Box2D

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What is the Box2D? Open-Source 2D Physics Engine

C++ by Erin Catto for the Game Developer’s Conference in 2006. Box2D site (http://www.box2d.org/) for reference Crayon Physics Angry Birds

JBox2D : Java Wrapper Box2D site (http://www.jbox2d.org//) for reference. Can be directly used in Processing

PBox2D : Processing Wrapper Make it easy to use JBox2D A layer library between JBox2D and Processing a Processing Box2D “helper” library ( a set of functions that help

you get your Box2D) PBox2D is not a Processing wrapper for all of Box2D https://github.com/shiffman/PBox2D

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Box2D Basics In old physical simulation

SETUP:1. Create all the objects in our world.

DRAW:1. Calculate all the forces in our world.2. Apply all the forces to our objects (F = M * A).3. Update the locations of all the objects based on their

acceleration.4. Draw all of our objects.

In Box2D SETUP:

1. Create all the objects in our world. DRAW:

1. Draw all of our objects.3

Five core parts for using Box2D1. World: Manages the physics simulation. It knows

everything about the overall coordinate space and also stores lists of every element in the world

2. Body: Serves as the primary element in the Box2D world. It has a location. It has a velocity. Sound familiar? The Body is essentially the class we’ve been writing on our own in our vectors and forces examples.

3. Shape: Keeps track of all the necessary collision geometry attached to a body.

4. Fixture: Attaches a shape to a body and sets properties such as density, friction, and restitution.

5. Joint: Acts as a connection between two bodies (or between one body and the world itself).

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A new Vector Class : Vec2

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PVector Vec2PVector a = new PVector(1,-1);PVector b = new PVector(3,4);a.add(b);

Vec2 a = new Vec2(1,-1);Vec2 b = new Vec2(3,4);a.addLocal(b);

PVector a = new PVector(1,-1);PVector b = new PVector(3,4);PVector c = PVector.add(a,b);

Vec2 a = new Vec2(1,-1);Vec2 b = new Vec2(3,4);Vec2 c = a.add(b);

PVector a = new PVector(1,-1);float n = 5;a.mult(n);

Vec2 a = new Vec2(1,-1);float n = 5;a.mulLocal(n);

PVector a = new PVector(1,-1);float n = 5;PVector c = PVector.mult(a,n);

Vec2 a = new Vec2(1,-1);float n = 5;Vec2 c = a.mul(n);

PVector a = new PVector(1,-1);float m = a.mag();a.normalize();

Vec2 a = new Vec2(1,-1);float m = a.length();a.normalize();

Box2D worlds

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How can we convert between them?

Helper functions

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Task FunctionConvert location from World to Pixels Vec2 coordWorldToPixels(Vec2 world)

Convert location from World to Pixels Vec2 coordWorldToPixels(float worldX, float worldY)

Convert location from Pixels to World Vec2 coordPixelsToWorld(Vec2 screen)

Convert location from Pixels to World Vec2 coordPixelsToWorld(float pixelX, float pixelY)

Scale a dimension (such as height, width, or ra-dius) from Pixels to World

float scalarPixelsToWorld(float val)

Scale a dimension from World to Pixels float scalarWorldToPixels(float val)

Set up the world

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PBox2D box2d;

void setup() { box2d = new PBox2D(this); //Initializes a Box2D world with default settings box2d.createWorld();}

Set up the Body Step 1: Define a body.

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//define the properties of the body we intend to makeBodyDef bd = new BodyDef();

Vec2 center = box2d.coordPixelsToWorld(width/2,height/2));bd.position.set(center);

bd.fixedRotation = true; //Fixed, never rotating objects

bd.linearDamping = 0.8; //Frictionbd.angularDamping = 0.9;

bd.bullet = true; //if it is FAST moving objectsbd.type = BodyType.DYNAMIC; // body type

Step 2: Configure the body definition.

