MSDN Magazine - April 2008 - (Page 77)

Figure 4 Building the Shape, from Graph to XAML tions take place, in the end a set of ordinary 2D polygons is being displayed on the monitor surface. If you are willing to do the 3D manipulations in your own code to calculate the coordinates of these polygons, a non-hardware-accelerated 3D rendering is possible. Provided you restrict yourself to a few dozen polygons, the performance is acceptable. (Subjectively, frame performance has improved moving from the Alpha to the Beta.) When a shape is selected, a two-dimensional unfolded template called a Net (nothing to do with .NET) is built. This is done in two phases, as illustrated in Figure 4. First a graph is built in memory with one GraphNode corresponding to each face in the shape (see Graph.cs). This graph is built by streaming in a set of connectivity information (indicating which face is connected to which faces) from one of the .shp resources that are embedded in the application assembly. For example, here are the contents of cube.shp: 1:3,4,5,2 2:1,5,6,3 3:2,6,4,1 4:3,6,5,1 5:1,4,6,2 6:2,5,4,3 when the shape is fully closed. It is, in theory, possible to calculate the dihedral angles directly using just the connectivity information in the .shp file. This is fairly easy for special cases such as any three faces meeting at a corner or any number of faces of the same type meeting at a corner. However the general case is somewhat more difficult, so I decided to store these angles in separate .dihedrals resources. For example, here’s an excerpt from cube.dihedrals: 1,3:1.5707965056551 1,4:1.57079661280793 1,5:1.57079614793469 1,2:1.57079604078186 2,1:1.57079604078186 2,5:1.57079628466395 2,6:1.57079661280793 2,3:1.57079636892585 3,2:1.57079636892585 3,6:1.57079614793469 3,4:1.57079628466395 Any GraphNode in the graph can then be chosen as the first node from which to start building the Net (see Net.cs in the code download for details). A two-dimensional FlatFace is then laid out with the same number of sides as the GraphNode has neighbors. Any neighboring GraphNode is then chosen to repeat the process, laying out the next FlatFace so it shares a side with the current one. Each GraphNode is only visited once, and the process continues until all GraphNodes have been visited, resulting in a tree structure of FlatFaces. In each frame of the 3D animation, a three-dimensional polyhedron (possibly partially closed) is built from the Net (see Polyhedron.cs later in this article for the details of this operation). This involves copying the tree of FlatFaces into a tree of Faces replacing 2D corner points with 3D points. (Figure 5 shows the two coordinate systems. Because I want to start in the horizontal plane, Y should start as zero. So I map from 2D to 3D by setting X=X, Y=0, and Z=Y.) Finally, recursion is used to visit each join in the tree of Faces to perform a fold. The amount of the fold is a fraction of the angle (called a dihedral angle) that the faces will have to each other The minute differences in these numbers, which should all be PI/2 in this case, are merely an artifact of the way they were calculated. The final process involves a series of transformations to project a view of the polyhedron onto the monitor as a set of polygons. You may have noticed that, during the folding animation, the polyhedron is also rotated about a vertical axis. To achieve this, I translate the pivot point to the origin, perform the rotation, and then translate to the viewpoint. Then I do a perspective transformation to ensure that far away objects (on the Z axis) appear smaller. (I’ll explain a bit more about how to do transformations in the next section.) Finally, a set of XAML polygons is generated corresponding to Figure 5 Transforming Points from 2D to 3D Silverlight Animations april2008 77 http://Graph.cs http://Graph.cs http://Net.cs http://Polyhedron.cs http://Polyhedron.cs

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

For optimal viewing of this digital publication, please enable JavaScript and then refresh the page. If you would like to try to load the digital publication without using Flash Player detection, please click here.