Let us assume that only a real object point O in water and the viewpoint V of an observer in air are known, as illustrated in Figure 4.2. According to the law of refraction, there is only one correct optical path that connects both points. Due to the unknown starting direction of the ray in water, the optical path can not be determined directly, making its recovery computationally expensive. This is equivalent to the problem of refractive forward projection. In contrast to the real object point O, the related VOP lies on the up to now unknown ray in air, which is passing through the viewpoint V . To make the best possible use of this later on, the relation between the virtual and the real object point should be as convenient as possible. To analyze the properties of this relation and due to the unknown starting direction of the ray in water, the VOPs will be determined and traced for a significant amount of all the possible directions of the rays that are emitted from the real object point O.
Traces of Virtual Object Points
The entirety of the possible directions of the emitted rays are limited by a right circular cone ∆w, with the object axis s1 as its axis and a radius that is limited by the occurring total reflection of the rays in water, as illustrated in Figure 4.2. For the transition from water to air, the maximal angle of incidence α is about 49 degrees. Rays outside this cone, such as w, exceed this maximal angle and get reflected at the interface Φ. The
ray in water, which forms the beginning of the optical path in question, is part of ∆w. Note that the cone ∆w from Figure 4.2 is reduced to its circular outline on the left side of Figure 4.3. Both images in Figure 4.3 are visualizations of the traces of the three possible VOP locations. These traces are the result of a representative movement of the point of refraction R of the ray w on the refractive interface Φ. This movement
consists of the circular outline of the base area of the cone ∆w and a straight line passing through the object axis s1 on the base area. It represents a significant amount of all the possible directions of the ray w. The traces are regulated by the previously presented
4 Virtual Object Points
Figure 4.3: Traces of the VOP locations in the case of one refractive interface. Left: 2D front
view. Right: Perspective 3D view.
determination processes of the three proposed locations ˇOm, ˇOs or ˇOM, respectively. In each case, the result is a 3D structure.
The left side of Figure4.3 illustrates the resulting traces of the VOP locations up to an optical path close to total reflection and the right side shows the resulting traces up to total reflection. This differentiation is done for visualization purposes of the trace ∆M. Both sides show the one optical path in question that truly connects O and V .
• In the case of ˇOM, the result is the circular area ∆M on the plane Γ. If the angle
of incidence of the ray w in water tends to total reflection, the radius of ∆M tends towards infinity.
• The result in the case of ˇOm is the 3D surface ∆m. Its extent along the object axis
s1 is limited at the bottom by a ray w with an angle of incidence that amounts to zero degrees and can therefore be computed by Equation 2.25. The upper end is limited by the rays whose angles of incidence tend to total reflection and therefore, by the flat refractive interface Φ. On Φ, the base area of ∆m coincides with the base area of the cone ∆w. At the bottom, the apex of the structure intersects the object axis s1.
• In the case of ˇOs, the result is the straight line segment ∆s along the object axis s1. The limits are the same as for ∆m. Therefore, the line segment reaches the
refractive interface Φ at the upper end and coincides with the apex of ∆m at the bottom.
These traces make clear that the assignment of VOPs to a single real object point is ambiguous if no further information on the optical path in question is known. The 3D structures ∆M, ∆m and ∆s must contain the VOPs that lie on the virtual part of the ray a in question. The most convenient case is indisputably ∆s. It has the least spacial extent and the simplest 3D structure. Furthermore, its range can be determined easily. An argument against ∆M is its infinite extent and an argument against ∆m is a resulting
4.2 Position Determination
Figure 4.4: VOP locations in the case of one refractive interface. Left: 2D front view. Right:
Perspective 3D view.
ambiguity. As can be seen on the right side of Figure 4.3, the trace ∆m intersects the ray a in question twice due to its symmetry. The respective traces can be considered as a search space for the determination of the true optical path between a given real object point O and a viewpoint V . Hence, the reduction of ∆s to being a single straight line is quite advantageous. Now let us take a look at the assignment of real object points to
VOPs. With a known viewpoint one can assign a real object point non-ambiguously to a VOP. This is possible since the ray passing through a viewpoint and a VOP allows the
direct computation of its angle of incidence. In the case of ˇOs, this in turn allows solving Equation 2.26 directly, which has been previously used to express the relation between the location of the virtual and the real object point.