MSDN Magazine - April 2008 - (Page 78)

Figure 6 Emulated DirectX Math Classes the projection of the 3D shape onto the monitor’s surface. Silverlight is forced to draw these polygons in back-to-front order by setting the XAML ZIndex property for each Polygon using the Z coordinate of the center of the 3D polygon (scaled to squeeze a float into an int). This is a fairly simplistic way of achieving 3D that will only work if the polygons behave themselves by, for example, not intersecting—something I can get away with in this sample. Smarter forms of 3D involve mechanisms such as A shape in 3D can be depth buffers. As these reidentified as a set of faces quire GPU access, they are out of scope for the Silverthat are identified as a light .NET sandbox. set of points corresponding As a last touch, XAML to corners. allows Polygons to be partially transparent (using the Opacity property), which gives the artistic effect of a partially seethrough polyhedron. And that, as they say, is all there is to it. I’m going to touch lightly on the math used to achieve the animation. If you want to go into more detail, there are many sites on the Web that will help, including wikipedia.com, euclideanspace.com, mathworld.wolfram.com, gamasutra.com, and gamedev.net. I have already introduced some 2D transformations, such as RotationTransform and ScaleTransform, in the XAML samples shown earlier. In these, Silverlight does all the work for you. If you use XAML in a WPF environment, you also have access to 3D transformations, but these are not available in Silverlight. WPF 78 msdnmagazine Silverlight Animations Emulating DirectX Math relies upon DirectX to do the low-level work for it. This makes sense, as DirectX has a nice set of classes for doing the math required for 3D transforms. To do some of the same transformations in Silverlight, I’ve produced a DirectX Math emulation of many of these classes so that they can be used in the Silverlight sandbox. If you have any experience coding in Direct3D®, you’ll be quite at home with the Microsoft implementations of Vector2, Vector3, Matrix, and the like (see Figure 6). If not, I’ll give a very brief and very non-rigorous overview and some examples. Locating a point in 2D requires two coordinates: X and Y. In 3D it requires three coordinates: X, Y, and Z. Although, strictly speaking, points are not vectors, 2D points can be stored in a Vector2 object and 3D points in a Vector3 object. (Vectors are actually entities with a magnitude and a direction, which you can illustrate by having an arrow that starts at the origin and ends at the point. In a sense, you can think of all vectors as starting from the origin.) Shapes in three dimensions can be identified as a set of faces. Faces, in turn, are identified as a set of points corresponding to corners. Nearly everything you might want to do involves transforming these points to another set of points by applying one or more operations such as rotation, translation (or moving), or scaling. Nearly every operation that you might want to do can be represented by a 4-by-4 grid of numbers called a matrix. (Matrices in general can actually have any number of rows and columns, but we’re only interested in the special ones used to manipulate 3D points with what are called homogeneous coordinates.) These operations are referred to as linear transformations because they don’t (overly) distort the shape of the object they are applied to. Very usefully, there is a way of combining two such matrices into one resulting matrix that has the same effect. This operation is called matrix multiplication and can be repeated several times. This means you can chain a whole set of transformations together into a single matrix with the same effect. This is far more efficient than applying each transformation separately. There’s no need to worry about how matrix multiplication is implemented or how it does its transformation magic. Just use it as a tool. To transform a source point into a target point, there is also a multiplication operation defined between a matrix and a vector object. To avoid confusion (with matrix-matrix multiplication), this is implemented in Vector3 using the TransformCoordinate method. A couple of little code snippets should make things more clear. First, the following code in polyhedron.cs is used to fold two faces along their common side: Vector3 axis = axisTo - axisFrom; Matrix foldTransform = Matrix.Translation(-axisFrom) * Matrix.RotationAxis(axis, proportion * _dihedralAngle) * http://wikipedia.com http://euclideanspace.com http://mathworld.wolfram.com http://gamasutra.com http://gamedev.net

Table of Contents Feed for the Digital Edition of MSDN Magazine - April 2008

MSDN Magazine - April 2008 Contents Toolbox CLR Inside Out Basic Instincts Cutting Edge Foundations Test Run Service Station Windows with C++ Going Places { End Bracket }

MSDN Magazine - April 2008

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