Body types Dynamic (BodyType.DYNAMIC)

a “fully simulated” body. A dynamic body moves around the world, collides with other bodies, and responds to the forces in its environment.

Static (BodyType.STATIC) cannot move (as if it had an infinite mass). We’ll use

static bodies for fixed platforms and boundaries. Kinematic(BodyType.KINEMATIC)

can be moved manually by setting its velocity directly. If you have a user-controlled object in your world, you can use a kinematic body. Note that kinematic bodies collide only with dynamic bodies and not with other static or kinematic ones.

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Body body = box2d.createBody(bd); Step 3: Create the body.

Set up the Body

Step 4: Set any other conditions for the body’s starting state.body.setLinearVelocity(new Vec2(0,3));body.setAngularVelocity(1.2);

Linking Shape with Body Body does not have geometry We have to attach geometry shape to

body with a Fixture We can attach multiple shapes to a

single body in order to create more complex forms.

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Step 1: Define a shapePolygonShape ps = new PolygonShape();float box2Dw = box2d.scalarPixelsToWorld(150);float box2Dh = box2d.scalarPixelsToWorld(100);ps.setAsBox(box2Dw, box2Dh);

Width & Height

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width

heightheight

width

Processing BOX2D

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Step 2: Create a fixture.FixtureDef fd = new FixtureDef();//The fixture is assigned the PolygonShape we just made.fd.shape = ps;fd.friction = 0.3; //The coefficient of friction for the shape, typically between 0 and 1fd.restitution = 0.5; //The Shape’s restitution (i.e. elasticity), typically between 0 and 1fd.density = 1.0; //The Shape’s density, measured in kilograms per meter squared

Step 3: Attach the shape to the body with the fixture.body.createFixture(fd); Or you can use the default fixturebody.createFixture(ps,1); // Creates the Fixture and attaches the Shape with a density of 1

In summary1. Define a body using a BodyDef object (set any

properties, such as location).2. Create the Body object from the body definition.3. Define a Shape object using PolygonShape,

CircleShape, or any other shape class.4. Define a fixture using FixtureDef and assign the

fixture a shape (set any properties, such as friction, density, and restitution).

5. Attach the shape to the body.

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In processing term.

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//Step 1. Define the body. BodyDef bd = new BodyDef(); bd.position.set(box2d.coordPixelsToWorld(width/2,height/2));

//Step 2. Create the body. Body body = box2d.createBody(bd);

//Step 3. Define the shape. PolygonShape ps = new PolygonShape(); float w = box2d.scalarPixelsToWorld(150); float h = box2d.scalarPixelsToWorld(100); ps.setAsBox(w, h)

//Step 4. Define the fixture. FixtureDef fd = new FixtureDef(); fd.shape = ps; fd.density = 1; fd.friction = 0.3; fd.restitution = 0.5;

//Step 5. Attach the shape to the body with the Fixture. body.createFixture(fd);

How can we keep tracking objects? Use the arrayList (Dynamic array) Exercise

Making boxes on the fly whenever we move the box and put it on an arrayList.

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// A list for all of our rectanglesArrayList<Box> boxes;

void setup() { size(800,200); smooth(); // Create ArrayLists boxes = new ArrayList<Box>();}

void draw() { background(255);

// When the mouse is clicked, add a new Box object if (mousePressed) { Box p = new Box(mouseX,mouseY); boxes.add(p); }

// Display all the boxes for (Box b: boxes) { b.display(); }}

// A rectangular boxclass Box {

float x,y; float w,h;

// Constructor Box(float x_, float y_) { x = x_; y = y_; w = 16; h = 16; }

// Drawing the box void display() { fill(127); stroke(0); strokeWeight(2); rectMode(CENTER); rect(x,y,w,h); }}

Class Design

Let’s put those boxes on the Box2D world Revise the Box class

Box size (8,8) fd.density = 1; fd.friction = 0.3; fd.restitution = 0.5;

