Fabrication of Moiré on Curved Surfaces

: Moiré is an appealing visual eﬀect observable when two or more repetitive patterns are superposed. Fabrication of moiré eﬀects has already proven to be useful in a range of applications, from art to engineering. Here, we introduce a method for designing and fabricating curved surfaces with level-line moiré shapes. These moiré shapes are obtained by superposing a partly absorbing layer and a layer formed by an array of cylindrical lenses or by two layers of cylindrical lenses. We formulate the problem of placing an array of cylindrical lenses on a curved surface as a design problem having a low number of dimensions. The range of possible solutions can therefore be explored by a human observer. We demonstrate the quality of our method by rendered simulations and by fabrication. The resulting static displays can be manufactured using diﬀerent fabrication techniques, from multi-material 3D printing to molding.


Introduction
Moiré is an intriguing phenomenon observable when spatial patterns are superposed [1].It has broad implications in a range of scientific disciplines, from optical interferometry [2] to atomic lattice interaction [3].Thanks to Moiré theory [1], one can create and fabricate fascinating visual effects.Physical realization of moiré patterns has many applications from art and decoration to anti-counterfeiting.However, synthesis and fabrication of moiré has until now strictly remained two-dimensional.We foresee a great potential in extending moiré applications to a wide range of real-life, non-planar objects thanks to recent advances in 3D fabrication techniques.
In the present contribution, we introduce curved surface moiré and explore different aspects of its design and fabrication.We express the layout of the lenses on the curved surface as a two-dimensional design space that can be explored subjectively.We simulate compelling moiré effects and fabricate them primarily using multi-material 3D printing.For a special single-material type of moiré, coined dual-lens moiré [4,5], we use additional fabrication techniques, such as molding, hinting at the potential of our method for mass manufacturing.

Related Work
We briefly review several papers that are most closely related to our work.
Moiré Synthesis Printing is the most common way of creating a wide range of moiré effects.Usually, a revealer layer (often a grating of transparent lines) is printed and superposed on a base layer incorporating repetitive structures.The superposition results in a message, motif or image that can be animated by having the revealer sampling successive positions of the base.
Orientations, periods as well as the moiré intensity profile of moiré fringes can also be modeled by considering the superposition of the layers as a multiplication in the spatial domain, and therefore as a convolution in the spatial-frequency domain [6], [ [1], Chapters 2,4].The so-called 1D moirés is obtained by smoothly shifting the revealing layer on top of the base layer.This yields as moirés animated symbols and motives having different speeds and orientations [7].Based on the theory of level line moiré [8], one can create appealing beating shapes by carefully designing a base layer from an input image.The use of 2D lens arrays to act as a revealer in order to obtain the moiré has been long known [9,10].Recently, micro-fabrication techniques have been used to realize moiré patterns where the revealing layer [11,12] or both the revealing and base layers [4,5] are made of microlens arrays.Our work can be thought of as the extension of lens-based, level-line moiré to curved surfaces.

Synthesis of Caustics
Caustics are obtained by devices that reflect or refract light into a surface according to a desired intensity profile, for example a multi-intensity image.Researchers fabricated slabs that guide the incoming light to form a desired distribution on a receiving surface through optimizing the slopes of a set of microfacets ( [13]).This work is later extended to refractive surface made of a collection of curved patches [14].Additional optimizations are introduced to stress the importance of surface continuity in order to improve the contrast of the resulting caustics [15][16][17].We will show that high-quality curved surface moirés also require the continuity of the underlying geometry.Unlike the static intensity images generated by these methods, the surface moiré exhibits a dynamic nature and can be embedded in a wide range of curved surfaces.

Light Field Display Fabrication
The moiré view-dependent characteristics can be best described using light fields.Our fabricated device belongs to the automultiscopic display category [18] making it, to the best of our knowledge, the first free-form automultiscopic display.Automultiscopic displays provide 3D vision cues, such as disparity and motion parallax, without cumbersome, special glasses.A predominant automultiscopic technology is based on parallax barriers that split the rays emitted towards the eyes.Our work is closely related to light-field displays embodied by lenticular prints [19].Compared to the opaque barriers, lenticular lenses offer a high light throughput efficiency.We draw insights from this work but show that its naive extension to surfaces does not yield high quality moirés.

