A perspective projection in Java

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A perspective projection
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d Focal point
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Projection plane, image plane
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In fact, you rst calculate this factor in the w dimension. For a focal point at the origin and a projection plane at distance d from the focal point, you set the w of every point to the ratio of its distance from the focal plane:
| | | | x y z w | | | | | = | | | 1 0 0 0 0 0 1 0 0 1 0 1/d 0 0 0 0 | | | | | | | | x y z w | | x | | y | = | z | | z/d | | | |
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Then, to make this perspective factor affect the image, you convert 4D to 3D, scaling down all the coordinates by w. So the 3D version of this vector is <x/w, y/w, z/w>. In the end, you throw away z and draw the vector at <x/w, y/w>, where w is z/d. For this trivial example, x = dx/z and y = dy/z. Look closely: you re just scaling down the points depending on their distance. And that s what perspective is all about! With a more complex projection matrix, you can change the eld of view and the position of the focal point. Why does this all work Because the four-dimensional space you work in is a homogeneous coordinate system. Its fourth dimension, w, extends all points in three space into a set of equivalent points in the
34: Geometric and Color Transformations
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homogeneous coordinate system. For any vector <x, y, z> in real coordinates, there exists an in nite equivalent set of points of the form <kx, ky, kz, k> (for all values of k) in homogeneous coordinates. The homogeneous coordinate space of dimension n+1 still represents points of dimension n, but it stretches out each point in the n-dimensional space to a line of all possible points in dimension n+1 in which the homogeneous coordinate is a common divisor of the other coordinates. That s why, when you project from the 4D homogeneous dimensions into real 3D space, you divide x, y, and z by w. And it s also why you can treat vectors in three dimensions as existing in four dimensions where w = 1. (You can think of the plane at w = 1 as overlapping the real coordinate space.) Another convenient property of homogeneous space is that it respects linear transformations. Its homogeneous properties are not broken when you apply a linear transformation in four dimensions. And, even better, af ne transformations in 3D space are linear transformations in 4D space. When you translate in 3D space (using a 4 4 transformation matrix), you re actually shearing 4D space! By scaling just w with regard to distance from the image plane (the camera), you re scaling all the original coordinates because of the ratio relationship they have to the homogeneous coordinate. If you nd it hard to visualize, think of w as a redundant, invisible dimension that is used to control the scale of the other coordinates. At least half its purpose is simply enlarging vectors to 4 1 so that they can be multiplied by 4 4 transformation matrices with a fourth column that enables translation through matrix multiplication. The other half is, as you ve seen, used in perspective projections.
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3D Transformations in ActionScript
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Up to this point, I ve carefully avoided mentioning how ActionScript 3.0 utilizes these basic principles of 3D transformations. As the language has grown somewhat organically to support rst 2D, then 3D geometry, there are some oddities that would have otherwise bogged down this introduction. To get started, I ll compare the different classes in use for 2D and 3D math. All the classes in Table 34-1 are in the flash.geom package. You ll notice that the naming scheme is a bit inconsistent. Point becomes Vector3D instead of Point3D; Point is able to behave like a vector; the Vector class refers to a data structure and not a mathematical vector. Although the class Vector in the default package and the class flash.geom.Vector could easily coexist thanks to namespaces, the Flash Player architects have spared you this particular confusion.
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TABLE 34-1
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