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. Solving the equation   = ry (t0) + λny (t0) (12) 2 2 2  rz (t0) + λnz (t0)  (1 + λnx (t0)) + (λny (t0)) + (λnz (t0)) = 1, (8) one can find and the sphere (11). Solving the equation 2 2 −2nx (t ) r (t ) = r (t ) + λn (t ) (13) λ 0 0 0 0 = 2 2 2 +nx (t0) ny (t0) + nz (t0) one obtains 2nx (t0) = − 2 , if n (t0) = 0. (9) n (t0) unedited 2n (t ) r (t ) + 2n (t ) r (t ) λ x 0 x 0 y 0 y 0 = − 2 2 2 Therefore, the direction of the reflected rays +nx (t0) ny (t0) + nz (t0) along this generator is 2nz (t0) rz (t0) − 2 2 2 +nx (t0) ny (t0) + nz (t0) 2nx(t0) 1 − n 2 nx (t0)  (t0)  2r (t0) n (t0) 2nx(t0) = − , if n (t0) = 0. (14) =d (t0) − n 2 ny (t0) 2  (t0)  n (t0)  2nx(t0)  − 2 nz (t0)  n(t0)  The reflected vector is 2nx (t0) = d − 2 n (t0) . (10) n (t0) 2r (t0) n (t0) =d (t0) r (t0) − 2 n (t0) , (15) 2.2 Light source in an affine point n (t0)

Without loss of generality of the foregoing com- which - together with the selected generator - deter- putation, we can assume that the source of light is mines the plane of reflected rays. in the origin of the coordinate system. We further 2.3 The family of planes of reflected rays and assume that none of the generators passes through their envelope surface this point. Again, let the selected generator of the surface be r (t0) + ug (t0). In the preceding subsections we have com- As we have observed, light rays intersecting this puted the vector d (t0) which is necessary to deter- generator form a plane, and the reflected light rays mine the plane of reflected rays at each generator 4 Hoffmann et al. / Front Inform Technol Electron Eng in press

r (t0) + ug (t0) of the developable surface. Forth- generators of the envelope surface passing through coming computations are independent of the actual the points of r (t). This direction can be computed position of the light source. as The reflected rays form a one-parameter family f (t) = g (t) × d (t) of planes ˙ × g˙ (t) × +d (t) g (t) × d (t) P (u, v, t) = r (t) + ug (t) + v d (t) , (16) = g (t) × dg (t) × ˙ (t) × d (t) ˙ R + g (t) × d (t) × g (t) × d (t) . (25) where t ∈ [a, b] is the family parameter and u, v ∈ . It is easy to see that the envelope surface of this To simplify this expression we apply the formula one-parameter family of reflected light planes is also developable (cf. Do Carmo (2016)). This envelope a × (b × c) = (a c) b − (a b) c. (26) surface is the caustic surface of the original mirror surface. Accordingly, the two terms of the expression (25) can In what follows, we will determine this caustic be written as surface in the standard form of developable surfaces g (t) × dg (t) × ˙ (t) × d (t) analogous to Eq. (2), i.e., in the following form: = g (t) × d (t) dg (t) ˙ (t) e (t, λ) = q (t) + λf (t) , t ∈ [a, b] , λ ∈ R, (17) − g (t) × dg (t) ˙ (t) d (t) where q (t) is some curve and f (t) is the direction = − g (t) × dg (t) ˙ (t) d (t) (27) of the generators passing through the points of this curve. We apply the computations and ˙ g (t) × d (t) × g (t) × d (t) ∂ P (u, v, t) = g (t) , (18) ˙ ∂u = g (t) × d (t) d (t) g (t) ∂ P (u, v, t) = d (t) , (19) − g (t) × d (t) g (t) d˙ (t) ∂v ∂ ˙ ˙ P (u, v, t) = ˙r (t) + u ˙g (t) + v d (t) , (20) = g (t) × d (t) d (t) g (t) . (28) ∂t unedited and Therefore, the direction of the generators of the en- velope surface can simply be written in the following det ∂ P (u, v, t) ∂ P (u, v, t) ∂ P (u, v, t) form: ∂u ∂v ∂t g t dr t ˙ t u ˙g t vd˙ t , ˙ = ( ) × ( ) ( ) + ( ) + 0 ( ) = f (t) = g (t) × d (t) d (t) g (t) (21) − g (t) × dg (t) ˙ (t) d (t) . (29) from which, we get the equality Now, if one can fix u and v in a way that the g (t) × dr (t) ˙ (t) + g (t) × dg (t) u ˙ (t) determinant Eq. (22) vanishes for ∀t ∈ [a, b], then ˙ u v + g (t) × d (t) v d (t) =0. (22) by substituting these values of and into Eq. (16) we obtain a curve (or a constant vector) q (t), based The normal vector of the plane P (u, v, t) is on which the caustic envelope surface can be written in the classic form of developable surfaces as g (t) × d (t) (23) e (t, λ) = q (t) + λf (t) , t ∈ [a, b] , λ ∈ R. (30) the derivative of which is ˙ ˙g (t) × +d (t) g (t) × d (t) . (24) 3 Caustics of developable surfaces

