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A polygon is a plane figure that is bounded by a closed circuit composed of a finite sequence of straight line segments.

This visualization features a few computational geometry algorithms that can be carried out on simple (non-crossing) polygons with 3 or more non-collinear points, such as determining their perimeters and areas, determining concavity or convexity, determining whether a point is inside or outside, and to cut them with a simple line.

Remarks: By default, we show e-Lecture Mode for first time (or non logged-in) visitor.
If you are an NUS student and a repeat visitor, please login.

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Vertices of a polygon can be ordered either in ClockWise (CW) or CounterClockWise (CCW) order. In this visualization, we prefer to use CCW order (although drawing polygon with vertices in CW order is also acceptable). Under the hood, we also set the first vertex = the last vertex to simplify implementation.

Note that we limit the drawn polygon to be a simple polygon, i.e. ,there is no edge intersection.

The number of vertices/corners of the polygon is stored in variable n. As polygon is a closed circuit, the number of edges/sides of the polygon is also n.

Pro-tip 1: Since you are not logged-in, you may be a first time visitor (or not an NUS student) who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown]/[PageUp] to go to the next/previous slide, respectively, (and if the drop-down box is highlighted, you can also use [→ or ↓/← or ↑] to do the same),and [Esc] to toggle between this e-Lecture mode and exploration mode.

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All available operations are listed in the left hand side menu as usual.

The first two are for giving simple input polygons and the next five are the computational geometry algorithms that you can run on the currently drawn polygon.

Pro-tip 2: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2021). We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (F11) to enjoy this setup. However, you can use zoom-in (Ctrl +) or zoom-out (Ctrl -) to calibrate this.

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In this visualization, you can draw any simple polygon (at least 3 points), without any collinear points (actually, it is possible to modify our implementation to allow collinear points, just that it will complicate a few operations). The smallest such polygon is a triangle.

The polygon that you draw can be either convex (line connecting any two points inside the polygon will remain inside the polygon) or concave.

If you do not close the loop (draw an edge from last vertex back to vertex 0), we will do that automatically for you.

Pro-tip 3: Other than using the typical media UI at the bottom of the page, you can also control the animation playback using keyboard shortcuts (in Exploration Mode): Spacebar to play/pause/replay the animation, / to step the animation backwards/forwards, respectively, and -/+ to decrease/increase the animation speed, respectively.

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We provide a few example polygons as a starting point.

Upon loading this visualization page, we will randomize the chosen example polygon.

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The perimeter of a polygon is simply the sum of the lengths (Euclidean distances) of consecutive line segments (polygon edges).

This routine works for both convex and concave polygons and runs in O(n).

Without further ado, let's compute the Perimeter of the currently drawn polygon.

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When the vertices of a polygon are given in a circular manner (CW or CCW), we can compute its area using the Shoelace Formula.

This Shoelace Formula returns the area, which is half the cross products of vectors defined by edge endpoints.

This formula is versatile as it works for both convex and concave polygons. It can be computed in O(n).

Without further ado, let's compute the Area of the currently drawn polygon.

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A polygon is called a Convex polygon if we draw a line between any two different points inside the polygon and the line always remain inside the polygon. Otherwise, the polygon is called Concave.

There is a far easier method to check if a given polygon (assume no three collinear points) is convex without using the direct definition above. We can check if all three consecutive vertices of the polygon form the same kind of turn (all CCWs or all CWs). This check is clearly O(n).

Without further ado, let's check if the currently drawn polygon IsConvex.

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There are a few algorithms for checking if a point (pt1) is inside a polygon or not. We reckon the most robust algorithm is the Winding Number algorithm that computes the sum of angles subtended by each edge/side of the polygon with pt1 as the origin. As there are only n such angles, this check also runs in O(n).

The input simple polygon can be as complicated as the currently displayed "MAZE" test case. Try InsidePolygon and OutsidePolygon test cases.

In Exploration Mode, you will be asked to provide the tested point (pt1) as additional input of this operation.

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We can cut a convex polygon with a straight line defined by two points (pt1, pt2). The result of the cut are two smaller but also convex polygons. This algorithm currently returns the smaller polygon on 'the left side' of the cutting line (pt1, pt2).

Note that although possible, cutting a Concave polygon is more complicated as it may result in more than two (and possibly degenerate) polygons. We allow such operation in this visualization but extra care must be exercised in the actual implementations.

Try Left Side to see the default version of this routine and Right Side where we swap pt1 and pt2 to get the other side of the cut.

In Exploration Mode, you will be asked to provide two points to define the cut line (pt1 and pt2) as additional input of this operation (to avoid degenerate case, these two points should be placed at different locations).

This routine also runs in O(n).

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There is one more computational geometry visualization in VisuAlgo: Convex Hull.

You can now use some of these algorithm on polygon routines to solve a few programming exercises: UVa 11265 - The Sultan's Problem and Kattis - robotprotection.

You are allowed to use/modify our implementation code for various polygon algorithms:
polygon.cpp | py | java | ml

You have reached the last slide. Return to 'Exploration Mode' to start exploring!

Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com.

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Visualisation Scale

Edit Polygon

Example Polygon

perimeter(P)

area(P)

isConvex(P)

inPolygon(pt, P)

cutPolygon(ln, P)

>

1.0x (Default)

0.5x (Minimal Details)

Convex

Concave

Mountain

Maze

Star

Goalkeeper

Fish

Initially conceived in 2011 by Associate Professor Steven Halim, VisuAlgo aimed to facilitate a deeper understanding of data structures and algorithms for his students by providing a self-paced, interactive learning platform.

