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Cubic tetrahedral algorithm

 

The hemicube algorithm replaced the hemisphere by a half cube, allowing the projection to be carried out on five planar rectangles, or side faces of the cube, instead of on a spherical surface. The number of planar surfaces can be decreased by using a cubic tetrahedron as an intermediate surface [BKP91], [BKP92].

  
Figure: Cubic tetrahedral method

An appropriate cubic tetrahedron may be constructed by slicing a cube by a plane that passes through three of its vertices, and placing the generated pyramid on surface i (see figure 1.6). A convenient coordinate system is defined with axes perpendicular to the faces of the tetrahedron, and setting scales to place the apex in point [1,1,1]. The base of the tetrahedron will be a triangle having vertices at [1,1,-2], [1,-2,1] and [-2,1,1].

  
Figure: Projection to the cubic tetrahedron

Consider the projection of a differential surface tex2html_wrap_inline1814 on a side face perpendicular to x axis, using the notations of figure 1.7. The projected area is:

equation476

The correction term, to provide the internal variable in the form factor integral, is:

equation481

Expressing the cosine of angles by a scalar product with tex2html_wrap_inline2244 pointing to the projected area:

equation487

Vector tex2html_wrap_inline2244 can also be defined as the sum of the vector pointing to the apex of the pyramid ([1,1,1]) and a linear combination of side vectors of pyramid face perpendicular to x axis:

equation493

This can be turned to the previous equation first, then to the formula of the correction term:

equation495

Because of symmetry, the values of this weight function -- that is the delta form factors   -- need to be computed and stored for only one-half of any face when the delta form factor table is generated. It should be mentioned that cells located along the base of the tetrahedron need special treatment, since they have triangular shape. They can either be simply ignored, because their delta form factors are usually very small, or they can be evaluated for the center of the triangle instead of the center of the rectangular pixel.


next up previous index
Next: Solution of the linear Up: Form factor calculation Previous: Hemicube algorithm

Szirmay-Kalos Laszlo
Mon Oct 21 14:07:41 METDST 1996