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Figure 2-40. Use the dot product to tell you when a collision has occurred and what the collision force is.
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Resolving the collision
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In Figures 2-35 through 2-39, the spaceship and the collision plane are aligned to the x and y axes. This makes the math really easy to understand and visualize. But in a real-world collision, it s likely that the ship and line will be colliding at weird angles. Fortunately, the math holds up no matter at what angle the collision happens. When the ship collides with the line, we need to know by how much it has crossed the line. This is important so we can move the ship out of the collision. We need to move it back to the exact point of collision with the line (the point where the dot product is zero). In Figure 2-39, it s easy to see that the ship has overshot the line by three cells. If we move the ship back by 3, it will be positioned exactly at the point of collision.
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We can say that 3 is the ship s collision vector. The collision vector has a magnitude of 3. If we know the direction of the collision and we know the magnitude, we can resolve the collision. As you ve seen, the dot product will always tell you the magnitude of the collision vector. We have that number in the bag. However, we don t always know the direction of the collision. Take a look at Figure 2-41 for an example. It shows how both the ship and the line are colliding at strange angles. We know that the dot product is 4 (rounded). That s the collision vector s magnitude. But we can t create a collision vector until we also know the direction of the collision.
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Figure 2-41. The ship has collided with the line. The magnitude of the collision will be the same as the value of the dot product. But what s the direction How can we figure out from which direction the ship is colliding with the line Surprise we already know it! It s the ship s motion vector, v1. Figure 2-42 shows the spaceship s motion vector, v1. It s definitely pointing in the right direction. The only problem is that it s too long. Its magnitude is 7 (rounded). What would Goldilocks do
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Figure 2-42. The ship s motion vector is pointing in the right direction, but its magnitude is too long. We need a vector that s pointing in the same direction as v1 but has the same magnitude as the dot product. Aha! That should ring a bell. We can use a unit vector to help solve this problem. Remember that normalized unit vectors are the smallest size a vector can be. They have the magical property of pointing in a specific direction, but can be scaled to any size. Unit vectors are the Drink Me! of Vectorland. How can we use a unit vector in this case
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1. Normalize the ship s motion vector, v1. That will allow us to keep its direction but scale it
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to any other size.
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2. Multiply v1 s unit vector by the value of the dot product, 4. The resulting vector is the
new collision vector.
Figure 2-43 shows how v1 s unit vector is scaled by the dot product to create the collision vector. Here s the code in the Collision example file that does this work. var collisionForce_Vx:Number = _v1.dx * Math.abs(dp2); var collisionForce_Vy:Number = _v1.dy * Math.abs(dp2);