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//v1: the ship's motion vector _v1.update ( _shipModel.xPos, _shipModel.yPos, (_shipModel.xPos + _shipModel.vx), (_shipModel.yPos + _shipModel.vy) ); //v2: the vector between the drag handles _v2.update(_handle1.x, _handle1.y, _handle2.x, _handle2.y); //v3: the vector between v1 and v2 _v3.update(_shipModel.xPos, _shipModel.yPos, _handle1.x, _handle1.y); Next, we calculate the ratio of the perp-dot product of v3 and v2, and of v1 and v2. var t:Number = VectorMath.perpProduct(_v3, _v2) / VectorMath.perpProduct(_v1, _v2); The VectorMath class has a perpProduct method that does this calculation for us. static public function perpProduct(v1:VectorModel, v2:VectorModel):Number { var perpProduct:Number = v1.ln.vx * v2.dx + v1.ln.vy * v2.dy; return perpProduct; } Next, we combine v1 s position and velocity, and multiply it by the ratio (t) to find the intersection point. var intersection_X:Number = _v1.a.x + _v1.vx * t; var intersection_Y:Number = _v1.a.y + _v1.vy * t; Finally, we position the intersection mark. _mark.x = intersection_X; _mark.y = intersection_Y; (The visible X that you see on the stage is created from the IntersectionMark class in the vectors package. It draws a simple crosshair.) There s one small problem with this example that I m sure you ve noticed. The intersection mark doesn t know that it should be limited to the magnitude of v2. It thinks that v2 goes on forever. As a result, it finds intersections beyond the limits of v2, as you can see in Figure 2-32. This isn t actually incorrect as it has been coded, but we would have a much more usable system if intersections were found only within the vector s point A and B.
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Figure 2-32. Intersections are found outside the limits of the vector. We need to conjure up a little more vector magic to help us solve this problem. In the chapter s source files, you ll find a folder called LimitingIntersection. Run the SWF, and you ll notice that the intersection point is now mapped only on the length of the vector, and not beyond it. This is very easy to implement. The source code is exactly the same as the previous example, except for these additions:
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1. Extend a vector from v2 s start point to the intersection point.
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var v4:VectorModel = new VectorModel (_v2.a.x, _v2.a.y, intersection_X, intersection_Y);
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2. Extend another vector running in the opposite direction from v2 s end point to the
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intersection point. var v5:VectorModel = new VectorModel (_v2.b.x, _v2.b.y, intersection_X, intersection_Y);
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3. If the magnitude of either of these vectors is larger than v2 s magnitude, then you know
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that the intersection point falls beyond v2 s bounds. if(v4.m > _v2.m || v5.m > _v2.m) { //The intersection point is outside the magnitude of v2 intersection_X = 0; intersection_Y = 0; } You can decide what actions to take on this. In this simple example, the intersection point is moved to position 0,0, which keeps it out of trouble. In a more complex game environment, you ll probably have many more variables to consider.
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VECTORS: GHOSTS IN THE MACHINE
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Figure 2-33 illustrates how this works. It s a very simple concept, and once you become more comfortable using vectors, it s the kind of thing you ll easily be able to figure out for yourself.
Figure 2-33. If the magnitude of either v4 or v5 is greater than v2, then the intersection point is not within the bounds of v2.
Collision and bounce
You can relax now! We ve covered almost everything you need to know about vectors for 2D game design. The theory class is over, and we can now put some of this new information to practical use. Now we re going to solve the classic problem of line-versus-point collision. This is collision between a two-dimensional line segment and a one-dimensional point. Points are objects with no height or width, just a position. They re often called particles, so that s how I ll be referring to them from now on. Collisions between lines and particles are the most basic form of collision detection that you can do. Once you understand these fundamentals, you ll find shape-versusshape collision, which we ll look at in the next chapters, much easier to comprehend.
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