The cross product of the normal vector in Eq. (23) As is well-known, there are three types of devel- and its derivative Eq. (24) is the direction of the opable surfaces: (generalized) cones, (generalized) Hoffmann et al. / Front Inform Technol Electron Eng in press 5 cylinders and the tangent surfaces of a spatial curves. Based on the calculations above, the caustic en- In this sections we specify the general computations velope surface of the cone can be written in the form of Section 2 to the three different types of developable R surfaces. e (t, λ) = p + λf (t) , t ∈ [a, b] , λ ∈ , (36)

3.1 Cones as mirrors where p is the apex and f (t) denotes the direction of generators. As we have discussed in the Introduction, con- An example can be seen in Fig. 1. sidering a cone as a mirror, defined by the curve r(t) and apex p, and light rays with direction d(t), the caustic surface, i.e., the envelope surface of the re- flected light rays with direction d(t) isalsoacone (see Pottman and Wallner (2000)). Herein present the exact computation of the caustic cone surface, and provide the solution in a closed form. In the case of a conic mirror surface the curve r(t) of Eq. (2) is an arbitrary planar or spatial curve, Fig. 1 The family of planes of reflected rays (yellow) while the generators can be written as and their envelope caustic surface (orange) in the case of a conic mirror (blue) defined by a cubic Bézier curve (red). Note: the direction of the incoming light g (t) = p − r (t) , t ∈ [a, b] (31) rays (yellow) is also shown. where p, the apex of the cone, is an arbitrary point 3.2 Cylinders as mirrors out of the curve r. The family of planes in Eq. (16) will be of the specific form In theory, it is known that considering a cylin- der, defined by the curve r(t) and direction a, as a P (u, v, t) = r (t) + u (p − r (t)) + v d (t) , (32) mirror and light rays with direction d(t), the caus- tic surface, i.e., the envelope surface of the reflected where t ∈ [a, b] and u, v ∈ R. light rays with direction d(t) is also a cylinder (see In this case, the determinant Eq. (22) will be Pottman and Wallner (2000)). In this section we (p − r (t)) × dr (tunedited) ˙ (t) present the computation of this caustic surface and provide a closed form of it. − (p − r (t)) × dr (t) u˙ (t) If the mirror surface Eq. (2) is a cylinder, then ˙ + (p − r (t)) × d (t) vd (t) the curve r(t) is an arbitrary planar or spatial curve, = (p − r (t)) × 1d (t) ( − u) ˙r (t) and the direction of the generators g(t) is constant, i.e. + (p − r (t)) × d (t) vd˙ (t) , (33) g (t) = a, t ∈ [a, b] , (37) which vanishes for ∀t ∈ [a, b] if u = 1, and v = 0. where a is a vector (not parallel to the plane of r Substituting these parameter values into Eq. (32) when r(t) is planar). The family of planes in Eq. we obtain (16) of the reflected rays can be written in the form

