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Getting the vector magnitude and orientation
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The vector creates a collision boundary for the ship. To create that boundary, we need to know the vector s magnitude and orientation. Where is it and how long is it The ship needs to know where the line starts and ends so that it can fly around it. v2 is the vector that you can see on the stage. An invisible vector called v3 runs between the ship and v2 s start point. The dot product of these vectors can tell us whether the ship is within the line s scope. If the dot product is greater than zero, the spaceship will be beyond v2 s scope, at the bottom. If the dot product is less than the negative of v2 s magnitude, then the ship is beyond v2 s scope at the top. If this sounds confusing, a picture will help. Take a look at Figure 2-35 and compare the value of dp1 with v2.m (the magnitude of the line). If dp1 is greater than 98 or less than 0, the ship is in a position to possibly collide with the line.
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Figure 2-35. dp1 can tell is whether the ship is within the scope of v2. Here s the code that figures this out and checks whether a collision might be possible: var dp1:Number = VectorMath.dotProduct(_v3, _v2); if(dp1 > -_v2.m && dp1 < 0) { a collision might be possible If this is false, the ship is free to fly around the top or bottom of the line. This relationship remains the same, even if v2 changes its magnitude or orientation, so it s an extremely useful bit of information.
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Detecting a collision
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dp2 is the dot product of v3 (the vector between the ship and the start point of v2) and v2 s normal. Earlier in the chapter, we looked at how we can use this dot product to find out on which side of the line the ship is. Let s review that again here, as it s crucial to understanding how vector-based collision works. Take a look at Figure 2-36. The spaceship is flying straight toward the line. It s probably going to hit it, so it will be useful to know when it hits and with how much force. In these examples, the collision plane is represented by a vector called v2. To get the information we need, we extend a vector from the spaceship to the collision plane s start point. In these examples, this new vector is called v3, and you can see it illustrated in Figure 2-37. This new vector will help us get a bit more extra information about the collision.
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Figure 2-36. A collision between the spaceship and the line is imminent.
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Figure 2-37. Extend a new vector between the spaceship and the collision plane s start point.
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There is a magic number that will help us easily solve the collision problem. That number is the dot product between the new vector, v3, and v2 s normal. (You can use either the left or right normal, the result will be the same.) Figure 2-38 shows how this dot product is found. In these examples, this dot product is called dp2.
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Figure 2-38. Find the dot product of v3 and v2 to help calculate the collision. The dot product in this example has a value of 3. Does that value seem significant in some way It is. Look carefully, and you ll see that that the spaceship is three cells away from the collision plane. As illustrated in Figure 2-39, the dot product can tell you exactly how far away the ship is from the line.
Figure 2-39. The dot product can tell you how far away the spaceship is from the collision plane. This remarkable coincidence is like a buried treasure hidden deep within the math. It holds true no matter what the angle of the collision plane is. Because you can use this information to tell how far the ship is away from the line, you can also use it to figure out if there has been a collision. Not only that, but it can tell you with how much force the ship has collided. Take a look at Figure 2-40, and you ll see what I mean. When the ship is exactly on the line, the dot product is zero. When it has crossed the line by three cells, the dot product is 3. This means that you know the ship is colliding with the line if the dot product is less than zero. That s the flag that triggers a collision. In the Collision example code, it s represented like this: if(dp2 <= 0) { // We have a collision! Now we have a way to detect a collision. Next, we need to find a way to resolve the collision.
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