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Reflections on Refractions Journal for Geometry and Graphics Volume VOL (YEAR), No. NO, 1{18. Reflections on Refractions Georg Glaeser, Hans-Peter SchrÄocker(Version September 5, 2001) Institute for Architecture, University of Applied Arts Vienna Oskar Kokoschka-Platz 2, A-1010 Wien, Austria email: [email protected] Abstract. In computer graphics, it is often an advantage to calculate refractions directly, especially when the application is time-critical or when line graphics have to be displayed. We specify formulas and parametric equations for the refraction on planes and spheres. The calculation of the \refrax" of a space point leads to algebraic equations. The counter image of straight lines with respect to refraction on a plane are speci¯c algebraic surfaces of degree four. Furthermore, we develop a general theory of refractions, with reflections as a spe- cial case. The manifold of all refracted rays is the normal congruence of certain algebraic surfaces. These surfaces and their diacaustic surfaces are investigated. Key Words: Refraction, reflection, curved perpectives, ¯sh-eye perspectives, dia- caustic, catacaustic, normal congruence, real-time rendering, underwater photog- raphy. 1. Introduction and state of the art Refractions are to be seen everywhere in daily life (+ example). They are also used for many technical purposes, most of all for eye glasses, optical lenses, etc. Despite of their importance, it seems that { apart from rather basic considerations { not much investigation has been done on the theoretical part of refractions. We will now briefly describe the results that are known so far. The mathamatical interest in refraction and reflection began more than 300 years ago, when Tschirnhaus and Huygens worked on this topic. They and { little later { Johann Bernoulli were especially interested in caustics, the hull curves of a one parameter set of rays that are reflected or refracted on a plane curve.1 Caustics produce nice optical e®ects because the light intensity is maximal along them. They also permit a deeper insight into reflection and refraction phenomena. The catacaustic of a pencil E(e) of rays with respect to a circle c, e.g., is an algebraic curve of class four. Thus, a circle (or a sphere) has theoretically four specular points.2 1The caustics of reflections are called catacaustics, those of refraction diacaustics. The term caustic refers to both cata- and diacaustics. 2It is quite remarkable that all four specular points can be of practical relevance (see [8]). ISSN 1433-8157/$ 2.50 c YEAR Heldermann Verlag, Berlin ° 2 G. Glaeser, H.P. SchrÄocker: Reflections on Refractions It also makes sense to investigate caustic surfaces, i.e., the focal surfaces of a two parameter set of rays refracted or reflected on a surface ([8, 10]). Luckily, the spatial problem can sometimes be reduced to a planar problem. The case were the set of rays being refracted is a pencil E(e) is of special interest. It serves as a 2D-model for human perception as well as for illumination of a scene with refracting objects. In [7] and [13], a general method of constructing caustics in this case was introduced: The refracted rays are all perpendicular to a hull curve h of certain circles. Thus, the caustic is the evolute of h.3 Since the days of Tschirnhaus and Huygens the caustics for many special cases were described by a number of authors ([4], [11], [14]). Caustics of higher order were also studied. The light rays are not only refracted once but twice or even more often on a certain curve or surface.4 In this context the theorem of Malus is important: A two parameter set of straight lines is called a line normal congruence when it is the set of normals of a surface. The theorem of Malus now states that a normal congruence remains normal after an arbitrary number of reflections or refractions.5 In [5], the caustics of a pencil of lines E(e) with respect to a plane curve c : : : ~x = ~x(t) were calculated in a general form. The same authors solved the problem of ¯nding the \anticaustic", e.g., the curve producing a certain given caustic in [6]. The given formulas, however, and the according di®erential equations are quite complicate. In [2], a new way of constructing the catacaustic of a pencil of lines E(e) was introduced: We regard the set of all conic sections osculating the reflecting curve c and having E as one focal point. Then the catacaustic is the locus of all focal points F = E of all conics. During the eighties the same authors (and some others as well) studied catacaustics6 from the completely di®erent point of