Programming Fundamentals  
Lecture 05 Vectors, Physics and Collision  
Detection  
Edirlei Soares de Lima  
<edirlei.lima@universidadeeuropeia.pt>  
What is a Vector?  
The word vector can mean a lot of different things…  
Biology/epidemiology: an organism that transmits infection from one host  
to another;  
Computer science: a one-dimensional array;  
Mathematics and physics: an entity that has both magnitude and  
direction.  
A vector is typically represented as an arrow, where the direction  
is indicated by where the arrow is pointing, and the magnitude  
by the length of the arrow itself.  
Bouncing Ball Without Vectors  
local x = 100  
local y = 100  
local xspeed = 300  
local yspeed = 500  
local radius = 30  
function love.update(dt)  
x = x + (xspeed * dt)  
y = y + (yspeed * dt)  
if (x > love.graphics.getWidth() - radius) or (x < radius) then  
xspeed = xspeed * -1;  
end  
if (y > love.graphics.getHeight() - radius) or (y < radius) then  
yspeed = yspeed * -1;  
end  
end  
function love.draw()  
love.graphics.circle("fill", x, y, radius, 30)  
end  
Bouncing Ball Without Vectors  
In the code, the ball has some properties that are represented  
by variables:  
Location  
Velocity  
x and y  
xspeed and yspeed  
Both location and velocity can be described as vectors:  
0
, 0  
Defining a Vector  
To better represent vectors in Lua, we can create a table to  
store both x and y:  
vector2 = {}  
vector2.lua  
function vector2.new(px, py)  
return {x = px, y = py}  
end  
We can save it in a separated module (file) and handle it as a  
new data type.  
What is a Table?  
A Lua table is defined by a set of pairs key-data, where the data  
is referenced by the key.  
The key (index) can be of any type of data (number, text, …);  
The data can also be of any type of data (number, text, function, image);  
With Lua tables we can associate names to its elements and  
create structures to store related data:  
player1 = {  
name = "John",  
score = 1000,  
lives = 3  
}
We can access an element of a table by its name:  
io.write(player1.score)  
Vector Addition  
Now that we have two vector objects (location and velocity),  
we are ready to implement the algorithm for motion:  
position = position + velocity  
Code without vectors:  
x = x + (xspeed * dt)  
y = y + (yspeed * dt)  
However, we are not allowed to do that directly, we need a  
function to add two vectors:  
=
function vector2.add(vec, inc)  
vector2.lua  
local result = vector2.new(0, 0)  
result.x = vec.x + inc.x  
result.y = vec.y + inc.y  
return result  
end  
Bouncing Ball With Vectors  
require "vector2"  
main.lua  
local position = vector2.new(100, 100)  
local velocity = vector2.new(300, 500)  
local radius = 30  
function love.update(dt)  
position = vector2.add(position, vector2.mult(velocity, dt))  
if (position.x > love.graphics.getWidth() - radius) or  
(
position.x < radius) then  
velocity.x = velocity.x * -1;  
end  
if (position.y > love.graphics.getHeight() - radius) or  
position.y < radius) then  
velocity.y = velocity.y * -1;  
end  
end  
(
function love.draw()  
love.graphics.circle("fill", position.x, position.y, radius, 30)  
end  
Vector Subtraction  
=
Implementation:  
function vector2.sub(vec, dec)  
local result = vector2.new(0, 0)  
result.x = vec.x - dec.x  
result.y = vec.y - dec.y  
return result  
vector2.lua  
end  
Vector Multiplication and Division  
=
Multiplication implementation:  
function vector2.mult(vec, n)  
local result = vector2.new(0, 0)  
result.x = vec.x * n  
result.y = vec.y * n  
return result  
vector2.lua  
end  
Division implementation:  
function vector2.div(vec, n)  
local result = vector2.new(0, 0)  
result.x = vec.x / n  
result.y = vec.y / n  
return result  
vector2.lua  
end  
Vector Magnitude  
Implementation:  
function vector2.magnitude(vec)  
return math.sqrt(vec.x * vec.x + vec.y * vec.y)  
end  
vector2.lua  
Vector Normalization  
Implementation:  
function vector2.normalize(vec)  
local m = vector2.magnitude(vec)  
vector2.lua  
if m ~= 0 then  
return vector2.div(vec, m)  
end  
return vec  
end  
Vector Motion: Acceleration  
Basic definitions:  
Velocity: the rate of change of location;  
Acceleration: the rate of change of velocity;  
Acceleration affects velocity, which in turn affects location.  
