IntroductionThe following tutorial attempts to tackle some of the challenges of writing a collision system for simple games.
SimulationWe begin with "unconstrained" motion based on simple Newtonian physics. Each object has a velocity vector that is added to its position using the formula:
velocity = velocity + gravity*dt position = position + velocity*dtThese calculations are performed during each step of the simulation for every moving (dynamic) object. For practical reasons, it's useful to limit the velocity of moving objects to a certain threshold (maxVelocity). It's also common in games to have non-moving or "static" objects. Static objects are not affected by gravity or collisions and can be used to represent platforms, walls and so on.
-- updates the simulation function update(dt) -- update velocity vectors local grav = gravity*dt for i = 1, #dynamics do local d = dynamics[i] -- gravity d.yv = d.yv + grav d.xv, d.yv = clampVec(d.xv, d.yv, maxVelocity) end -- move dynamic objects for i = 1, #dynamics do local d = dynamics[i] moveShape(d, d.xv*dt, d.yv*dt) -- check and resolve collisions for j, s in ipairs(statics) do checkCollision(d, s) end for j = i + 1, #dynamics do checkCollision(d, dynamics[j]) end end endThe technique can be summed up as:
1. Update the velocities and positions of all moving objects
2. Iterate all pairs of intersecting objects
a) Move one of the objects (so the two no longer intersect)
b) Adjust the velocities of both objects
DetectionFirst, we want to check if two objects are intersecting. We'll be looking at axis-aligned rectangles in particular since they are simple and sufficient for most old-school games. One approach is to represent each rectangle shape as a center position with half-width/height extents.
Axis-aligned rectangle with position and extents
This way, we can simply compare the distance between the centers of two rectangles against the sum of their extents. The following algorithm checks the X and Y axis separately:
function testRectRect(a, b) -- distance between the rects local dx, dy = a.x - b.x, a.y - b.y local adx = math.abs(dx) local ady = math.abs(dy) -- sum of the extents local shw, shh = a.hw + b.hw, a.hh + b.hh if adx >= shw or ady >= shh then -- no intersection return end -- intersection
SeparationSuppose that the previous algorithm detects an intersection between two rectangles.
Intersection between two rectangles
Next, we have to figure out how much the first rectangle (a) needs to be displaced (sx, sy) so that the two shapes are no longer intersecting. The result is sometimes called the "shortest separation" vector.
-- shortest separation local sx, sy = shw - adx, shh - ady -- ignore longer axis if sx < sy then if sx > 0 then sy = 0 end else if sy > 0 then sx = 0 end end -- correct sign if dx < 0 then sx = -sx end if dy < 0 then sy = -sy end return sx, sy endThe "separation" vector gives us two important pieces of information about the intersecting shapes:
1. magnitude (penetration depth)
2. direction (penetration axis or the "collision normal")
-- checks for collision function checkCollision(a, b) local sx, sy = testRectRect(a, b) if sx and sy then solveCollision(a, b, sx, sy) end end -- resolves collision function solveCollision(a, b, sx, sy) -- find the collision normal local d = math.sqrt(sx*sx + sy*sy) local nx, ny = sx/d, sy/d -- relative velocity local vx, vy = a.xv - (b.xv or 0), a.yv - (b.yv or 0) -- penetration speed local ps = vx*nx + vy*ny -- objects moving towards each other? if ps <= 0 then -- separate the two objects moveShape(a, sx, sy) end end
ResponseGenerally speaking, "realistic" physics simulation is a difficult problem. So a lot of games have some collision response (like pushable blocks) that although not physically accurate, make a fun addition to the gameplay. We can produce a variety of effects by modifying the velocities of one or both objects during a collision.
First, we subtract the velocities of the two objects.
This gives us the relative velocity during impact.
Subtracting the velocities of objects a and b allows us to treat shape b as if it is stationary.