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class Box { Body body; float w,h;

Box(float x, float y) { //initial position w = 16; h = 16; // following the steps //Step 1. Define the body. //Step 2. Create the body.(bodyType=DYNAMIC) //Step 3. Define the shape. // Step 4. Define the fixture. // Step 5. Attach the shape to the body with the Fixture }

display() { Vec2 pos = box2d.getBodyPixelCoord(body); float a = body.getAngle();

pushMatrix(); translate(pos.x,pos.y);

// Box2D coordinate system considers rotation in the opposite direction from Processing

rotate(-a); fill(127); stroke(0); strokeWeight(2); rectMode(CENTER); rect(0,0,w,h); popMatrix(); }}

Main function

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import pbox2d.*;import org.jbox2d.collision.shapes.*;import org.jbox2d.common.*;import org.jbox2d.dynamics.*;

// A list for all of our rectanglesArrayList<Box> boxes;

PBox2D box2d;

void setup() { size(800, 200); smooth(); // Initialize and create the Box2D world box2d = new PBox2D(this); box2d.createWorld();

// Create ArrayLists boxes = new ArrayList<Box>();}

void draw() { background(255);

// We must always step through time! box2d.step();

// When the mouse is clicked, add a new Box object Box p = new Box(mouseX, mouseY); boxes.add(p);

// Display all the boxes for (Box b: boxes) { b.display(); }

}

Let’s put some boundaries.

To create boundaries, we need to set BodyDef object’s type to STATIC.

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class Boundary {

// A boundary is a simple rectangle with x,y,width,and height float x; float y; float w; float h; // But we also have to make a body for box2d to know about it Body b;

Boundary(float x_,float y_, float w_, float h_) { x = x_; y = y_; w = w_; h = h_; // following the steps //Step 1. Define the body. //Step 2. Create the body.(bodyType=STATIC) //Step 3. Define the shape. //(Step 4. Define the fixture) (Skip this step and use the default fixture!) // Step 5. Attach the shape to the body with the Fixture

}

// Draw the boundary, if it were at an angle we'd have to do something fancier void display() { fill(0); stroke(0); rectMode(CENTER); rect(x,y,w,h); }

}

Let’s design Boundary class

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// A reference to our box2d worldPBox2D box2d;

// A list we'll use to track fixed objectsArrayList<Boundary> boundaries;// A list for all of our rectanglesArrayList<Box> boxes;

void setup() { size(800, 200); smooth(); // Initialize and create the Box2D world box2d = new PBox2D(this); box2d.createWorld();

// Create ArrayLists boxes = new ArrayList<Box>(); boundaries = new ArrayList<Boundary>(); // Add a bunch of fixed boundaries boundaries.add(new Boundary(width/4,height-5,width/2-50,10)); boundaries.add(new Boundary(3*width/4,height-50,width/2-50,10));}

Let’s design main function

One problem of this example Computationally expensive as we

increase the number of boxes It is getting slow as it goes. How can we fix it?

If the box is out of screen, remove it!!

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void killBody() { box2d.destroyBody(body); //ask box2d to destroy the body }

// Is the particle ready for deletion? boolean done() { // Let's find the screen position of the particle Vec2 pos = box2d.getBodyPixelCoord(body); // Is it off the bottom of the screen? if (pos.y > height+h) { killBody(); return true; } return false; }

Let’s modify Box class and main function

for (int i = boxes.size()-1; i >= 0; i--) { Box b = boxes.get(i); if (b.done()) { boxes.remove(i); //remove from the array } }

BOX classDraw() in main function

Curvy Boundary If you want a fixed boundary that is

a curved surface (as opposed to a polygon)

Use the ChainShape

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polygonShape : Convex moving object chainShape : static game world

Steps Step 1: Define a body.