velocity = vector2.add(velocity, vector2.mult(acceleration, dt))  
position = vector2.add(position, vector2.mult(velocity, dt))  
Acceleration values accumulate over time in the velocity,  
therefore, a function to limit the velocity is required:  
function vector2.limit(vec, max)  
local result = vec  
vector2.lua  
if (vector2.magnitude(vec) > max) then  
result = vector2.normalize(vec)  
result = vector2.mult(result, max)  
end  
return result  
end  
Example 1: Constant Acceleration  
require "vector2"  
main.lua  
local position = vector2.new(400, 100)  
local velocity = vector2.new(0, 0)  
local radius = 30  
local maxspeed = 500  
function love.update(dt)  
local acceleration = vector2.new(-10,100)  
velocity = vector2.add(velocity, vector2.mult(acceleration, dt))  
velocity = vector2.limit(velocity, maxspeed)  
position = vector2.add(position, vector2.mult(velocity, dt))  
if (position.x > love.graphics.getWidth()) then  
position.x = 0  
elseif (position.x < 0) then  
position.x = love.graphics.getWidth()  
end  
...  
Example 1: Constant Acceleration  
..  
.
main.lua  
if (position.y > love.graphics.getHeight()) then  
position.y = 0  
elseif (position.y < 0) then  
position.y = love.graphics.getHeight()  
end  
end  
function love.draw()  
love.graphics.circle("fill", position.x, position.y, radius, 30)  
end  
Example 2: Acceleration Towards the Mouse  
require "vector2"  
main.lua  
local position = vector2.new(400, 100)  
local velocity = vector2.new(0, 0)  
local radius = 30  
local maxspeed = 250  
function love.update(dt)  
local mouseposition = vector2.new(love.mouse.getX(),love.mouse.getY())  
local mousedirection = vector2.sub(mouseposition, position)  
mousedirection = vector2.normalize(mousedirection)  
local acceleration = vector2.mult(mousedirection, 700)  
velocity = vector2.add(velocity, vector2.mult(acceleration, dt))  
velocity = vector2.limit(velocity, maxspeed)  
position = vector2.add(position, vector2.mult(velocity, dt))  
end  
function love.draw()  
love.graphics.circle("fill", position.x, position.y, radius, 30)  
end  
Forces  
A force is a vector that causes an object with mass to accelerate.  
Newton’s Laws of Motion:  
1
2
3
. An object at rest stays at rest and an object in motion stays in motion.  
. Force equals mass times acceleration.  
. For every action there is an equal and opposite reaction.  
Apply Force  
Implementation:  
vector2.lua  
function vector2.applyForce(force, mass, acceleration)  
local f = vector2.div(force, mass)  
return vector2.add(acceleration, f)  
end  
.
..  
local mass = 1  
..  
main.lua  
.
Example: Gravity and Wind Forces  
require "vector2"  
main.lua  
local position = vector2.new(400, 100)  
local velocity = vector2.new(0, 0)  
local radius = 30  
local maxspeed = 5  
local mass = 1  
local wind = vector2.new(100,0)  
local gravity = vector2.new(0, 500)  
function love.update(dt)  
local acceleration = vector2.new(0, 0)  
acceleration = vector2.applyForce(wind, mass, acceleration)  
acceleration = vector2.applyForce(gravity, mass, acceleration)  
velocity = vector2.add(velocity, vector2.mult(acceleration, dt))  
position = vector2.add(position, vector2.mult(velocity, dt))  
...  
Example: Gravity and Wind Forces  
main.lua  
...  
if (position.x > love.graphics.getWidth() - radius) then  
velocity.x = velocity.x * -1  
elseif (position.x < radius) then  
velocity.x = velocity.x * -1  
end  
if (position.y > love.graphics.getHeight() - radius) then  
velocity.y = velocity.y * -1  
end  
end  
function love.draw()  
love.graphics.circle("fill", position.x, position.y, radius, 30)  
end  
Exercise 1  
1
) Write the code to accelerate a spaceship towards the mouse  
position.  
You need to use the applyForcefunction  
to move the spaceship.  
Hint:  
o First, you need to calculate a vector that points  
from the spaceship to the mouse location  
(direction).  
The spaceship must rotate towards the  
direction of its movement.  
