The math to move the position of the camera around so that you rotate around the X axis is just a parametric equation of a circle:

where t is an angle from zero to 2 pi.
Imagine that you are standing on the street looking at a house. The coordinates of the camera should follow in a circle around the house, and the latitude and longitude are constantly changing to stay at the same distance from the house. Therefore, there is no value that you can change to make it spin.
Once you know the position of the camera, the direction is simply the difference between the start and position of the camera.
All this is easy to calculate, but there is an easier way. Instead, hold the camera still and rotate the subject . This makes the animation easier. Here's an MSDN article that contains examples of this approach, including animation:
This article is for WPF and Visual Studio, but you can easily adapt the same ideas to Expression Blend.
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