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Next: Hemicube algorithm Up: Form factor calculation Previous: Analytic form factor computation

Hemisphere algorithm

    

First of all the result of Nusselt is proven using figure  1.3, which shows that the inner form factor integral can be calculated by a double projection of tex2html_wrap_inline1824 , first onto the unit hemisphere centered above tex2html_wrap_inline1812 , then to the base circle of this hemisphere in the plane of tex2html_wrap_inline1812 , and finally by calculating the ratio of the double projected area and the area of the unit circle ( tex2html_wrap_inline1906 ). By geometric arguments, or by the definition of solid angles, the projected area of a differential area tex2html_wrap_inline1814 on the surface of the hemisphere is tex2html_wrap_inline1914 . This area is orthographically projected onto the plane of tex2html_wrap_inline1812 , multiplying the area by factor tex2html_wrap_inline2047 . The ratio of the double projected area and the area of the base circle is:

equation355

Since the double projection is a one-to-one mapping, if surface tex2html_wrap_inline1824 is above the plane of tex2html_wrap_inline2051 , the portion, taking the whole tex2html_wrap_inline1824 surface into account, is:

equation359

This is exactly the formula of an inner form factor integral.

  
Figure: Hemispherical projection of a planar polygon

  Now we turn to the problem of the hemispherical projection of a planar polygon. To simplify the problem, consider only one edge line of the polygon first, and two vertices, tex2html_wrap_inline2024 and tex2html_wrap_inline2057 , on it (figure 1.4). The hemispherical projection of this line is a half great circle. Since the radius of this great circle is 1, the area of the sector formed by the projections of tex2html_wrap_inline2024 and tex2html_wrap_inline2057 and the center of the hemisphere is simply half the angle of tex2html_wrap_inline2024 and tex2html_wrap_inline2057 . Projecting this sector orthographically onto the equatorial plane, an ellipse sector is generated, having the area of the great circle sector multiplied by the cosine of the angle of the surface normal tex2html_wrap_inline1994 and the normal of the segment ( tex2html_wrap_inline2069 ).

The area of the doubly projected polygon can be obtained by adding and subtracting the areas of the ellipse sectors of the different edges, as is demonstrated in figure 1.4, depending on whether the projections of vectors tex2html_wrap_inline2024 and tex2html_wrap_inline2057 follow each other clockwise. This sign value can also be represented by a signed angle of the two vectors, expressing the area of the double projected polygon as a summation:

equation379

Having divided this by tex2html_wrap_inline1906 to calculate the ratio of the area of the double projected polygon and the area of the equatorial circle, equation 1.26 can be generated.

These methods have supposed that surface tex2html_wrap_inline1824 is above the plane of tex2html_wrap_inline1812 and is totally visible. Surfaces below the equatorial plane do not pose any problems, since we can get rid of them by the application of a clipping algorithm. Total visibility, that is when visibility term tex2html_wrap_inline1818 is everywhere 1, however, is only an extreme case in the possible arrangements. The other extreme case is when the visibility term is everywhere 0, and thus the form factor will obviously be zero.

When partial occlusion occurs, the computation can make use of these two extreme cases according to the following approaches:

  1. A continuous (object precision) visibility algorithm is used in the form factor computation to select the visible parts of the surfaces. Having executed this step, the parts are either totally visible or hidden from the given point on surface i.
  2. The visibility term is estimated by firing several rays to surface element j and averaging their 0/1 associated visibilities. If the result is about 1, no occlusion is assumed; if it is about 0, the surface is assumed to be obscured; otherwise the surface i has to be subdivided, and the whole step repeated recursively [Tam92].

next up previous index
Next: Hemicube algorithm Up: Form factor calculation Previous: Analytic form factor computation

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