P (u, v, t) = r (t) + p − r (t) = p, (34) P (u, v, t) = r (t) + ua + vtd (t) , ∈ [a, b] , u, v ∈ R. (38) i.e., the caustic envelope surface is also a cone with The direction of generators of the caustic enve- apex p. The direction of the generators can be writ- lope surface will be ten as f (t) = a × d (t) d˙ (t) a a d t 0 d t f (t) = (p − r (t)) × d (t) d˙ (t) (p − r (t)) − × ( ) ( ) ˙ + (p − r (t)) × dr (t) ˙ (t) d (t) . (35) = a × d (t) d (t) a, (39) 6 Hoffmann et al. / Front Inform Technol Electron Eng in press i.e., f (t) is parallel to a, ∀t ∈ [a, b]. This means that surface of the reflected light rays with direction d(t) the caustic surface of a cylindrical mirror is also a has been proven to be a tangent surface (see Pottman cylinder, and the generators of the two cylinders are and Wallner (2000)), but exact construction of this parallel. surface is not known. Now we present the computa- Determinant (22) becomes tion of the caustic surface and provide the solution in a closed form. 0 = a × dr (t) ˙ (t) + a × d (t) u0 If the developable mirror surface in Eq. (2) is a ˙ + a × d (t) vd (t) general tangent surface of a spatial curve, then r(t) is an arbitrary spatial curve, while the direction of = a × dr (t) ˙ (t) + a × d (t) v d˙ (t) , (40) the generators fulfills the relationship from which one can express v as g (t) = ˙r (t) , t ∈ [a, b] . (45) − a × dr (t) ˙ (t) = a × d (t) v d˙ (t) , (41) In this case, the determinant (22) is of the form a × dr (t) ˙ (t) v = − , (42) ˙ a × d (t) d˙ (t) ˙r (t) × dr (t) u¨(t) + ˙r (t) × d (t) v d (t) , (46) if a × d (t) 0d˙ (t) = . and it vanishes for ∀t ∈ [a, b] at u = v = 0. This im- mediately yields the somewhat surprising fact that the caustic envelope surface contains the original curve r(t). The normal vector of the plane P (u, v, t) is

˙r (t) × d (t) , (47)

