Hoffmann et al. / Front Inform Technol Electron Eng in press 1 Frontiers of Information Technology & Electronic Engineering www.jzus.zju.edu.cn; engineering.cae.cn; www.springerlink.com ISSN 2095-9184 (print); ISSN 2095-9230 (online) E-mail: [email protected] Caustics of developable surfaces∗ Miklós HOFFMANN‡1,2, Imre JUHÁSZ3, Ede TROLL1 1Institute of Mathematics and Computer Science, Eszterházy Károly University, Eger, Hungary 2Department of Computer Graphics and Image Processing, University of Debrecen, Hungary 3Department of Descriptive Geometry, University of Miskolc, Hungary E-mail: hoff[email protected]; [email protected]; [email protected] Received Nov. 6, 2020; Revision accepted Mar. 17, 2021; Crosschecked Abstract: While considering a mirror and light rays coming either from a point source or from infinity, the reflected light rays may have an envelope, called a caustic curve. In this paper we study developable surfaces as mirrors. These caustic surfaces, described in closed form in the paper, are also developable surfaces of the same type as the original mirror surface. We provide efficient, algorithmic computation to find the caustic surface of each of the three types of developable surfaces (cone, cylinder and tangent surface of a spatial curve). We also provide a potential application of the results in contemporary free-form architecture design. Key words: Caustics; Developable surface; Reflected light rays; Curve of regression https://doi.org/10.1631/FITEE.2000613 CLC number: 1 Introduction design process (Tang et al., 2016). Optical studies frequently apply these free-form surfaces in lens de- When light rays are reflected from a curved mir- sign (Liu et al., 2012; Ponce-Hernández et al., 2020) ror, the following optical phenomenonunedited may be ob- or light-emitting diode (LED) illumination research served: the reflected light rays may possess an enve- (Wu et al., 2013). lope, called a caustic curve or surface. These caustics can appear in our everyday experience, such as on In engineering and architecture, this question is the surface of coffee in our coffee cup, but the also especially relevant in terms of developable surfaces, play an important role in the sciences, from physics i.e. curved surfaces that can be unfolded to (and to computer graphics (Arnold et al., 1985; Lock and therefore created from) a planar shape. There are ba- Andrews, 1992). Scientists – from the ancient Greeks sically three different types of developable surfaces. through Huygens to contemporary opticians, engi- Two of them are the well-known cone and cylinder, neers and geometers – have studied reflected and re- while the third one is a more general type, namely, fracted light rays from the theoretical point of view the tangent surfaces of spatial curves. From the com- as well as through various applications. putational point of view, this latter type is the most challenging one in engineering applications, but at In recent years these surfaces have been usually the same time this type provides much more freedom described as parametric free-form surfaces, typically in engineering design than the classical cones and as Bézier or B-spline surfaces in various applications cylinders (Seguin et al., 2021). These surfaces are to provide more freedom for users in the interactive used, among other applications, for creating special ‡ Corresponding author developable mechanisms (see, e.g., Greenwood et al. * This Rresearch was supported by the European Union and (2019) for cylindrical developables and Hyatt et al. cofinanced by the European Social Fund vide grant number EFOP-3.6.3-VEKOP-16-2017-00002 (2020) for conical developables). Non-developable c Zhejiang University Press 2021 surfaces are also frequently approximated by devel- 2 Hoffmann et al. / Front Inform Technol Electron Eng in press opable ones, for instance, to ease fabrication from 2 Developable surfaces as mirrors sheet metal (Liu et al., 2016). For an excellent overview of developable Bézier surfaces see the pa- As it is well-known, considering the curve per by Zhang and Wang (2006). In this paper we too rx (t) follow this construction. r (t) = ry (t) , t ∈ [a, b] (1) One of the most spectacularly evolving appli- rz (t) cation of developable surfaces can be found in ar- g t , t a, b chitecture, more precisely in the so-called free-form and direction ( ) ∈ [ ], the surface architecture. Due to evident mechanical and ma- s (t, u) = r (t) + ug (t) , u ∈ R (2) terial restrictions this field pay special attention to developable surfaces (Pottmann et al., 2015; Martín- is a ruled surface. The given curve is the directrix Pastor, 2019). of the surface, while for any fixed t0 ∈ [a, b] the lines The caustics of classical planar curves are well- r (t0) + ug (t0) are called