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BodyDef bd = new BodyDef();Body body = box2d.world.createBody(bd);

Step 2: Define the Shape.ChainShape chain = new ChainShape();

Step 3: Configure the Shape.Vec2[] vertices = new Vec2[2];vertices[0] = box2d.coordPixelsToWorld(0,150);vertices[1] = box2d.coordPixelsToWorld(width,150);chain.createChain(vertices, vertices.length);

Steps Step 4: Attach the Shape to the body with a

Fixture.

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FixtureDef fd = new FixtureDef();fd.shape = chain;fd.density = 1;fd.friction = 0.3;fd.restitution = 0.5;body.createFixture(fd);

Define a Surface

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class Surface { // We'll keep track of all of the surface points ArrayList<Vec2> surface;

Surface() { surface = new ArrayList<Vec2>(); // Here we keep track of the screen coordinates of the chain surface.add(new Vec2(0, height/2)); //surface.add(new Vec2(width/2, height/2+50)); surface.add(new Vec2(width, height/2));

// This is what box2d uses to put the surface in its world ChainShape chain = new ChainShape();

// We can add 3 vertices by making an array of 3 Vec2 objects Vec2[] vertices = new Vec2[surface.size()]; for (int i = 0; i < vertices.length; i++) { vertices[i] = box2d.coordPixelsToWorld(surface.get(i)); }

chain.createChain(vertices, vertices.length);

// The edge chain is now a body! BodyDef bd = new BodyDef(); Body body = box2d.world.createBody(bd); // Shortcut, we could define a fixture if we // want to specify frictions, restitution, etc. body.createFixture(chain, 1); }

How to draw it?

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void display() { strokeWeight(1); stroke(0); fill(0); beginShape(); for (int i=0; i<surface.size(); i++) { Vec2 v = surface.get(i); vertex(v.x,v.y); } vertex(width, height); vertex(0, height); endShape(CLOSE); }

(0,height) (width,height)

vertex(v.x,v.y); vertex(v.x,v.y);

Exercise Use the sin() to draw the mountains Use the mouse to create circles Use the CircleShape cs = new CircleShape() for

body

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Body type and parameters

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float r = random(4,8); CircleShape cs = new CircleShape(); cs.m_radius = box2d.scalarPixelsToWorld(r); FixtureDef fd = new FixtureDef(); fd.shape = cs; fd.density = 1; fd.friction = 0.01; fd.restitution = 0.3; . . . body.setLinearVelocity(new Vec2(random(-10f,10f),random(5f,10f)));

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Building Mountain : use the sin function

float angle = 0;float angleVel = 0.2;float amplitude = 100;size(400,200);background(255);smooth();stroke(0);strokeWeight(2);noFill();

beginShape();for (int x = 0; x <= width; x += 5) {

//map one value from the current range to another range float y = map(sin(angle),-1,1,0,height); vertex(x,y); angle +=angleVel;}endShape();

Complex Shapes How to create more complex

shapes?

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build a completely custom shape as a series of connected vertices

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Vec2[] vertices = new Vec2[4];// An array of 4 vectorsvertices[0] = box2d.vectorPixelsToWorld(new Vec2(-15, 25));vertices[1] = box2d.vectorPixelsToWorld(new Vec2(15, 0));vertices[2] = box2d.vectorPixelsToWorld(new Vec2(20, -15));vertices[3] = box2d.vectorPixelsToWorld(new Vec2(-10, -10));PolygonShape ps = new PolygonShape();ps.set(vertices, vertices.length);

1. Order of vertices! If you are thinking in terms of pixels (as above) the vertices should be defined in counterclockwise order.