Hint:  
o Use the math.atan2 function to calculate the  
arc tangent value from the y and x variables of  
the direction of the velocity.  
Friction  
Friction is a dissipative force, which is a type of force where the  
total energy of the system decreases when an object is in  
motion.  
= coefficient of friction (strength of a friction force for a particular surface)  
N = normal force (the force perpendicular to the object’s motion along a surface)  
= velocity unit vector  
Example: Friction Force  
require "vector2"  
main.lua  
local position = vector2.new(400, 100)  
local velocity = vector2.new(0, 0)  
local radius = 30  
local maxspeed = 5  
local mass = 1  
local frictioncoefficient = 100  
local wind = vector2.new(100,0)  
local gravity = vector2.new(0, 500)  
function love.update(dt)  
local friction = vector2.mult(velocity, -1)  
friction = vector2.normalize(friction)  
friction = vector2.mult(friction, frictioncoefficient)  
local acceleration = vector2.new(0, 0)  
acceleration = vector2.applyForce(friction, mass, acceleration)  
acceleration = vector2.applyForce(wind, mass, acceleration)  
acceleration = vector2.applyForce(gravity, mass, acceleration)  
...  
Example: Friction Force  
...  
main.lua  
velocity = vector2.add(velocity, vector2.mult(acceleration, dt))  
position = vector2.add(position, vector2.mult(velocity, dt))  
if (position.x > love.graphics.getWidth() - radius) then  
position.x = love.graphics.getWidth() - radius  
velocity.x = velocity.x * -1  
elseif (position.x < radius) then  
position.x = radius  
velocity.x = velocity.x * -1  
end  
if (position.y > love.graphics.getHeight() - radius) then  
position.y = love.graphics.getHeight() - radius  
velocity.y = velocity.y * -1  
end  
end  
function love.draw()  
love.graphics.circle("fill", position.x, position.y, radius, 30)  
end  
Example: Player Movement and Jump  
require "vector2"  
main.lua  
local position = vector2.new(400, 100)  
local velocity = vector2.new(0, 0)  
local playerwidth = 30  
local playerheight = 60  
local maxspeed = 400  
local mass = 1  
local frictioncoefficient = 300  
local onGround = false  
function love.update(dt)  
local acceleration = vector2.new(0, 0)  
local gravity = vector2.new(0, 500)  
acceleration = vector2.applyForce(gravity, mass, acceleration)  
if (onGround) then  
local friction = vector2.mult(velocity, -1)  
friction = vector2.normalize(friction)  
friction = vector2.mult(friction, frictioncoefficient)  
acceleration = vector2.applyForce(friction, mass, acceleration)  
...  
Example: Player Movement and Jump  
...  
main.lua  
if love.keyboard.isDown("right") then  
local move = vector2.new(500, 0)  
acceleration = vector2.applyForce(move, mass, acceleration)  
end  
if love.keyboard.isDown("left") then  
local move = vector2.new(-500, 0)  
acceleration = vector2.applyForce(move, mass, acceleration)  
end  
if love.keyboard.isDown("up") then  
velocity.y = -450  
onGround = false  
end  
end  
velocity = vector2.add(velocity, vector2.mult(acceleration, dt))  
velocity = vector2.limit(velocity, maxspeed)  
position = vector2.add(position, vector2.mult(velocity, dt))  
...  
Example: Player Movement and Jump  
...  
main.lua  
if (position.x > love.graphics.getWidth() - playerwidth) then  
position.x = love.graphics.getWidth() - playerwidth  
velocity.x = (velocity.x * -1) / 2  
elseif (position.x < 0) then  
position.x = 0  
velocity.x = (velocity.x * -1) / 2  
end  
if (position.y > love.graphics.getHeight() - playerheight) then  
position.y = love.graphics.getHeight() - playerheight  
velocity.y = 0  
onGround = true  
end  
end  
function love.draw()  
love.graphics.rectangle("fill", position.x, position.y,  
playerwidth, playerheight)  
end  
Exercise 2  
2
) Modify the code of the player movement and jump example  
to allow the player to double jump.  
The player must be able to jump only two  
times.  
After double jumping, a new jump can only  
be performed when the player touches the  
ground.  
Hint:  
o Use the callback love.keyreleased(key,  
scancode)to detect when the player releases  
the jump key, which indicates that he/she can  
jump again (if it was the first jump).  
Dot Product  
ꢂ ∙ =,, =+ꢅ  
The dot product has several applications in game programming.  