the generators (or rulings). known and widely studied (see, e.g., Yates (1947) The surface is developable, if the normals of the tan- and Lockwood (1967)). In the case of surfaces, the- gent planes along the generators are of constant di- oretical results are also known. For a given surface rection, i.e., considering the partial derivatives a somewhat similar notion is the focal surface, the surface of centers formed by taking the centers of ∂ s (t, u) = ˙r (t) + u ˙g (t) , (3) the curvature spheres. The relationship between the ∂t ∂ focal and caustic surfaces is established (see, e.g., s (t, u) = g (t) (4) Pottman and Wallner (2000)) and it has been proven ∂u that the focal surface of a developable surface will be the normal vector developable of the same type as the original surface ∂ ∂ n (t, u) = s (t, u) × s (t, u) (Pottman and Wallner, 2000). This result, in the- ∂t ∂s ory, yields the same consequence in terms of caustic = (˙r (t) + u ˙g (t)) × g (t) (5) surfaces too, but these theoretical outcomes do not provide exact, constructive and algorithmic solutions does not depend on u for any t. In other words, to compute and display these surfaces in practical the tangent planes of the surface along its generators applications. For calculatingunedited the caustics of a given coincide. surface only numerical solutions have been provided Now let us consider a developable surface as a in a previous paper (Schwartzburg et al., 2014). mirror. Here we assume that the light source is point- Instead of numerical calculation, in this paper like, either being at infinity or not, and none of the we provide the exact computation and closed formu- generators of the developable surface goes through lae for the caustics of developable surfaces. These the light source. For any generator r (t0) + ug (t0) of caustic surfaces are of utmost importance in contem- the surface, incoming light rays meeting the mirror porary architecture (Pottmann et al., 2015), where surface along this generator are coplanar, and – due caustics may appear, e.g., as an outcome of the re- to the fixed tangent plane along the generator – the flected sunshine beams. reflected light rays are also coplanar. The plane of The structure of the paper is as follows. In Sec- reflected light rays is to be computed first. This plane tion 2, after a brief introduction to the basics of is fully determined by the selected generator and one developable surfaces, a general computation of the single reflected light ray. In the following subsections reflected light rays is provided for these surfaces. In the computation is presented separately for the case Section 3, this computation is specified and caustics when the light source is at infinity (yielding parallel are presented for each of the three different types of light rays) and for the case when the source of light developable surfaces: cones (Subsection 3.1), cylin- is a point of the affine space. ders (Subsection 3.2) and tangent surface of a spatial 2.1 Light source at infinity curves (Subsection 3.3). An application of our results in architectural design is presented in Section 4, and Without loss of generality of the forthcoming the Conclusions section closes the paper. computation, we can assume that the direction of Hoffmann et al. / Front Inform Technol Electron Eng in press 3 T parallel light rays is d = 1 0 0 . We further will also form a plane passing through this generator. assume that none of the generators is parallel to the To determine this plane it is enough to reflect one given direction. Let the selected generator of the single light ray, e.g., the one intersecting the directrix surface be r (t0) + ug (t0). curve r (t) at this generator. In other words, we The direction d(t0) ofthereflectedrayscanbe have to reflect the vector r (t0) (this is the direction computed by reflecting d with respect to the tan- of a light ray coming from the origin, i.e. r (t0) = gent plane along the selected generator with normal d (t0)) with respect to the normal vector n (t0) of the vector n (t0). Since the endpoint of d is on the unit tangent plane. sphere The endpoint of r (t0) = d (t0) is on the sphere 2 2 2 x + y + z = 1 (6) 2 2 2 2 x + y + z = r (t0) , (11) the endpoint of the reflected image d(t0) will also be on this sphere. Thus, we have to compute the therefore the endpoint of the reflected vector d (t0) intersection point of the unit sphere and the line also has to be on this sphere. Thus, we have to compute the intersection point x 1 nx (t0) r t of the line passing through ( 0) with the direction d + λn (t ) = y = 0 + λ ny (t ) 0 0 vector n (t0), i.e., of the line z 0 nz (t0) rx (t0) nx (t0) 1 + λnx (t0) r (t0) + λn (t0) = ry (t0) + λ ny (t0) = λny (t0) , (7) rz (t0) nz (t0) λnz (t0) rx (t0) + λnx (t0) where λ ∈ R.