2. Convex shapes only! A concave shape is one where the surface curves inward. Convex is the opposite

Complex shapes We can attach several shapes on a

single body

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BodyDef bd = new BodyDef();bd.type = BodyType.DYNAMIC;bd.position.set(box2d.coordPixelsToWorld(center));body = box2d.createBody(bd);

PolygonShape ps = new PolygonShape();float box2dW = box2d.scalarPixelsToWorld(w/2);float box2dH = box2d.scalarPixelsToWorld(h/2);sd.setAsBox(box2dW, box2dH);

CircleShape cs = new CircleShape();cs.m_radius = box2d.scalarPixelsToWorld(r);

body.createFixture(ps,1.0);body.createFixture(cs, 1.0);

Body center

Moving

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Vec2 offset = new Vec2(0,-h/2);//Converting the vector to Box2D worldoffset = box2d.vectorPixelsToWorld(offset);//Setting the local position of the circlecircle.m_p.set(offset.x,offset.y);

m_p : local center of shape

Setting shapes on body Setting proper shapes on the body is

really important for correct simulation. If not, the collision detection does not

work and simulation has errors.

39Overlapping each other !!

Exercise : Revise the old example

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void display() { Vec2 pos = box2d.getBodyPixelCoord(body); float a = body.getAngle(); rectMode(CENTER); pushMatrix(); translate(pos.x,pos.y); rotate(-a); fill(175); stroke(0); //First the rectangle at (0,0) rect(0,0,w,h); //Then the ellipse offset at (0,-h/2) ellipse(0,-h/2,r*2,r*2); popMatrix();}

Draw()

Box2D Joints Connecting one body to another

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Three different joints distance joints revolute joints “mouse” joints

Distant Joints A joint that connects two

bodies with a fixed length Steps

Step 1. Make sure you have two bodies ready to go.

Step 2. Define the joint. Step 3. Configure the joint’s properties

(What are the bodies? Where are the anchors? What is its rest length? Is it elastic or rigid?)

Step 4. Create the joint.

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Example Assume that we have a particle

body

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Particle p1 = new Particle();Particle p2 = new Particle();

DistanceJointDef djd = new DistanceJointDef();

djd.bodyA = p1.body;djd.bodyB = p2.body;

//if our rest length is in pixels, we need to convert it!djd.length = box2d.scalarPixelsToWorld(10);

Joint dj = (DistanceJoint) box2d.world.createJoint(djd);

Revolute Joints connects two Box2D

bodies at a common anchor point,

Steps Step 1. Make sure you have two

bodies ready to go. Step 2. Define the joint. Step 3. Configure the joint’s

properties (What are the bodies? Step 4. Create the joint.

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Example Assume that we have a particle

body

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Box box1 = new Box();Box box2 = new Box();

RevoluteJointDef rjd = new RevoluteJointDef();

//the anchor point (i.e. where they are connected)is set with the functioninitialize().

rjd.initialize(box1.body, box2.body, box1.body.getWorldCenter());

rjd.motorSpeed = PI*2; // how fast?rjd.maxMotorTorque = 1000.0; // how powerful?rjd.enableMotor = false; // is it on?

rjd.enableLimit = true;rjd.lowerAngle = -PI/8;rjd.upperAngle = PI/8;

Joint joint = box2d.world.createJoint(rjd);

Exercise: Let’s making a windmill

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class Windmill {

RevoluteJoint joint; Box box1; Box box2;

Windmill(float x, float y) {

// Initialize locations of two boxes box1 = new Box(x, y-20, 120, 10, false); box2 = new Box(x, y, 10, 40, true);

// Define joint as between two bodies RevoluteJointDef rjd = new RevoluteJointDef(); rjd.initialize(box1.body, box2.body, box1.body.getWorldCenter());

// Turning on a motor (optional) rjd.motorSpeed = PI*2; // how fast? rjd.maxMotorTorque = 1000.0; // how powerful? rjd.enableMotor = false; // is it on?

// Create the joint rjd.enableLimit = true; rjd.lowerAngle = -PI/8; rjd.upperAngle = PI/8; }

void display() { box2.display(); box1.display();

// Draw anchor just for debug Vec2 anchor = box2d.coordWorldToPixels(box1.body.getWorldCenter()); fill(0); noStroke(); ellipse(anchor.x, anchor.y, 8, 8); }}

Create a ball whenever you click the mouse!