Paper Overview
Of particular interest in the context of this manuscript, is a special class of moiré, namely, the level-line moiré, see [20] and [ [1], Chap.11].Readers unfamiliar with the level-line moiré are referred to Appendix A. In Section 2, we explain the principles of lens-based level-line moiré design, accounting for free parameters such as lens radius or focal length.We describe our method for surface moiré design in Section 3. In Section 4 we introduce our method of positioning the base and revealing layer surfaces relative to each other.In Section 5, we show that our framework is extendable to a special type of single-material moiré where both base and revealing moiré layers are made of cylindrical lens arrays [4,5].In Section 6 we show a variety of examples, both simulated and fabricated, demonstrating high-quality moiré effects on different curved surfaces.
Given the dynamic nature of the moiré, we recommend the reader to watch the accompanying video for viewing the simulated and fabricated results.

Lens-based Planar Moiré
In level-line moiré, the base layer is calculated from an elevation profile extracted from a letter, motif or image (Figure 1a).It is formed by horizontally laid out replicated line patterns, shifted vertically in proportion to the elevation at that location (Figure 1b).Superposing a rectilinear line grating made of transparent lines (Figure 1c) on this base layer produces a moiré whose constant colors or intensities follow the level set of the elevation profile, i.e. its level lines (Figure 1d).
In the lens-based planar moiré, the base layer (Figure 2b) is made of unidirectional structures having the same orientation as the revealer's cylindrical lenses, shifted in proportion to an  elevation profile.An enlargement of the base-layer structure is shown in Figure 2c.The base must have the same repetition period as the lens pitch, resulting in a one to one correspondence between cylindrical lenses and base layer bands.The superposition of the cylindrical lens array on the base layer yields the moiré.In this moiré, the level lines of the elevation profile are displayed as constant intensity lines (Figure 2, bottom row).This type of moiré is intriguing, since by tilting the device formed by base and revealer, the moire shape is subject to a dynamic beating effect.A lens array (Figure 3a) reveals the moiré with a much higher light efficiency than a black and clear grating.While the grating blocks a portion of the light, the lens array only redirects it.Recently, the use of flat cylindrical lens arrays for creating high-quality 1D moirés [4,11] and level-line moirés [5] has been demonstrated.Here, we review the relevant equations of a lens system and derive the relationship between lens parameters [5].
In a cylindrical lens (Figure 3b), there is a relationship between the lens radius , the lens pitch (or width) and its sag-height ℎ: The focal length is given by [21]: where and are the indices of refraction of the lens material and of the air, respectively.For many polymers, including our materials, we can assume a refractive index of 1.5, thus = 3 .
For the synthesis of a revealer, once the lens pitch has been set according to the base layer repetition period, the only additional parameter that can be freely chosen within certain bounds is the lens curvature radius .The curvature radius defines the angular field of view, see Section 3.2.
The sag-height ℎ enables obtaining the center of the lens surface, useful for creating the mesh that is used for fabrication.The sag-height is obtained from Equation 1: In contrast to moirés on curved surfaces, only a single cylindrical lens pitch is present on level-line moirés laid out on a planar surface.

From 2D to 3D
The input to our method is a 2D base layer derived from an elevation profile of a source image, and a curved surface.In the present work, we limit the discussion to parametric surfaces.In the 3D settings, we call the virtual surface that goes through the intersection points of lens arcs (blue dots in Figure 4a) the lens-pitch surface or simply the pitch surface.We call the surface of the cylindrical lens array, i.e. the top interface of the moiré device with air, the lens-arc surface or simply the lens surface.Finally, the base-layer surface or base surface is the moiré base-layer in 3D.Our proposed surface moiré design takes three steps: 1. Constructing the lens-pitch surface by mapping the planar lens-pitch to the desired surface (step 1 in Figure 4a).
2. Constructing the lens-arc surface by fitting circular arcs to neighboring lens-pitch samples (step 2 in Figure 4a).
3. Constructing the base-layer surface from the lens-pitch surface (step 3 in Figure 4a).
In the following, we explain each step in more details.

Lens-pitch Surface
In the first step of creating the surface moiré, assuming the input surface is parametric, the lens-pitch surface P is described by where f(u,v) is a vector function mapping 2D planar positions ( , ) onto ( , , ) locations in 3D space.When mapping the equidistant samples of the planar lens-pitch from the parameter domain ( , ) into the lens-pitch surface, distortions are introduced to the formerly equal pitches, resulting in lenses with different focal properties on the surface.It is well known from differential geometry [22] that an isometric parameterization exists only for developable surfaces.Therefore, this distortion is in general unavoidable.

