![]() Thus, the scalar multiples = doesn't seem so weird, since this is just like saying our character from the example is moving farther away from the point we're looking at. Now if we move this screen around (example, since this is a computer science book: the head of a character in an FPS looking around) the image we see will change, even though it represents the same set of points. Notice that the magnitude of the cross product is always the same. ![]() If all of this seems confusing and you're asking why we use this weird 3D-style space for 2D projective space, think of it like this: everything in 3D space we're going to "project" onto a 2D screen. As you change these vectors, observe how the cross product (the vector in red),, changes. This also gives a nice reason why the single vector gives the line between them, since the point is actually a line in 3D space. For example, the product rule is used to derive Frenet. Picking a method depends on the problem at hand. ![]() Or you can use the product rule, which works just fine with the cross product: d d t ( u × v) d u d t × v + u × d v d t. Thus, because 2D projective geometry relies on three coordinates to specify a "point" we can use the usual cross product we know and love from 3D space. The vector c that is equal to the cross product of non-zero vectors a and b, is perpendicular to these. You can evaluate this expression in two ways: You can find the cross product first, and then differentiate it. Since they're all scalar multiples of one another. In homogeneous coordinates, the point given by Let me better explain that last statement. Homogeneous coordinates mean that "points" in this space are actually what we could call "lines" in normal 3D geometry (although this is projective 2D geometry). The : between coordinates is to let you know that we're using homogeneous coordinates. In other words, The sign of the 2D cross product tells you whether the second vector is on the left or right side of the first vector (the direction of the first vector being front ). You cant do a cross product with vectors in 2D space. ![]() Where all three of these coordinates can't be zero. What they're doing is using homogeneous coordinates, which take the form of ![]()
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