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Exercise: main functionimport pbox2d.*;import org.jbox2d.common.*;import org.jbox2d.dynamics.joints.*;import org.jbox2d.collision.shapes.*;import org.jbox2d.collision.shapes.Shape;import org.jbox2d.common.*;import org.jbox2d.dynamics.*;import org.jbox2d.dynamics.contacts.*;

// A reference to our box2d worldPBox2D box2d;

// An object to describe a Windmill (two bodies and one joint)Windmill windmill;

// An ArrayList of particles that will fall on the surfaceArrayList<Particle> particles;

void setup() { size(800,200); smooth();

// Initialize box2d physics and create the world box2d = new PBox2D(this); box2d.createWorld();

// Make the windmill at an x,y location windmill = new Windmill(width/2,175);

// Create the empty list particles = new ArrayList<Particle>();

}

void mousePressed(){ float sz = random(4,8); //radius particles.add(new Particle(mouseX,mouseY,sz));}

Create a particle class!!Class Particle{

//following the steps for creating body and shape

}

Applying Forces When we want to apply any forces other than

gravity, call applyForce() function

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class Box { Body body;

void applyForce(Vec2 force) { Vec2 pos = body.getWorldCenter(); //Calling the Body's applyForce() function body.applyForce(force, pos); }

}

Our earlier forces examples assumed that the force was always applied at the mover’s center. Here we get to specify exactly where on the body the force is applied

Collision Detection How do we know when two objects

collide? beginContact() is triggered when two

shapes collide

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void setup() { box2d = new PBox2D(this); box2d.createWorld(); Add this line if you want to listen for collisions. box2d.listenForCollisions();}

void beginContact(Contact cp) { println("Something collided in the Box2D World!");}

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Collision Call-back functions1. beginContact() —Triggered whenever two shapes first

come into contact with each other.2. endContact() —Triggered over and over again as long as

shapes continue to be in contact.3. preSolve() —Triggered before Box2D solves the outcome

of the collision, i.e. before beginContact(). It can be used to disable a collision if necessary.

4. postSolve() —Triggered after the outcome of the collision is solved. It allows you to gather information about that “solution” (known as an “impulse”).

endContact() works identically to beginContact(), the only difference being that it occurs the moment bodies separate.

Set-up steps : Given a particle class

Step 1: Contact, could you tell me what two things collided? Box2D detects collisions between shapes, but shapes

are attached by fixtures. So, we need to find two fixtures collided

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Fixture f1 = cp.getFixtureA();Fixture f2 = cp.getFixtureB();

Step 2: Fixtures, could you tell me which body you are attached to?

Body b1 = f1.getBody();Body b2 = f2.getBody();

Set-up steps : Given a particle class

Step 3: Bodies, could you tell me which Particles you are associated with? How can the body know about the particle class? Box2D provides a function that allows us to attach our

Processingobject (a Particle) to a Box2D body via the setUserData() and getUserData() methods.

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class Particle { Body body; Particle(float x, float y, float r) { BodyDef bd = new BodyDef(); bd.position = box2d.coordPixelsToWorld(x, y); bd.type = BodyType.DYNAMIC; body = box2d.createBody(bd); CircleShape cs = new CircleShape(); cs.m_radius = box2d.scalarPixelsToWorld(r); body.createFixture(fd,1);

body.setUserData(this);}

beginContact() call-back function

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void beginContact(Contact cp) { Fixture f1 = cp.getFixtureA(); Fixture f2 = cp.getFixtureB(); Body b1 = f1.getBody(); Body b2 = f2.getBody();

// When we pull the “user data” object out of the Body object, we have to remind our program // that it is a Particle object. Box2D doesn’t know this.

