fixed messages

[vor.git] / sprite.c

diff --git a/sprite.c b/sprite.c
index abe9fb3..447e482 100644 (file)
--- a/sprite.c
+++ b/sprite.c
@@ -336,26 +336,113 @@ sprite_mass(Sprite *s)
        else return 0;
 }
 
+/*
+ * BOUNCE THEORY
+ *
+ * ******************  In 1 Dimension  *****************
+ *
+ * For now we will imagine bouncing A and B off each other in 1 dimension (along
+ * a line). We can safely save the other dimension for later.
+ *
+ * A and B are the same weight, and are both traveling 1m/sec, to collide right
+ * at the origin. With perfect bounciness, their full momentum is reversed.
+ *
+ * If we cut the weight of A down by half, then the center of our colision will
+ * drift towards A (the speeds of A and B are not simply reversed as in our last
+ * example.) However, there is always a place between A and B on the line (I'll
+ * call it x) such that the speeds of A and B relative to x, are simply
+ * reversed. Thus we can find the new speed for A like so:
+ *
+ *     new A = x -(A - x)
+ *
+ *     new B = x -(B - x)
+ * 
+ * or, simply:
+ *
+ *     new A = 2x - A
+ *
+ *     new B = 2x - B
+ *
+ *
+ * this point x is the sort of center of momentum. If, instead of bouncing, A
+ * and B just globbed together, x would be center of the new glob.
+ *
+ * x is the point where there's an equal amount of force coming in from both
+ * sides. ie the weighted average of the speeds of A and B.
+ *
+ * average force = (A force + B force) / total mass
+ *
+ * x.speed = (a.speed * a.mass + b.speed * b.mass) / (a.mass + b.mas)
+ *
+ * then we apply the formula above for calculating the new A and B.
+ *
+ *
+ *
+ *
+ * ******************  In 2 Dimensions  *****************
+ *
+ * OK, that's how we do it in 1D. Now we need to deal with 2D.
+ * 
+ * Imagine (or draw) the two balls just as they are bouncing off each other.
+ * Imagine drawing a line through the centers of the balls. The balls are
+ * exerting force on each other only along this axis. So if we rotate
+ * everything, we can do our earlier 1D math along this line.
+ *
+ * It doesn't matter what direction the balls are going in, they only exert
+ * force on each other along this line. What we will do is to compute the part
+ * of the balls' momentum that is going along this line, and bounce it according
+ * to our math above. The other part is unaffected by the bounce, and we can
+ * just leave it alone.
+ *
+ * To get this component of the balls' momentum, we can use the dot product.
+ *
+ *     dot(U, V) = length(U) * length(V) * cos(angle between U and V)
+ *
+ * If U is a length 1 vector, then dot(U, V) is the length of the component of V
+ * in the direction of U.  So the components of V are:
+ *
+ *     U * dot(U, V)      parallel to U
+ *
+ *     V - U * dot(U, V)  perpendicular to U
+ *
+ * To do the actual bounce, we compute the unit vector between the center of the
+ * two balls, compute the components of the balls' speeds along this vector (A
+ * and B), and then bounce them according to the math above:
+ *
+ *     new A = 2x - A
+ *
+ *     new B = 2x - B
+ *
+ * But we rewrite it in relative terms:
+ *
+ *     new A = A + 2(x-A)
+ *
+ *     new B = B + 2(x-B)
+ */
+
 void
 bounce(Sprite *a, Sprite *b)
 {
-       float x, y, n;
-       float va, vb;
-       float ma, mb, mr;
+       float x, y, n;  // (x, y) is unit vector from a to b.
+       float va, vb;   // va, vb are balls' speeds along (x, y)
+       float ma, mb;   // ma, mb are the balls' masses.
+       float vc;       // vc is the "center of momentum"
 
        // (x, y) is unit vector pointing from A's center to B's center.
        x = (b->x + b->w / 2) - (a->x + a->w / 2);
        y = (b->y + b->h / 2) - (a->y + a->h / 2);
        n = sqrt(x*x + y*y); x /= n; y /= n;
 
-       // velocities along (x, y), or 0 if already moving away.
-       va = max(x*a->dx + y*a->dy, 0);
-       vb = min(x*b->dx + y*b->dy, 0);
+       // velocities along (x, y)
+       va = x*a->dx + y*a->dy;
+       vb = x*b->dx + y*b->dy;
+       if(vb-va > 0) return;  // don't bounce if we're already moving away.
 
-       // mass ratio
+       // get masses and compute "center" speed
        ma = sprite_mass(a); mb = sprite_mass(b);
-       if(ma && mb) mr = mb/ma; else mr = 1;
+       vc = (va*ma + vb*mb) / (ma+mb);
 
-       a->dx += x*(mb*vb - ma*va)/ma; a->dy += y*(mb*vb - ma*va)/ma;
-       b->dx += x*(ma*va - mb*vb)/mb; b->dy += y*(ma*va - mb*vb)/mb;
+       // bounce off the center speed.
+       a->dx += 2*x*(vc-va); a->dy += 2*y*(vc-va);
+       b->dx += 2*x*(vc-vb); b->dy += 2*y*(vc-vb);
 }