while its derivative is

¨r (t) × +d (t) ˙r (t) × d˙ (t) . (48) Fig. 2 The family of planes of reflected rays (yellow) and their envelope caustic surface (orange) in case In this case the direction of the generators of the of cylindric mirror (blue). Note: The direction of caustic envelope surface can be written as incoming light rays (yellow) is also shown. uneditedf (t) = ˙r (t) × d (t) dr˙ (t) ˙ (t) Substituting the calculated v and u = 0 into − ˙r (t) × (d (t) t) ¨r (t) d (t) , (49) Eq. (22), it vanishes for ∀t ∈ [a, b]. Substituting the same v and u = 0 to the family of plane Eq. (38) the and the caustic envelope surface is of the form following curve is obtained: e (t, λ) = r (t) + λf (t) , t ∈ [a, b] , λ ∈ R. (50) a × dr (t) ˙ (t) h (t) = r (t) − d (t) . (43) ˙ a × d (t) d (t) Based on the calculations presented above, the caus- tic surface of the cylinder can be written in the form e (t, λ) = h (t) + λa, t ∈ [a, b] , λ ∈ R. (44) A cylindrical mirror and its caustic surface can be seen in Fig. 2. Fig. 3 The family of planes of reflected rays (yellow) and their envelope caustic surface (orange) in the case 3.3 Tangent surface of a spatial curve as a mir- of a tangent surface of a spatial curve as a mirror ror (blue). Note: The direction of incoming light rays (yellow) is also shown. Considering a general tangent surface of a spa- tial curve r(t) as a mirror and light rays with di- It is clear from this expression that the caustic rection d(t), the caustic surface, i.e., the envelope surface is a ruled surface. However, our aim is to Hoffmann et al. / Front Inform Technol Electron Eng in press 7 prove that it is a developable surface, more specif- therefore ically, that it is a tangent developable surface of a −˙r (t) ¨r (t) × d (t) spatial curve. λ (t) = , (58) ˙ ˙ To reach this aim, we have to find the curve of f (t) ¨r (t) × +d (t) ˙r (t) × d (t) regression, i.e., a curve c (t) , t ∈ [a, b], the tangents of which are parallel to the directions f (t) of the whenever generators of the surface for ∀t ∈ [a, b]. ˙ f˙ (t) ¨r (t) × +d (t) ˙r (t) × 0d (t) = (59) Since each curve on the ruled surface in Eq. (50) can be considered a functional translation of holds. Here, we prove that this scalar product cannot the curve r (t) along the rulings, we search for the be equal to zero. curve c(t) in the following form: The first factor of the scalar product in Eq. (59), f˙ t 0 c (t) = r (t) + λ (t) f (t) , (51) i.e., ( ) = , since the generators of the surface are not parallel (it is not a cylinder). Thus, f˙ (t) can be where the function λ (t) , t ∈ [a, b] is to be deter- written in the following form: mined. Moreover, the derivative of this curve with d ˙ respect to t is f˙ (t) = r˙ (t) × d (t) dr (t) ˙ (t) dt ˙ ˙ ˙ + ˙r (t) × d (t) dr (t) ¨(t) ˙c (t) = ˙r (t) + λ (t) f (t) + λ (t) f (t) , (52) d − ˙r (t) × dr (t) ¨(t) d (t) which must be parallel to f (t), i.e., ˙c (t) must be or- dt thogonal to vectors (47) and (48). These conditions − ˙r (t) × dr (t) ¨(t) d˙ (t) yield the following system of equations: d ˙ = r˙ (t) × d (t) dr (t) ˙ (t) ˙c (t) ˙r (t) × 0d (t) = , (53) dt d ˙ − ˙r (t) × dr (t) ¨(t) d (t) (60) ˙c (t) ¨r (t) × +d (t) ˙r (t) × 0d (t) = . (54) dt ˙ + ˙r (t) × d (t) dr (t) ¨(t) From the first equation, i.e., Eq.53, we obtain the ˙ following equality: − ˙r (t) × dr (t) ¨(t) d (t) (61) unedited 0 = ˙r (t) + λ˙ (t) f (t) + λ (t) f˙ (t) ˙r (t) × d (t) Since Eq. (60) is the linear combination of ˙r (t) and d (t), this term is parallel to the plane = λ (t) f˙ (t) r˙ (t) × d (t) , (55) spanned by ˙r (t) and d (t), which is actually the plane while from the second expression, we get P (u, v, t). Applying identity (26) to expression (61), one ˙ ˙ can find 0 = ˙r (t) + λ (t) f (t) + λ (t) f (t) ˙ ¨r (t) × +d (t) ˙r (t) × d˙ (t) ˙r (t) × d (t) dr (t) ¨(t) ˙ ˙ = r˙ (t) ¨r (t) × +d (t) ˙r (t) × d (t) − ˙r (t) × dr (t) ¨(t) d (t) ˙ ˙ ˙ = r˙ (t) × dr (t) × ¨(t) × d (t) . (62) + λ (t) f (t) ¨r (t) × +d (t) ˙r (t) × d (t) = r˙ (t) ¨r (t) × d (t) + λ (t) f˙ (t) However, this cross product is evidently orthogonal ˙ to one of its factors, i.e., ¨r (t) × +d (t) ˙r (t) × d (t) , (56) ˙r (t) × dr (t) × ¨(t) × dr˙ (t) ⊥ ˙ (t) × d (t) , which further yields (63) − ˙r (t) ¨r (t) × d (t) therefore, Eq. (61) is also parallel to the plane (or- ˙ thogonal to the normal vector of the plane) spanned = λ (t) f˙ (t) ¨r (t) × +d (t) ˙r (t) × d (t) , (57) by ˙r (t) and d (t), i.e., to P (u, v, t). We obtained that 8 Hoffmann et al. / Front Inform Technol Electron Eng in press both terms of f˙ (t) (Eqs. (60) and (61)), thus f˙ (t) it- self, is parallel to the plane P (u, v, t) and orthogonal to its normal vector, i.e., f˙ (t) ⊥ ˙r (t) × d (t) . For the second factor of the scalar product in Eq. (59), the equality