Particle p1 = (Particle) b1.getUserData(); Particle p2 = (Particle) b2.getUserData();

//Once we have the particles, we can do anything to them. Here we just call a function that // changes their color.

p1.change(); p2.change();}

void change() { col = color(255, 0, 0); }

What if the colliding objects are not particle object? What if the colliding objects are

boundary, box or any other class? Use the query function

You cannot create or destroy Box2D entities inside of beginContact(), endContact()

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Object o1 = b1.getUserData();//Asking that object if it’s a Particleif (o1.getClass() == Particle.class) { Particle p = (Particle) o1; p.change();}

Integration & Differentiation Integration(적분 ), Differentiation(미분 ) Differentiation : The derivative of a function is a

measure of how a function changes over time. Velocity can be described as the “derivative” of

location Acceleration is the “derivative” of velocity.

Integration : inverse of differentiation, How can we update location from given velocity? Location is the integral of velocity Velocity is the integral of acceleration

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Euler integration The simplest form of integration and very easy

to implement in our code

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velocity.add(acceleration);location.add(velocity);

But, not accurate! : Real world is continuous, but our world is discrete

Real world is curveBut, our world is line segments

Better integration methods semi-explicit Euler : Box2D use

http://en.wikipedia.org/wiki/Symplectic_Euler_method

Runge-Kutta : other game engine Verlet integration

Do not store velocity, we always know the all past locations and current locations.

Library toxiclibs(http://toxiclibs.org/)58

Toxiclib

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Verlet Physics with toxiclibs toxiclibs includes a suite of other wonderful packages

that help with audio, color, geometry, and more designed specifically for use with Processing

coordinate system that we’ll use for the physics engine is the coordinate system of Processing

all of the physics simulations and functions work in both two and three dimensions

Does not have collision detection Best work if you have lots of particles flying around the

screen. Sometimes they attract each other. Sometimes they repel each other. And sometimes they are connected with springs. High performance due to the Verlet integration algorithm

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Comparison

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Feature Box2D toxiclibs Verlet-Physics

Collision geometry Yes No3D physics No YesParticle attraction /repulsion forces

No Yes

Spring connections Yes YesOther connections: revolute,pulley, gear, pris-matic

Yes No

Motors Yes NoFriction Yes No

VerletPhysics toxiclibs (http://toxiclibs.org/) core elements

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Box2D toxiclibs VerletPhysicsWorld VerletPhysicsBody VerletParticleShape Nothing! toxiclibs does not handle

shape geometryFixture Nothing! toxiclibs does not handle

shape geometryJoint VerletSpring

A new vector class for Vec2D

(http://toxiclibs.org/docs/core/toxi/geom/Vec2D.html) Vec3D

(http://toxiclibs.org/docs/core/toxi/geom/Vec3D.html)

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PVector Vec2DPVector a = new PVector(1,-1);PVector b = new PVector(3,4);a.add(b);

Vec2D a = new Vec2D(1,-1);Vec2D b = new Vec2D(3,4);a.addSelf(b);

PVector a = new PVector(1,-1);PVector b = new PVector(3,4);PVector c = PVector.add(a,b);

Vec2D a = new Vec2D(1,-1);Vec2D b = new Vec2D(3,4);Vec2D c = a.add(b);

PVector a = new PVector(1,-1);float m = a.mag();a.normalize();

Vec2D a = new Vec2D(1,-1);float m = a.magnitude();a.normalize();

Basic setup

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import toxi.physics2d.*;import toxi.physics2d.behaviors.*;import toxi.geom.*;

VerletPhysics2D physics;void setup() { //Creating a toxiclibs Verlet physics world physics=new VerletPhysics2D();

//put hard boundaries physics.setWorldBounds(new Rect(0,0,width,height));

//set the gravity physics.addBehavior(new GravityBehavior(new Vec2D(0,0.5)));}

void draw() { // This is the same as Box2D’s “step()” function physics.update();}

Example codesI will give several example source codes that use the toxiclib.

If you are interested in this code, take a look at those codes and use for your project

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