Example: sight/view angle  
Example: View Angle and Modularization  
require "vector2"  
player.lua  
local playerposition = vector2.new(400, 300)  
local playervelocity = vector2.new(0, 0)  
local playerrotation = 0  
local playermaxspeed = 100  
local playermass = 1  
local playerimage  
function LoadPlayer()  
playerimage = love.graphics.newImage("spaceship.png")  
end  
function UpdatePlayer(dt)  
local acceleration = vector2.new(0, 0)  
local mouseposition = vector2.new(love.mouse.getX(),  
love.mouse.getY())  
local mousedirection = vector2.sub(mouseposition, playerposition)  
local velocitydirection = vector2.normalize(playervelocity)  
mousedirection = vector2.normalize(mousedirection)  
...  
Example: View Angle and Modularization  
..  
.
player.lua  
playerrotation = math.atan2(mousedirection.y, mousedirection.x)  
if (love.mouse.isDown(1)) then  
local engineForce = vector2.mult(mousedirection, 100)  
acceleration = vector2.applyForce(engineForce, playermass,  
acceleration)  
end  
playervelocity = vector2.add(playervelocity,  
vector2.mult(acceleration, dt))  
playervelocity = vector2.limit(playervelocity, playermaxspeed)  
playerposition = vector2.add(playerposition,  
vector2.mult(playervelocity, dt))  
end  
function DrawPlayer()  
love.graphics.draw(playerimage, playerposition.x, playerposition.y,  
playerrotation, 0.3, 0.3, playerimage:getWidth() / 2,  
playerimage:getHeight() / 2)  
end  
function GetPlayerPosition()  
return playerposition  
end  
Example: View Angle and Modularization  
require "vector2"  
enemy.lua  
local enemyposition = vector2.new(700, 500)  
local enemyvelocity = vector2.new(0, 0)  
local enemydirection = vector2.new(-1, 0)  
local enemyrotation = math.atan2(enemydirection.y, enemydirection.x)  
local enemymaxspeed = 100  
local enemymass = 1  
local enemyviewangle = 30  
local enemychasing = false  
local enemyimage  
function LoadEnemy()  
enemyimage = love.graphics.newImage("spaceship.png")  
end  
function UpdateEnemy(dt, pposition)  
local acceleration = vector2.new(0, 0)  
if (CanSee(pposition)) then  
enemychasing = true  
end  
...  
Example: View Angle and Modularization  
..  
.
player.lua  
if (enemychasing) then  
local pdirection = vector2.normalize(vector2.sub(pposition,  
enemyposition))  
local engineForce = vector2.mult(pdirection, 100)  
acceleration = vector2.applyForce(engineForce, enemymass,  
acceleration)  
enemyrotation = math.atan2(enemyvelocity.y, enemyvelocity.x)  
end  
enemyvelocity = vector2.add(enemyvelocity,  
vector2.mult(acceleration, dt))  
enemyvelocity = vector2.limit(enemyvelocity, enemymaxspeed)  
enemyposition = vector2.add(enemyposition,  
vector2.mult(enemyvelocity, dt))  
end  
function DrawEnemy()  
love.graphics.draw(enemyimage, enemyposition.x, enemyposition.y,  
enemyrotation, 0.3, 0.3, enemyimage:getWidth() / 2,  
enemyimage:getHeight() / 2)  
end  
...  
Example: View Angle and Modularization  
..  
.
