Honours Project Report

Transcription

1 1 Honours Project Report Natural Looking Graph Generator Author : Eashan Kumar Singh Supervisor : Dr. David Mould, School of Computer Science, Carleton University Date : 11th December 2015

2 2 Abstract The aim of this project was to create pseudo random graphs that look like something that would exist in nature, like the pattern on insect wings. It achieves this by modelling the random graphs using linear and angular springs on the nodes to make the graph more curvy rather than straight and jagged. This can be used to create randomly generated textures for objects that exist in nature. The main conclusion drawn was that cohesive circular structures looked more natural and graphs with nodes anchored by multiple edges produced the best results. Acknowledgements I d like to thank Dr. David Mould for supervising my honours project and for helping me with my project all the way from the selection of my topic to helping with the problems in the code.

3 3 Table of Contents Abstract 2 Acknowledgements 2 Table of Contents 3 Table of Figures 4 Outline 5 Introduction 6 Problem Statement 6 Motivation 6 Goals 7 Objectives 7 Background 8 Processing 8 Running the Project 8 Report 9 Generating Completely Random Graphs 9 Subdividing Edges 10 Anchoring Nodes 11 Edges That Try To Make Circular Patterns 12 Edges That Try To Make Sine Curves 13 Combining The Two Types Of Curves 14 Adding Variety In Angles And Anchoring Primary Vertices 15 Conclusion 17 Looking Forward 17 Appendix A Graphs generated for the statistics in the conclusion 18

4 4 List of Figures FIGURE 1 Bubbles under ice 6 FIGURE 2 A Completely Random Graph 9 FIGURE 3 Graph with edges subdivided 10 FIGURE 4 Graph with anchored edges 11 FIGURE 5 Graph with circular patterns 12 FIGURE 6 Graph with sine curves 13 FIGURE 7 Graph with sine curves and circular curves 14 FIGURE 8 Graph with variable circular curves and fixed primary vertices 15 FIGURE 9 Graph with variable sine curves and fixed primary vertices 16 FIGURE 10 Natural looking sine curve 1 18 FIGURE 11 Natural looking sine curve 2 19 FIGURE 12 Natural looking sine curve 3 19 FIGURE 13 Natural looking circular curve 1 20 FIGURE 14 Natural looking circular curve 2 20 FIGURE 15 Natural looking circular curve 3 21

5 5 Outline This document will talk about the Natural Looking Graph Generator project. The document will start off with an introduction which will present the problem statement being challenged, the motivation for the production of this project, the goals that need to be achieved and the objectives of the project. The background will give a small introduction to Processing and explain how to run the project. Finally, the report will chronicle all the major advancements in the code, explain the rationale behind each of them and analyze a screenshot from that version of the project.

6 6 Introduction Problem Statement When modelling graphs, is it possible to randomly generate graphs and alter them using linear and angular springs to make something more natural looking, e.g., the pattern on an insect wing, or the bubbles under ice in FIGURE 1. FIGURE 1 Bubbles under ice Motivation Most randomly generated graphs have straight edges with intersections that look jagged but when we look at patterns that exist in nature, we rarely see such structures. When we want to generate random structures, the easiest structures to generate are graphs since we only need to randomly generate some nodes and randomly generate which pairs of those nodes will have edges between them. The purpose of this project was to find a way to randomly generate graphs and modify them using linear and angular springs to make them look natural.

7 7 Goals 1. Randomly generate graphs that would work well with springs. 2. Add linear(hookean) springs between each edge. 3. Add angular springs between each pair of edges that is connected. 4. Study what values work for both types of springs. 5. Study what constraints on graph generation work well with spring modeling. Objectives 1. Make a completely randomly generated graph. a. Randomly generate coordinates for the nodes. b. Randomly assign edges between nodes. c. Add a linear spring with a random desired length on each edge. d. Add an angular spring with a random desired angle between each pair of edges that share a parent. 2. Study the completely random graphs to find desirable traits. 3. Make a Pseudorandomly generated graph with constraints dependant on the desirable traits 4. Study the Pseudorandomly generated graphs to find desirable traits and move to step 3 (Repeat as many times as desired)

8 8 Background Processing Processing is an open source programming language and integrated development environment (IDE) built for the electronic arts, new media art, and visual design communities with the purpose of teaching the fundamentals of computer programming in a visual context. The language builds on the Java language, but uses a simplified syntax and graphics programming model.this project uses Processing because it is an incredibly simple language for creating purely graphical programs. The simplicity of the language allows the code to be written much faster and leaves time to focus on the research aspects of this project. Running the project This project uses processing To install processing 2.2.1, 1. Go to 2. Look for under stable releases and download the version for your operating system 3. Extract the package that was downloaded To run the project, 1. Open the directory that was extracted 2. Execute processing.exe 3. In the menubar, select File > Open 4. Navigate to the project directory and select NaturalLookingGraphGenerator.pde 5. Press the run button 6. In order to activate the linear springs, press the left mouse button 7. In order to activate the angular springs, press any key on the keyboard