-- resolves collision function solveCollision(a, b, sx, sy) -- find the collision normal local d = math.sqrt(sx*sx + sy*sy) local nx, ny = sx/d, sy/d -- relative velocity local vx, vy = a.xv - (b.xv or 0), a.yv - (b.yv or 0)Next, we have to figure out how the moving shape should react to the collision (bounce, slide or stop completely). This is done be "splitting" the relative velocity into penetration and tangent components.
v: velocity (black)
vp: penetration component (red)
vt: tangent component (yellow)
n: collision normal axis (blue)
The dot product (ps) between the relative velocity and the collision normal (vx*nx + vy*ny) gives us the speed at which the two objects are moving toward each other. This product (ps) may be positive or negative depending on whether the objects are moving apart or toward each other.
-- penetration speed local ps = vx*nx + vy*ny -- objects moving apart? if ps > 0 then return end -- penetration component local px, py = nx*ps, ny*psUsually, we calculate the tangent component by rotating the collision normal 90 degrees and using the dot product:
-- tangent speed local ts = vx*ny - vy*nx -- tangent component local tx, ty = ny*ts, -nx*tsHowever, since we already know the penetration speed (ps) and velocity (px, py) at which the two objects are colliding, we can just subtract it from the relative velocity (vx, vy):
-- penetration speed local ps = vx*nx + vy*ny -- penetration component local px, py = nx*ps, ny*ps -- tangent component local tx, ty = vx - px, vy - py
Restitution or bounceThe restitution coefficient (or bounce) describes how "elastic" is the collision. This coefficient is a value between 0 and 1, where 0 means "inelastic" and 1 means "perfectly elastic".
Example of an elastic (green) versus non-elastic collision (pink).
Restitution always acts along the penetration axis (parallel to the collision normal). Remember that the "collision normal" is basically the axis of shortest separation between two colliding shapes.
-- penetration speed local ps = vx*nx + vy*ny -- penetration component local px, py = nx*ps, ny*ps -- restitution local r = 1 + math.max(a.bounce, b.bounce or 0) -- change the velocity of shape a a.xv = a.xv - px*r a.yv = a.yv - py*r
FrictionThe friction coefficient describes how much velocity is "lost" when the edges of two shapes are touching. Friction always acts along the tangent axis (perpendicular to the collision normal).
Note that the two objects may have different friction or restitution (bounce) coefficients. Combining each two coefficients "realistically" involves additional math so we use the "min" and "max" functions as a simplification.
-- penetration speed local ps = vx*nx + vy*ny -- penetration component local px, py = nx*ps, ny*ps -- tangent component local tx, ty = vx - px, vy - py -- restitution local r = 1 + math.max(a.bounce, b.bounce or 0) -- friction local f = math.min(a.friction, b.friction or 0) -- change the velocity of shape a a.xv = a.xv - px*r + tx*f a.yv = a.yv - py*r + ty*fFinaly, we multiply the penetration component by the restitution (r), the tangent component by the friction (f) and add them together to get the resulting change in velocity of the moving shape.
Mass and momentumRoland Yonaba pointed out that the equations above do not take mass into consideration. Therefore a small rectangle can easily push large rectangles when they collide. In a sense, all rectangles (regardless of their size) are treated as if they have equal mass. This is usually fine for old-school games and makes the equations a little bit easier to understand. Mass can be introduced to the simulation as follows:
-- how mass affects impulse ChangeInVelocity = FinalVelocity - InitialVelocity Impulse = Mass*ChangeInVelocity -- how mass affects momentum Momentum = Velocity*Mass -- how mass affects force Acceleration = ChangeInVelocity/Time Force = Mass*Acceleration
Coulomb's law and frictionThe code in this tutorial makes some obvious simplifications to the real laws of physics. However a somewhat subtle error is allowed in the way friction is simulated. With real life collisions: "the force of friction is always less than or equal to the normal [separation] force multiplied by some constant (whose value depends on the materials of the objects)". Without taking Coulomb's law into consideration objects with friction tend to behave like they're "sticky". I mention this perhaps to stir your interest in real physics and how many of its rich and interesting equations could be simulated in games.
Demo: simple collision response
- N Tutorial by Raigan Burns and Mare Sheppard
- Love2D Community
- How to create a custom physics engine by Randy Gaul