MSDN Magazine - April 2008 - (Page 79)

Matrix.Translation(axisFrom); Vector3[] p= Vector3.TransformCoordinate( face.Points,foldTransform); More to Explore Here the common side is defined as a line from axisFrom to axisTo. Rotations are always defined about a line through the origin, so, before I rotate, I need to move one point of the side to the origin and then move it back afterward. To do this, I’m chaining together three linear transformations to build the matrix that I use to transform the face coordinate: move to the origin, rotate, and then move back again. There’s one other little trick here: the vector representing the axis of rotation also has to start from the origin, so I subtract axisFrom, moving (axisFrom,axisTo) to (origin, axisTo-AxisFrom). If your head is not too full of math now, let’s try one more example (based on projector.cs) that is used to transform the polygons of the polyhedron into what you would expect to see from the screen viewpoint: Matrix projection = Matrix.Translation(-pivot) * Matrix.RotationY(yaw) * Matrix.RotationX(angle) * Matrix.Translation(viewPoint) * Geometry.Perspective(5); Vector3[] p= Vector3.TransformCoordinate(face.Points,foldTransform); Although not used in Polyhedra, quaternions are also available in the emulation library. They are a good way to combine a set of 3D rotations into a single rotation. They are also great for smoothly changing from one orientation to another. They were discovered by Sir William Rowan Hamilton (from my hometown of Dublin, Ireland) while walking along the street, and he promptly scribbled the equation on the side of the Broome Bridge Quaternions are also so he wouldn’t forget it (go available in the emulation to www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html). I’d like to claim that Remember that the shape is being rotated in the horizontal plane while it is being folded. To do this, a point on the pivot axis is moved to the origin, then a rotation takes place about the vertical (Y) axis through the yaw angle. (If you’ve flown an aircraft in reality or simulation, you will have come across the other two possible independent rotations called pitch and roll.) This time, instead of moving back to the start position, a move takes place to the viewpoint. Finally, a perspective transformation takes place to shrink the stuff that is far away on the Z axis. There’s a lot going on here, yet it can all be combined into a single 4-by-4 matrix, which is pretty impressive. all our graffiti is equally erudite, but I don’t suppose you’d believe me. Quaternions also avoid a problem called gimbal lock. This played an important part in gyroscope navigation during the Apollo moon missions and was featured in a movie about the Apollo 13 mission. There’s plenty of other stuff that I haven’t gone into, such as vector dot products and cross products, but hopefully I’ve given you a flavor of what’s possible with the DirectX Math emulation. I originally developed this emulation to do transformations on a hosted server running ASP.NET where I couldn’t install DirectX. You may find other applications for it in this type of environment. Polyhedron should give you a taste of how Silverlight provides a useful, extensible tool for Web UIs that really grab the user’s attention and get information across in fast and memorable ways. You can build on your existing .NET experience in all sorts of creative ways, and, unlike JavaScript, you don’t have to be obsessive about the number of lines of code being executed on the client. library. They are a good way to combine a set of 3D rotations into a single rotation. Silverlight Animations april2008 79 http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html http://projector.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

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