d Fig. 4 The caustic surface (orange) in the case of a ¨r (t) × +d (t) ˙r (t) × d˙ (t) = (r˙ (t) × )d (t) dt tangent surface of a spatial curve as a mirror (blue) (64) defined by a cubic Bézier curve (red) and the regres- holds; therefore, this term describes the change of sion curve (orange) of the caustic surface. Note: The direction of incoming light rays (yellow) is also shown. the normal vector ˙r (t) × d (t) of the plane P (u, v, t). As a consequence, the scalar product Eq. (59) could be equal to zero only if either the second factor, 4 An application: developable surfaces i.e., the derivative of the normal vector ˙r (t) × d (t), in architecture is the nullvector or the normal vector is parallel to Developable surfaces, especially the tangent sur- its derivative. Both would yield that the planes faces of a curve are commonly used in modern archi- P (u, v, t) coincide or they are parallel for ∀t, i.e., the tecture (for an overview, see, e.g., Pottmann et al. curve r (t) is a planar curve. But, this contradicts (2015); Glaeser and Gruber (2007)). If the surface our assumption that we are studying the tangent of such a construction is made from a reflective ma- surfaces of spatial curves. terial (e.g. metal sheets, such as at the Guggenheim Back to the system of Eqs. (53) and (54), we Museum in Bilbao), then the surface may behave like have expressed λ (t) from Eq. (54), but this must a mirror. When a free-form building or a sculpture fulfill Eq. (53) as well. reflects light rays, some places around the structure In other words, – where light rays are concentrated – can be of high temperature. This especially holds around the cusp 0 = λ (t) f˙ (t) ˙r (t) × d (t) (65) of the intersection curve of the caustic surface of the building and the ground plane. Our aim is to find those "hot" points of the ground plane. Without the must hold. But, this equationunedited indeed holds, since loss of generality, we can assume that this plane is we have just seen that f˙ (t) ⊥ ˙r (t) × d (t) , i.e., the the coordinate plane [x, y]. scalar product above equals zero for ∀t. The intersection of the caustic surface e (t, λ) Based on the computation presented above, the and the [x, y] plane is a curve, as follows: caustic surface of a tangent surface can be computed as a tangent surface of the spatial curve c (t) = r (t)+ e (t, λ) ∩ [x, y] = ec(t). (68) λ (t) f (t), where

f (t) = ˙r (t) × d (t) dr˙ (t) ˙ (t) − ˙r (t) × dr (t) ¨(t) d (t) , (66) and

Fig. 5 The caustic surface (orange) in the case of a −˙r (t) ¨r (t) × d (t) tangent surface of a spatial curve as a mirror (blue) λ (t) = . (67) ˙ ˙ defined by a cubic Bézier curve (red) and the regres- f (t) ¨r (t) × +d (t) ˙r (t) × d (t) sion curve (orange) of the caustic surface. Notes: The caustic surface intersects the [x, y] plane in the curve ec(t) (green). The curve of regression hits the [x, y] An example of this type of mirror and its caustic plane in the cusp of the curve ec(t). The direction of surface can be seen in Fig. 4. the incoming light rays (yellow) is also shown. Hoffmann et al. / Front Inform Technol Electron Eng in press 9