player.lua  
function CanSee(pposition)  
local pdirection = vector2.normalize(vector2.sub(pposition,  
enemyposition))  
local angle = math.acos(vector2.dot(enemydirection, pdirection))  
if (math.deg(angle) < 30) then  
return true  
end  
return false  
end  
Example: View Angle and Modularization  
require "player"  
require "enemy"  
main.lua  
function love.load()  
LoadPlayer()  
LoadEnemy()  
end  
function love.update(dt)  
UpdatePlayer(dt)  
UpdateEnemy(dt, GetPlayerPosition())  
end  
function love.draw()  
DrawPlayer()  
DrawEnemy()  
end  
Collision Detection  
No Collision  
Collision  
Box2  
Box2  
Box1  
Box1  
Collision Detection  
No Collision  
Collision  
Circle2  
Circle2  
Circle1  
Circle1  
Collision Detection (Box)  
function love.load()  
player1 = {  
x = 390,  
y = 300,  
width = 50,  
height = 50,  
collided = false  
}
box1 = {  
x = 100,  
y = 300,  
width = 50,  
height = 50  
}
box2 = {  
x = 650,  
y = 275,  
width = 100,  
height = 100  
}
end  
function CheckBoxCollision(x1,y1,w1,h1,x2,y2,w2,h2)  
return x1 < x2+w2 and x2 < x1+w1 and y1 < y2+h2 and y2 < y1+h1  
end  
function love.update(dt)  
if love.keyboard.isDown("left") then  
player1.x = player1.x - (120 * dt)  
end  
if love.keyboard.isDown("right") then  
player1.x = player1.x + (120 * dt)  
end  
if love.keyboard.isDown("up") then  
player1.y = player1.y - (120 * dt)  
end  
if love.keyboard.isDown("down") then  
player1.y = player1.y + (120 * dt)  
end  
if CheckBoxCollision(player1.x, player1.y, player1.width,  
player1.height, box1.x, box1.y, box1.width, box1.height) or  
CheckBoxCollision(player1.x, player1.y, player1.width,  
player1.height, box2.x, box2.y, box2.width, box2.height) then  
player1.collided = true  
else  
player1.collided = false  
end  
end  
function love.draw()  
love.graphics.setColor(255,255,255)  
love.graphics.rectangle("fill", box1.x, box1.y,  
box1.width, box1.height)  
love.graphics.rectangle("fill", box2.x, box2.y,  
box2.width, box2.height)  
if player1.collided == true then  
love.graphics.setColor(255,0,0)  
end  
love.graphics.rectangle("fill", player1.x, player1.y,  
player1.width, player1.height)  
end  
Collision Detection (Circle)  
function love.load()  
player1 = {  
x = 390,  
y = 300,  
radius = 25,  
collided = false  
}
circle1 = {  
x = 100,  
y = 300,  
radius = 25  
}
circle2 = {  
x = 650,  
y = 300,  
radius = 50  
}
end  
function CheckCircularCollision(ax, ay, ar, bx, by, br)  
local dx = bx - ax  
local dy = by - ay  
local dist = math.sqrt(dx * dx + dy * dy)  
return dist < ar + br  
end  
function love.update(dt)  
if love.keyboard.isDown("left") then  
player1.x = player1.x - (120 * dt)  
end  
if love.keyboard.isDown("right") then  
player1.x = player1.x + (120 * dt)  
end  
if love.keyboard.isDown("up") then  
player1.y = player1.y - (120 * dt)  
end  
if love.keyboard.isDown("down") then  
player1.y = player1.y + (120 * dt)  
end  
if CheckCircularCollision(player1.x, player1.y, player1.radius,  
circle1.x, circle1.y, circle1.radius) or  
CheckCircularCollision(player1.x, player1.y, player1.radius,  
circle2.x, circle2.y, circle2.radius) then  
player1.collided = true  
else  
player1.collided = false  
end  
end  
function love.draw()  
love.graphics.setColor(255,255,255)  
love.graphics.circle("fill",circle1.x, circle1.y, circle1.radius, 100)  
love.graphics.circle("fill",circle2.x, circle2.y, circle2.radius, 100)  
if player1.collided == true then  
love.graphics.setColor(255,0,0)  
end  
love.graphics.circle("fill",player1.x, player1.y, player1.radius, 100)  
end  
Example 1: Collision Detection with Physics  
Example 1: Collision Detection with Physics  
require "vector2"  
main.lua  
local position = vector2.new(400, 100)  
local velocity = vector2.new(0, 0)  
local playersize = vector2.new(30, 60)  
local maxspeed = 400  
local mass = 1  
local frictioncoefficient = 300  
local onGround = false  
local groundposition = vector2.new(50, 550)  
local groundsize = vector2.new(700, 50)  
local box1position = vector2.new(500, 450)  
local box1size = vector2.new(50, 50)  
function love.update(dt)  
local acceleration = vector2.new(0, 0)  
local gravity = vector2.new(0, 500)  
acceleration = vector2.applyForce(gravity, mass, acceleration)  
...  
