Computer-Generated Holography (CGH)

PHYS4412 (Computational Science II)
Department of Physics & Astronomy
Louisiana State University

III.A Constructing Surfaces from VRML Files

General Concept

What I would like to do now is combine my newly gained understanding of how CGH works with my understanding of how the surfaces of 3D objects have traditionally been constucted using voxel-based or wire-frame modeling and ray-tracing techniques to create an extremely efficient means of generating digital holograms of arbitrarily complex surfaces. The first thing to realize along these lines is that most modern 3D-rendering tools expect to receive input in the form of long lists of vertices and polygons. This is, for example, how the virtual-reality markup language (VRML) works. I would like to take advantage of this standard input format and design a CGH algorithm that can compute the holographic image of any surface that is defined by a group of variously connected polygons. With this in mind, § B of this chapter describes how the vertices and polygons of an object's surface are usually specified within a VRML file.

I will also take note of the fact that VRML files usually contain only the simplest form of polygons, namely, triangles. That is to say, I will expect that my complex surface will be broken down into variously connected triangles. Building on the results of earlier chapters, then, in § C of this chapter we derive an analytical expression that specifies the holographic pattern of an arbitrarily shaped triangular aperture. The derived expression is expressed entirely in terms of the intrinsic properties of the triangle except that its final expression depends on the coordinates of the single point (xa, ya, za) in space from which the triangle is being viewed. The coordinates of this viewing point will be expressed in terms of a Cartesian coordinate system that is affixed to the triangular aperture in such a way that the unit vector ka is normal to the surface of the triangle and points away from the lighted surface of the triangle; one of the edges of the triangle shall point in the ja direction; and the "origin" of this coordinate system is chosen so as to minimize the calculations required in the determination of the hologram.

Finally, in § D of this chapter [not yet written!] we combine the results of §§ B & C and demonstrate how to transform from the object coordinate system that is used in VRML to specify the location and orientation of each (triangular) polygon on the surface of an object to a coordinate system that is affixed to the triangular aperture.

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