Lens-arc Surface
Having computed the spatially-varying lens width, determined by the points of the lens-pitch surface, the next step is to fit the lens-arc surface to this underlying grid.For each individual lens pitch from the previous step, there is still one degree of freedom: the lens radius .At the first glance, one can use this degree of freedom to fit equi-radius arcs to the spatially-varying widths (Figure 4b).This has the desirable effect of making the focal length of all lenses equal.The downside, however, is that this leads to lenses with a wide range of angular fields of view that degrade the quality of the resulting moiré.
We use instead this degree of freedom to fit arcs with radii that generate lenses with equal angular field of views (AFOV or ) (Figure 4c).For a fixed angular field of view , Eq. ( 5) shows that the lens radius is proportional to the lens width .The focal length is therefore also proportional to the lens width .According to Fig. 3b, . (5)

Base-layer Surface
Having synthesized the lens-arc surface, the final and most important step for creating the surface moiré is to create the base-layer surface.The main challenge is that the inevitable distortions in the lens-pitch surface have an effect on the lens-surface, resulting in spatially-varying lens radii.
A careful look at designing flat lenticular (cylindrical) prints [19] reveals a similar challenge: how to adjust the lenses to the spatial frequency of their base?They adapt the focal length of each lens to its radius while keeping the base-layer facets perfectly parallel to the planar lens-pitch layer facets.Although the curved lens-pitch surface of curved surface moirés differs from the flat lens-pitch surface of planar lenticular prints, the problem is similar.
We therefore compute the base-layer surface from the lens surface in the same manner as in the case of adaptive lenticular prints [19], by placing the surface patches parallel to the corresponding lens-pitch elements at a distance corresponding to their focal length.A typical cross section of the the resulting construction is shown in Figure 5a, in this case for a saddle.The rendered simulation of the resulting lens-based moiré is shown in Figure 5b (details of our renderings are discussed in Section 6).
Despite each individual lens focusing on a parallel surface segment, the discretization in the base layer introduces aliasing-like artifacts in the form of undesired moiré noise.In the next section, we put forward our solution for designing optimal, smooth base-layer surfaces.

Optimal Base-layer Surface
The observations in Section 3.3 leads to our main insight in designing surface moiré: the optimal base-layer surface must be parallel to the lens surface and stay in focus while avoiding discrete jumps.However, parallelity and variable focality, for most continuous surfaces, are conflicting objectives (Figure 5c).A notable exception is the plane where the base layer is both parallel and perfectly in focus, explaining the high quality of the planar moiré.Thus, for a given lens surface, we propose to compute its corresponding focal surface, i.e. the surface that is smoothly in focus as well as its offset surface(s), i.e. surfaces whose elements are strictly parallel to the corresponding lens-pitch surface elements.We conjecture that the optimal base-layer surface lies in between these two surfaces.Therefore, we introduce a formulation that embeds the optimal base surface in a low dimensional design space thereby enabling its rapid exploration.

Offset Surface
Given our lens-pitch surface P (Equation 4), the offset surface Q of P is defined as a surface whose points are at a distance from their corresponding points on P where the parameter implies that Q is not unique, and ( , ) is the unit surface normal vector of P defined as with = / and = / being the partial derivatives of .

Focal Surface
For a given lens surface, its corresponding focal surface is a smooth surface continuously in focus.We compute the focal surface as a continuous function of the lens-pitch surface assuming that any point on the lens-pitch surface is the midpoint of the two end points of a lens.If all such lenses on the lens surface had the same default radius (and default pitch ), the focal surface would be equal to one of the offset surfaces with a distance that could be calculated using Equation 2. Due to the distortion, the radius changes along the surface depending on the surface curvature.Given the AFOV is fixed in the lens-arc surface construction stage, both the standard pitch and its corresponding radius are distorted with the same factor.According to geometric considerations (Figure 3b), we have / = / .Therefore the focal surface R is formulated as:

Interpolated Base Surface
We explore the space between the offset surfaces Q and the focal surface R using a bi-linear interpolation in order to find the base surface S with the most visible moiré effect.Given that the focal surface is uniquely defined, the design space is made of two free parameters, the relative weight ( ) between offset and focal surface and the offset ( ).We have therefore The most important implication of Equation 8is that the base layer's design space has only two dimensions.This is very convenient as, in the absence of a reference, the user can simply walk this space and find the two parameters that create the most satisfying effect.According to Figure 7, the moiré with the base surface identical with the offset surface (i.e.= 0) seems superior to the other moirés.