Example 1: Collision Detection with Physics  
local friction = vector2.mult(velocity, -1)  
friction = vector2.normalize(friction)  
main.lua  
friction = vector2.mult(friction, frictioncoefficient)  
acceleration = vector2.applyForce(friction, mass, acceleration)  
local movedirection = vector2.new(0, -1)  
if love.keyboard.isDown("right") then  
local move = vector2.new(500, 0)  
acceleration = vector2.applyForce(move, mass, acceleration)  
movedirection.x = 1  
end  
if love.keyboard.isDown("left") then  
local move = vector2.new(-500, 0)  
acceleration = vector2.applyForce(move, mass, acceleration)  
movedirection.x = -1  
end  
if love.keyboard.isDown("up") and (onGround) then  
velocity.y = -450  
movedirection.y = 1  
onGround = false  
end  
Example 1: Collision Detection with Physics  
local futurevelocity = vector2.add(velocity,  
vector2.mult(acceleration, dt))  
futurevelocity = vector2.limit(futurevelocity, maxspeed)  
local futureposition = vector2.add(position,  
vector2.mult(futurevelocity, dt))  
local collisiondir1 = GetBoxCollisionDirection(futureposition.x,  
futureposition.y, playersize.x, playersize.y,  
groundposition.x, groundposition.y, groundsize.x,  
groundsize.y)  
if not (collisiondir1.x == 0 and collisiondir1.y == 0) then  
if collisiondir1.y == movedirection.y then --down collision  
velocity.y = 0  
acceleration.y = 0  
onGround = true  
elseif collisiondir1.y == 1 then --up collision  
velocity.y = 0  
acceleration.y = 0  
elseif movedirection.x ~= collisiondir1.x then --side collision  
velocity.x = 0  
acceleration.x = 0  
end  
end  
Example 1: Collision Detection with Physics  
local collisiondir2 = GetBoxCollisionDirection(futureposition.x,  
futureposition.y, playersize.x, playersize.y,  
box1position.x, box1position.y, box1size.x,  
box1size.y)  
if not (collisiondir2.x == 0 and collisiondir2.y == 0) then  
if collisiondir2.y == movedirection.y then --ground collision  
velocity.y = 0  
acceleration.y = 0  
onGround = true  
elseif collisiondir2.y == 1 then --up collision  
velocity.y = 0  
acceleration.y = 0  
elseif movedirection.x ~= collisiondir2.x then --side collision  
velocity.x = 0  
acceleration.x = 0  
end  
end  
velocity = vector2.add(velocity, vector2.mult(acceleration, dt))  
velocity = vector2.limit(velocity, maxspeed)  
position = vector2.add(position, vector2.mult(velocity, dt))  
end  
Example 1: Collision Detection with Physics  
function GetBoxCollisionDirection(x1,y1,w1,h1,x2,y2,w2,h2)  
local xdist = math.abs((x1 + (w1 / 2)) - (x2 + (w2 / 2)))  
local ydist = math.abs((y1 + (h1 / 2)) - (y2 + (h2 / 2)))  
local combinedwidth = (w1 / 2) + (w2 / 2)  
local combinedheight = (h1 / 2) + (h2 / 2)  
if(xdist > combinedwidth) then  
return vector2.new(0, 0)  
end  
if(ydist > combinedheight) then  
return vector2.new(0, 0)  
end  
local overlapx = math.abs(xdist - combinedwidth)  
local overlapy = math.abs(ydist - combinedheight)  
local playerdir = vector2.normalize(vector2.sub(vector2.new(x1,y1),  
vector2.new(x2,y2)))  
local collisiondir  
if overlapx > overlapy then  
collisiondir = vector2.normalize(vector2.new(0, playerdir.y *  
overlapy))  
elseif overlapx < overlapy then  
collisiondir = vector2.normalize(vector2.new(playerdir.x *  
overlapx, 0))  
Example 1: Collision Detection with Physics  
else  
collisiondir = vector2.normalize(vector2.new(playerdir.x *  
overlapx, playerdir.y * overlapy))  
end  
return collisiondir  
end  
function love.draw()  
love.graphics.setColor(0.2, 0.2, 0.9)  
love.graphics.rectangle("fill", position.x, position.y,  
playersize.x, playersize.y)  
love.graphics.setColor(0.2, 0.8, 0.2)  
love.graphics.rectangle("fill", groundposition.x, groundposition.y,  
groundsize.x, groundsize.y)  
love.graphics.rectangle("fill", box1position.x, box1position.y,  
box1size.x, box1size.y)  
end  
Exercise 3  
3
) Using the code of the collision detection with physics example,  
implement a scene with at least four platforms (like the one  
illustrated bellow).  
Avoid code repetitions by using an  
array to store the platforms.