view of singularity theory ([???],[???]). A lot of recent books and publications on computer graphics deal with the topic of reflection and refraction. Apart from rather basic considerations, however, they usually rely on ray- tracing methods or approximating calculations and hardly ever make use of the profound (but rather old and not well-known) theoretical background. Therefore we think it is time to present a new theory of reflection and refraction adapted to the needs of modern computer graphics. [MEHR LITERATURZITATE; ABBILDUNG (KONSTRUKTION EINER DIAKAUSTIK. ZWEI KONSTRUKTIONEN EINER KATAKAUSTIK] 2. The physical approach: Snell's Law, Fermat's principle In Euclidean 2-space IE2, we choose a straight refracting line s, an eye point E = s and a positive real fraction ratio r. For easier writing, [s; r; E] shall denote the refraction2 on s with ratio r with respect to E. (This notation is generalizationR of the notation introduced in [8] for reflections.) Physically speaking, r has the following meaning: When light propagates with speed c1 on the side of E, it propagates with speed c2 = c1=r on the other side of s. For r > 1, the side on E is \optically less dense". In Fig. 1, r 1:33 was chosen for the ratio of the light speed ¼ in the atmosphere (to the right) and water (to the left). 3In case of a reflection h is the orthonomic of the reflecting curve with respect to E. 4The problem of the n-th caustic of reflection on a circle, e.g., was solved in [9]. 5For a proof of this theorem, see [17]. Of course, an analogous theorem holds for the plane case. G. Glaeser, H.P. SchrÄocker: Reflections on Refractions 3 Figure 1: Refraction on a straight line With [s; r; E], we connect a Cartesian coordinate system as follows (Fig. 1): E is a point on theRpositive x-axis (position vector ~e = (e; 0)T ) and s is the y-axis. Due to the physical law of refraction (Snell's law), a straight line b1 (incidence angle ®1 to the normal of s) is refracted into a straight line b2 = [s; r; E](b1) through B = b1 s with R \ incidence angle ®2 according to the equation sin ®1 = r sin ®2: (1) ¼ ¼ Though in principle we have ®1; ®2 [ 2 ; 2 ], we have a restriction on either ®1 or ®2, 2 ¡ 1 depending on the ratio r: For r > 1, we have ®2 arcsin r , else we have ®1 arcsin r. j j max· ± j j · E.g., for r 0:75 (water air) we have ®1 ® = 48:5 . For small angles ®1, the ¼ ! j j · 1 percentage rate of reflected light is also small. When ®1 approaches the limiting angle, this max percentage rate increases, until, for ®1 ® , we have total reflection on s. j j ¸ 1 Figure 2: Who can see whom? As a consequence, ¯sh A in a calm pool (Fig. 2) will see \everyhing" outside the pool, though partly very distorted. The refracted image ¯lls a ² circle c on the surface that stems from a cone a revolution ¡ with apex angle 2 48:5±; £ 4 G. Glaeser, H.P. SchrÄocker: Reflections on Refractions the total reflections of those parts of the pool that are outside the reflected cone ¡¤ ² (e.g., ¯sh C); very dim reflections of the rest of the pool (e.g., ¯sh B) inside c as a result of partial ² reflection. When a person outside the pool takes a picture of the pool (e.g., from the spring board), the image will show all the ¯shes. 3. Refracting projecting rays through space points Snell's law does not explicitely require the position of the eye point. Nevertheless, we will now take into accont such a point, since we usually oberserve with our eye (or even two eyes, of course). Therefore, we will distinguish between projection rays through E and general rays. Let us take a simple example: From the border of a pool, we are watching a ¯sh swimming around. We all know that the ¯sh is not at the position we see it. Our goal is now to solve the two problems: 1. Given the position S of a point on the ¯shes surface, we are looking for the projection ray r through our eye that runs through S after being reflected on the pools plane surface σ. The intersection point R = r σ will be the key for the determination of r. \ 2. Given the point R, can we say anything about the spatial position of S? Well, of course we cannot with only one eye, but what if we look two-eyed? Speaking of ¯shes: We call the photographic images created by refracting optical ultra- wideangle lenses \¯sheye-perspectives". The creation of such curved perspectives is another motivation for the investigation of refractions. To cut longer sentences short, we will henceforth use a new word: De¯nition 1. The \image" R © of a point S under the influence on a refraction on a surface © R = [©; r; E](S) is2called refrax of S on ©.
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