Featuring numerous advanced algorithms discussed in Dr. Steven Halim's book, 'Competitive Programming' — co-authored with Dr. Felix Halim and Dr. Suhendry Effendy — VisuAlgo remains the exclusive platform for visualizing and animating several of these complex algorithms even after a decade.

While primarily designed for National University of Singapore (NUS) students enrolled in various data structure and algorithm courses (e.g., CS1010/equivalent, CS2040/equivalent (including IT5003), CS3230, CS3233, and CS4234), VisuAlgo also serves as a valuable resource for inquisitive minds worldwide, promoting online learning.

Initially, VisuAlgo was not designed for small touch screens like smartphones, as intricate algorithm visualizations required substantial pixel space and click-and-drag interactions. For an optimal user experience, a minimum screen resolution of 1366x768 is recommended. However, since April 2022, a mobile (lite) version of VisuAlgo has been made available, making it possible to use a subset of VisuAlgo features on smartphone screens.

VisuAlgo remains a work in progress, with the ongoing development of more complex visualizations. At present, the platform features 24 visualization modules.

Equipped with a built-in question generator and answer verifier, VisuAlgo's "online quiz system" enables students to test their knowledge of basic data structures and algorithms. Questions are randomly generated based on specific rules, and students' answers are automatically graded upon submission to our grading server. As more CS instructors adopt this online quiz system worldwide, it could effectively eliminate manual basic data structure and algorithm questions from standard Computer Science exams in many universities. By assigning a small (but non-zero) weight to passing the online quiz, CS instructors can significantly enhance their students' mastery of these basic concepts, as they have access to an almost unlimited number of practice questions that can be instantly verified before taking the online quiz. Each VisuAlgo visualization module now includes its own online quiz component.

VisuAlgo has been translated into three primary languages: English, Chinese, and Indonesian. Additionally, we have authored public notes about VisuAlgo in various languages, including Indonesian, Korean, Vietnamese, and Thai:

id, kr, vn, th.

#### Team

Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

CDTL TEG 1: Jul 2011-Apr 2012: Koh Zi Chun, Victor Loh Bo Huai

Final Year Project/UROP students 1
Jul 2012-Dec 2013: Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy
Jun 2013-Apr 2014 Rose Marie Tan Zhao Yun, Ivan Reinaldo

CDTL TEG 2: May 2014-Jul 2014: Jonathan Irvin Gunawan, Nathan Azaria, Ian Leow Tze Wei, Nguyen Viet Dung, Nguyen Khac Tung, Steven Kester Yuwono, Cao Shengze, Mohan Jishnu

Final Year Project/UROP students 2
Jun 2014-Apr 2015: Erin Teo Yi Ling, Wang Zi
Jun 2016-Dec 2017: Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir
Aug 2021-Apr 2023: Liu Guangyuan, Manas Vegi, Sha Long, Vuong Hoang Long, Ting Xiao, Lim Dewen Aloysius

Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

Final Year Project/UROP students 3
Aug 2023-Apr 2024: Xiong Jingya, Radian Krisno, Ng Wee Han

List of translators who have contributed ≥ 100 translations can be found at statistics page.

Acknowledgements
NUS CDTL gave Teaching Enhancement Grant to kickstart this project.

For Academic Year 2023/24, a generous donation from Optiver will be used to further develop VisuAlgo.

VisuAlgo is generously offered at no cost to the global Computer Science community. If you appreciate VisuAlgo, we kindly request that you spread the word about its existence to fellow Computer Science students and instructors. You can share VisuAlgo through social media platforms (e.g., Facebook, YouTube, Instagram, TikTok, Twitter, etc), course webpages, blog reviews, emails, and more.

Data Structures and Algorithms (DSA) students and instructors are welcome to use this website directly for their classes. If you capture screenshots or videos from this site, feel free to use them elsewhere, provided that you cite the URL of this website (https://visualgo.net) and/or the list of publications below as references. However, please refrain from downloading VisuAlgo's client-side files and hosting them on your website, as this constitutes plagiarism. At this time, we do not permit others to fork this project or create VisuAlgo variants. Personal use of an offline copy of the client-side VisuAlgo is acceptable.

Please note that VisuAlgo's online quiz component has a substantial server-side element, and it is not easy to save server-side scripts and databases locally. Currently, the general public can access the online quiz system only through the 'training mode.' The 'test mode' offers a more controlled environment for using randomly generated questions and automatic verification in real examinations at NUS.

List of Publications

This work has been presented at the CLI Workshop at the ICPC World Finals 2012 (Poland, Warsaw) and at the IOI Conference at IOI 2012 (Sirmione-Montichiari, Italy). You can click this link to read our 2012 paper about this system (it was not yet called VisuAlgo back in 2012) and this link for the short update in 2015 (to link VisuAlgo name with the previous project).

Bug Reports or Request for New Features

VisuAlgo is not a finished project. Associate Professor Steven Halim is still actively improving VisuAlgo. If you are using VisuAlgo and spot a bug in any of our visualization page/online quiz tool or if you want to request for new features, please contact Associate Professor Steven Halim. His contact is the concatenation of his name and add gmail dot com.