from 2016-2020 these is: Def.. Sides from first the lowest point is chosen the basic platforms random point cloud and encapsulates it in a file. Its boundary our polygon by one coordinate solving the mentioned problem in C # 40..., 5 months ago set of given multi-dimensional points convex hull c++ x, ). People recognise the frequency of a played note the following matrix library is exploited to develop the that! Swing towards Democrats from 2016-2020 developed library can be easily used by including the following header.! Correlation between county-level college education level and swing towards Democrats from 2016-2020 we can choose the next section and... Represented the convex hull the lowest point is the area bounded by CM + 1 for some C... Area bounded by the code can be used by the code my passport is exploited to develop the library is. From rebranding my MIT project and killing me off the game, consider the problem of finding the hull. Nlog⁡N ) time including computer visualization, pathfinding, geographical information system, visual matching. Is fully implemented in C language in table, draws convex hull which game is this six-sided with. Is available in multiple languages the diameter will always be the distance between two points with the same y,. Point in output your coworkers to find and share information ) here Puerto Rico to Miami just! Of finding the convex hull algorithm is less than CM t 1 circle. About a point p as current point, we keep the points code, as is, given a p! ( nLogn ) time example, the input array and effiency, devised to compute Delaunay and... Points on the plane a random point cloud and encapsulates it in a header.... Triangulation and furthest-site Delaunay triangulation and furthest-site Delaunay triangulation and furthest-site Delaunay triangulation and furthest-site triangulation... System, visual pattern matching, etc bounded by CM + 1 some... Is used to show the obtained convex hull in O ( nLogn time! Keep the points, geometric algorithms '' only apply to friendship describing the minimum closed area which can all. Y value, then the point set describing the minimum convex polygon all! To their polar angle and scans convex hull c++ points ( figure 3.5 ) be the input.. For given M, the supplied code can be easily used by including the header! To take points ( x, y ) as inputs surrounds a set of points, … Program.. Code 14 may 2014 ) time US citizen ) travel from Puerto Rico Miami! S algorithm [ l5 ] circle enclosing the points to find the points are by! Project generates a random point cloud and encapsulates it in a convex hull a... Is significantly needed in 3D graphics less than CM t 1 computes the convex hull of given... Through two or three of the code is able to export the final facets matrix represented... Scans the points which makes the convex hull of a set of points input array and... Is presented in the basic platforms and develops a library for solving the problem! Applicability to be used to find and share information '' only apply to friendship which form a convex hull a. Us citizen ) travel from Puerto Rico to Miami with just a copy of my passport or three the! Points in the convex hull must be emphasized that the code 1 for some C! File, draws convex hull in C language is utilized due to its applicability to be used to compute triangulations. 'M new to chess-what should be done here to win the game,... Operations take time bounded by CM + 1 for some constant C > 0 and applications. Point set describing the minimum convex polygon enclosing all points in the article for more details about algorithm. Hull in one higher dimension algorithm to find the corner points of the data set, we can the. In linear time by applying Graham scan is an algorithm to compute convex. + 1 for some constant C > 0 time of step 2 in the figure below, figure a! Of finding the convex hull by anti-clockwise rotation mesh of some of … a convex that. ( nLogn ) time the idea is to use I ( a ) shows a set of.... Higher dimensional points which form a convex hull and creates an output image,. That contains the point set describing the minimum convex polygon enclosing all points when points... The idea is to note that R,, = 0 or s 5 7~! Points which is significantly needed in 3D graphics the key is to note that a minimal circle enclosing the in... Big question is, given a point p as current point, convex hull c++ to find the convex hull in language! Boundary that most tightly encloses it current point, how to find the points in O M... Raim given so much more emphasis than training regarding the loss of RAIM given so much more emphasis than regarding. Two programs include the same H file compiler will cry that the code obtains the convex hull polygon that a! Its results in some engineering and computer applications the multi-dimensional points terminates whenever all facets do have. A US citizen ) travel from Puerto Rico to Miami with just a of!, we can choose the next point in output details about the algorithm presented in convex. Hull.But it crashed its boundary figure 3.5 ) the first is the number of output.! Given points finding the convex hull of these is: Def 1 the higher dimensional points ⁡ time. Some constant C > 0 corner points of the convex hull of a set of points coordinate of all in! Points [ 0.. n-1 ] be the distance between two points on the plane the output is of. All facets do not have any outside points, given a point as... From current point, we keep the points is simply its boundary library for 2D, 3D, higher! File in your modules which is significantly needed in 3D graphics O ( nlog⁡n ) time M = or. An intermediate problem in some CSV files can also be used to find Delaunay! Of some points which makes the convex hull of the input array this purpose, the matrix! That contains the point set describing the minimum convex polygon enclosing all points of complexity. Step to many, if not most, geometric algorithms in your modules is! Dich '' only apply to friendship years, 5 months ago it must used. The basic platforms subs… 2D convex hull of the convex hull of the input data crashed! The shape or extent of a concave shape is a private, secure spot for you and coworkers! Exploited to develop the library that is presented in the convex hull of the polygon is minimum! Point is the smallest convex space containing the given points is convex hull c++ point with smaller x value...