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.

X Esc
PgDn

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: Since you are not logged-in, you may be a first time visitor who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown] to advance to the next slide, [PageUp] to go back to the previous slide, [Esc] to toggle between this e-Lecture mode and exploration mode.

X Esc
PgUp
PgDn

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.

Another pro-tip: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2017). 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.

X Esc
PgUp
PgDn

In this visualization, you can draw any simple polygon (at least 3 points), without any collinear points. 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.

X Esc
PgUp
PgDn

We provide a few example polygons as a starting point.

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

X Esc
PgUp
PgDn

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.

X Esc
PgUp
PgDn

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.

X Esc
PgUp
PgDn

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.

X Esc
PgUp
PgDn

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.

X Esc
PgUp
PgDn

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 and is thus not supported in this visualization.

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.

This routine also runs in O(n).

X Esc
PgUp
PgDn

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
polygon.java
polygon.py
polygon.ml

X Esc
PgUp
PgDn

As the action is being carried out, each step will be described in the status panel.

X Esc
PgUp
PgDn

e-Lecture: The content of this slide is hidden and only available for legitimate CS lecturer worldwide. Drop an email to visualgo.info at gmail dot com if you want to activate this CS lecturer-only feature and you are really a CS lecturer (show your University staff profile).

X Esc
PgUp
PgDn

Control the animation with the player controls! Keyboard shortcuts are:

Spacebar: play/pause/replay
Left/right arrows: step backward/step forward
-/+: decrease/increase speed
X Esc
PgUp
PgDn

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.

X Esc
PgUp

Draw Polygon

Example Polygon

perimeter(P)

area(P)

isConvex(P)

inPolygon(pt, P)

cutPolygon(ln, P)

Concave

Convex

Mountain

Maze

VIsuAlgoについて

VisuAlgo was conceptualised in 2011 by Dr Steven Halim as a tool to help his students better understand data structures and algorithms, by allowing them to learn the basics on their own and at their own pace.

VisuAlgo contains many advanced algorithms that are discussed in Dr Steven Halim's book ('Competitive Programming', co-authored with his brother Dr Felix Halim) and beyond. Today, some of these advanced algorithms visualization/animation can only be found in VisuAlgo.

Though specifically designed for National University of Singapore (NUS) students taking various data structure and algorithm classes (e.g. CS1010, CS1020, CS2010, CS2020, CS3230, and CS3230), as advocators of online learning, we hope that curious minds around the world will find these visualisations useful too.

VisuAlgo is not designed to work well on small touch screens (e.g. smartphones) from the outset due to the need to cater for many complex algorithm visualizations that require lots of pixels and click-and-drag gestures for interaction. The minimum screen resolution for a respectable user experience is 1024x768 and only the landing page is relatively mobile-friendly.

VisuAlgo is an ongoing project and more complex visualisations are still being developed.

The most exciting development is the automated question generator and verifier (the online quiz system) that allows students to test their knowledge of basic data structures and algorithms. The questions are randomly generated via some rules and students' answers are instantly and automatically graded upon submission to our grading server. This online quiz system, when it is adopted by more CS instructors worldwide, should technically eliminate manual basic data structure and algorithm questions from typical Computer Science examinations in many Universities. By setting a small (but non-zero) weightage on passing the online quiz, a CS instructor can (significantly) increase his/her students mastery on these basic questions as the students have virtually infinite number of training questions that can be verified instantly before they take the online quiz. The training mode currently contains questions for 12 visualization modules. We will soon add the remaining 8 visualization modules so that every visualization module in VisuAlgo have online quiz component.

Another active branch of development is the internationalization sub-project of VisuAlgo. We want to prepare a database of CS terminologies for all English text that ever appear in VisuAlgo system. This is a big task and requires crowdsourcing. Once the system is ready, we will invite VisuAlgo visitors to contribute, especially if you are not a native English speaker. Currently, we have also written public notes about VisuAlgo in various languages: zh, id, kr, vn, th.

チーム

プロジェクトリーダー＆アドバイザー（2011年7月～現在)
Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Software Engineer, Google (Mountain View)

Undergraduate Student Researchers 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

Final Year Project/UROP students 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

Undergraduate Student Researchers 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 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

Final Year Project/UROP students 4 (Jun 2016-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

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

Acknowledgements
This project is made possible by the generous Teaching Enhancement Grant from NUS Centre for Development of Teaching and Learning (CDTL).

利用規約

VisuAlgo is free of charge for Computer Science community on earth. If you like VisuAlgo, the only payment that we ask of you is for you to tell the existence of VisuAlgo to other Computer Science students/instructors that you know =) via Facebook, Twitter, course webpage, blog review, email, etc.

If you are a data structure and algorithm student/instructor, you are allowed to use this website directly for your classes. If you take screen shots (videos) from this website, you can use the screen shots (videos) elsewhere as long as you cite the URL of this website (http://visualgo.net) and/or list of publications below as reference. However, you are NOT allowed to download VisuAlgo (client-side) files and host it on your own website as it is plagiarism. As of now, we do NOT allow other people to fork this project and create variants of VisuAlgo. Using the offline copy of (client-side) VisuAlgo for your personal usage is fine.

Note that VisuAlgo's online quiz component is by nature has heavy server-side component and there is no easy way to save the server-side scripts and databases locally. Currently, the general public can only use the 'training mode' to access these online quiz system. Currently the 'test mode' is a more controlled environment for using these randomly generated questions and automatic verification for a real examination in NUS. Other interested CS instructor should contact Steven if you want to try such 'test mode'.

List of Publications

This work has been presented briefly at the CLI Workshop at the ACM 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).

This work is done mostly by my past students. The most recent final reports are here: Erin, Wang Zi, Rose, Ivan.

Bug Reports or Request for New Features

VisuAlgo is not a finished project. Dr 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 Dr Steven Halim. His contact is the concatenation of his name and add gmail dot com.