9 9 Report Generating completely random graphs At the start of the project, the graph generation was incredibly unsophisticated, it was just as a sanity check for the graph generation framework actually working. The results were mostly insignificant but there were two very important findings from it. Firstly, the graph was incredibly jagged so edges had to be divided into a lot of sub edges to make curves. Secondly, dangling edges make the graph look more sloppy so every node in the graph had to be anchored to the edges of the screen at two points. It also definitively proved that the graph manipulation framework was working as intended. FIGURE 2 A Completely Random Graph

10 10 Subdividing edges To subdivide the edges, the program divides each edge large enough into two edges by inserting a vertex between them and recursively sends both edges through the same process. After a couple of iterations, it seemed like keeping the threshold at 10 pixels worked out pretty well, keeping the length of all edges between 5 to 10 pixels. While the problem of the graph looking jagged was fixed, the anchoring problem became even more apparent. The dangling edges could not get to a rest state with the angular springs because they would just keep swinging with nothing to keep them anchored. Two other problems also arose at this point. The first was that the randomly generated angles were making slightly odd looking structures which could be fixed by making the primary edges have some cohesive structure that the sub edges tried to follow, like a sine curve. The second was that intersections didn t look very nice when the physics were being applied. This could be fixed by making a node at the intersection point and dividing the two edges intersecting there into four edges converging on that node. FIGURE 3 Graph with edges subdivided

11 11 Anchoring nodes The first few attempts at anchoring all nodes to two points on the edge of the screen ended in the program running into an infinite loop until I realized that for nodes to be anchored, there needed to be a minimum of two points on the edge of the screen. The graphs that had only two points on the edge of the screen looked very biased toward one corner of the screen if the points were on adjacent edges of the screen, or very narrow if the points were on the opposite sides of the screen. This problem got fixed by mandating a node on each edge of the screen. This led to more structured graphs but made the cohesive edge angles problem more apparent. FIGURE 4 Graph with anchored edges

12 12 Edges that try to make circular patterns To make all edges try to follow circular patterns, I set all the desired angles for angular springs to the same value. This made them all curve evenly making a perfect semicircle. FIGURE 4 was a huge success for the project because the left hand side of it has a pattern that has more of a natural look to it than anything that had been produced till now. This was a breakthrough for the project. That being said, the rest of the graph was still very bare and bland. FIGURE 5 Graph with circular patterns

13 13 Edges that try to make sine curves The next experiment was to make a graph where all the edges were trying to form a sine curve. This was done by assigning the first two subdivisions of an edge the values 1 and 1 and multiplying the fixed desired angles with the sign of the edge. FIGURE 5 brings an image that closely resembles a butterfly wing at the center which signified that the project was moving in the right direction. The rest of the graph was still very unremarkable. FIGURE 6 Graph with sine curves Remark In order to make the graph generate sine curves instead of circular curves, change the value of the boolean variable SINE to true.

14 14 Combining the two types of curves I was curious as to how the two types of curves mixed into a graph so I made edges have a 50% chance of being one of the two curves to see how it would turn out. After looking at a lot of randomly generated graphs, FIGURE 6 is the best graph that was produced. I think the graph became less natural looking by using both types of curves. FIGURE 7 Graph with sine curves and circular curves

15 15 Adding variety in angles and anchoring primary vertices In all the previous iterations with sine curves and circular curves, all of the angular springs were using the same angles regardless of which primary edge they were a part of. The next experiment was to change the angles each of the primary edges made. Another test I did here was that I made all the primary vertices immovable. This made the angles exert more force within their constraints which I think made the angles look more pronounced. I tried this with both circular curves and sine curves. FIGURE 7 and 8 show that adding the variety in angles and anchoring the primary vertices in both types of curve graphs made the graphs more natural looking. FIGURE 8 Graph with variable circular curves and fixed primary vertices

16 FIGURE 9 Graph with variable sine curves and fixed primary vertices 16

17 17 Conclusion The project set out to randomly generate natural looking graphs and I made a lot of progress on that front but ultimately the program only partially succeeds in that endeavour. In order to get three natural looking sine curve graphs, it took the program 13 tries and to get three natural looking circular curve graphs, it took the program 15 tries. There are a couple of ideas that I have listed in the section titled Looking Forward that I believe would have made the program have a better success rate. Given the start point, the program has evolved a decent bit and is capable of generating some beautiful graphs but the success rate is too low for me to label it a success. Looking Forward These are a couple of features I wanted to implement in the project but did not get the time to 1. Line intersections At every point of intersection, add a new node and split the two edges into four edges that converge at that node. 2. Coulombs s Law I wanted to solve the problem of edges crossing over each other due to the spring forces by making nodes have charges based on their parent edge that would repel all vertices that belong to other edges. 3. Bezier Curves I wanted to make a keypress function that made the physics simulation halt at the current point and model the current state of the graph using Bezier curves.

18 18 Appendix A Graphs generated for the statistics in the conclusion FIGURE 10 Natural looking sine curve 1

19 19 FIGURE 11 Natural looking sine curve 2 FIGURE 12 Natural looking sine curve 3

20 20 FIGURE 13 Natural looking circular curve 1 FIGURE 14 Natural looking circular curve 2

21 FIGURE 15 Natural looking circular curve 3 21