This curve can be described as Do Carmo MP, 2016. Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition. ec(t) = e (t, λ (t)) , (69) Courier Dover Publications. Glaeser G, Gruber F, 2007. Developable surfaces in contem- where λ (t) is the solution of the equation e (t, λ) = 0 porary architecture. Journal of Mathematics and the Arts, 1(1):59-71. for the parameter t. This curve usually has a cusp Greenwood JR, Magleby SP, Howell LL, 2019. Developable e (t0, λ0) which can be very hot due to physical rea- mechanisms on regular cylindrical surfaces. Mechanism sons. Applying standard methods, in most cases this and Machine Theory, 142:1-13. Hyatt LP, Magleby SP, Howell LL, 2020. Developable mech- cusp cannot be directly computed exactly. Based on anisms on right conical surfaces. Mechanism and Ma- our results, however, this cusp can simply be com- chine Theory, 149:1-13. puted as the intersection of the curve of regression of Liu P, Wu Rm, Zheng Zr, et al., 2012. Optimized design of led the caustic surface and the [x, y] plane (see Fig. 5), freeform lens for uniform circular illumination. Journal of Zhejiang University SCIENCE C, 13(12):929-936. as follows: Liu X, Li S, Zheng X, et al., 2016. Development of a flattening system for sheet metal with free-form surface. e (t0, λ0) = c (t) ∩ [x, y] . (70) Advances in Mechanical Engineering, 8(2):1-12. Lock JA, Andrews JH, 1992. Optical caustics in natural phenomena. American journal of physics, 60(5):397- 5 Conclusions 407. Lockwood EH, 1967. A book of curves. Cambridge Univer- The caustics of developable surfaces have been sity Press. studied in this paper. In the theory of developable Martín-Pastor A, 2019. Augmented graphic thinking in surfaces, it is known that the caustics of each of the geometry. developable architectural surfaces in experi- mental pavilions. Graphic Imprints, :1065-1075. three different types of developable surfaces are also Ponce-Hernández O, Avendaño-Alejo M, Castañeda L, 2020. developable surfaces of the same type. These caus- Caustic surface produced by a simple lens consider- tic surfaces have been expressed in an exact, closed ing a point source placed at arbitrary position along form. In the case of the tangent surface of a spatial the optical axis. Nonimaging Optics: Efficient De- sign for Illumination and Solar Concentration XVII, curve, the curve of regression of the caustic surface 11495:114950A. has also been computed. This curve is applied in Pottman H, Wallner J, 2000. Computational Line Geometry. a practical computation of finding the cusp of the Springer. Pottmann H, Eigensatz M, Vaxman A, et al., 2015. Archi- intersection curve of a caustic surface and a plane, tectural geometry. Computers & Graphics, 47:145-164. which may appear in applicationsunedited such as free-form Schwartzburg Y, Testuz R, Tagliasacchi A, et al., 2014. High- architecture. contrast computational caustic design. ACM Transac- tions on Graphics (TOG), 33(4):74. Seguin B, Chen YC, Fried E, 2021. Bridging the gap be- Contributors tween rectifying developables and tangent developables: The authors, Miklós HOFFMANN, Imre JUHÁSZ and a family of developable surfaces associated with a space Ede TROLL contributed equally, on a shared basis to the curve. Proceedings of The Royal Society A, 477:1-14. Tang C, Bo P, Wallner J, et al., 2016. Interactive design of research, jointly developing the mathematical computations developable surfaces. ACM Transactions on Graphics and creating the experimental figures. (TOG), 35(2):1-12. Wu RM, Liu P, Zhang YQ, et al., 2013. Ray targeting Compliance with ethics guidelines for optimizing smooth freeform surfaces for led non- rotational illumination. Journal of Zhejiang University The authors declare that they have no conflict of inter- SCIENCE C, 14(10):785-791. est. Yates RC, 1947. A handbook on curves and their properties. JW Edwards, Ann Arbor. References Zhang XW, Wang GJ, 2006. A new algorithm for design- Arnold VI, Guse˘ın-Zade SM, Varchenko A, 1985. Clas- ing developable bézier surfaces. Journal of Zhejiang sification of critical points, caustics and wave fronts. University-SCIENCE A, 7(12):2050-2056. Springer.