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
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
-- 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")Once we know how to find the "separation vector", it's easy to move the two colliding objects apart. This alone is enough for games (like the original Super Mario, Megaman or Contra) where you just want to ensure that the player cannot pass through the static environment.
-- 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
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.
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 "bounce"
The 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
The 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.
-- 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*AccelerationCoulomb's law and friction