Dual-lens Surface Moiré
An interesting variant of the level-line moiré consists in using an array of cylindrical lenses also for the base layer [5].The resulting setup combining the two layers of cylindrical lenses has a number of advantages.There is only little light attenuation by the base.The moiré that can be seen in transmission mode shows level lines having the colors of the incident light rays, for example the blue light reflected from the sky or the green light reflected from the trees.In addition, the device can be 3D printed with a single material and manufacturing methods such as casting can be used.Let us consider a base formed by horizontal cylindrical lenses.The trajectory of the center line of the lens surface is shifted vertically according to the elevation profile.The shift of the base cylindrical lens grating in respect to the revealer cylindrical lens grating has the effect of selecting incoming light having a different orientation (Figure 6).The change of orientation of the incoming light rays that pass through the device and reach the eye of the observer creates the level line moiré effect.Simulations of the dual lens surface moiré are shown in Figure 10 and the fabricated dual-lens moirés are shown in Figure 12.

Simulation and Fabrication
In this section, we show a variety of surface moirés both simulated and fabricated with different elevation profiles and different geometries.As we restrict ourselves to print with only two materials at the same time, as in [19], we choose to use a transparent (clear) material for lenses and a black material for the absorbing parts of the base layer halftone.Therefore, all our results are transmission-based devices and should be illuminated from behind.In all our exploration of different geometries, we found values near 0 to give the best results (see Figure 7).This highlights the fact that parallel arrangement of lens and base surfaces is of utmost importance, as also hinted by [5].
Figure 8 shows a set of different elevation profiles used in rendering and fabrication of our results.

Simulations
We use Cycles, a physically-based rendering engine based on unidirectional path tracing [23], for all our simulations (128 samples, default settings).As we are rather interested in simulating the dynamism of moiré, and not necessarily the exact appearance, we don't measure the appearance of the employed materials.We use two default materials, close to our black and clear materials, namely the "Diffuse BSDF" with roughness 1 and refractive index 1.5 for the black material, and "Glass BSDF" with roughness 0 and refractive index 1.5 for the clear material.For the rendering of multi-material moiré the lighting of the scene is ambient light, while for dual-lens surface moiré lateral area lights have been used.Figure 9 and 10 show different simulations with various elevation profiles and parametric surfaces.As it can be seen in these figures, and in the accompanying videos, our designs create dynamically beating moiré shapes as well as highlight and dark areas that move along the surface when viewing the device from different angles.We don't notice undesirable moirés, indicating the usefulness of our framework that comprises constant field of view lens-arc surfaces and the selection of the optimal base surface by design space exploration.

Fabrication
The new additive fabrication technology is enabling a lot of applications.In addition to creating moirés on various kinds to geometries, its multi-material capabilities create unprecedented opportunities for designing new systems, e.g.displays [19].Multi-material 3D printing is our primary fabrication platform for moiré surfaces.While the current additive technology is still mostly used for prototyping, we also show that there is a potential for mass manufacturing of moiré using molding techniques.

Multi-material 3D Printing
Our multi-material 3D printer is an inkjet-based, phase-shift, photopolymer printer [24].We print all our samples in the glossy mode and align the lens's longer axis with the printing direction for a better lens profile reproduction.Figures 11, 12a and 12b show the samples fabricated with our printer.The movement and the width of dark/highlight moiré bands are very well predicted in our simulation.

Molding
For designing the mold, we follow the steps proposed by [25] and create the two-piece molds.We use SYLGARD 184 from Dow chemicals, a very clear silicone whose flexibility can be controlled by the ratio of its two components.The high transmittance of the silicone leads to a very clear moiré (Figure 12c), with considerably cleaner beating effects than its 3D printed counterpart (Figure 12a and 12b).

Conclusion
We introduced a simple and practical method of extending moiré to curved surfaces.Our results enable creating a novel high-quality effect on surfaces which can be fabricated with different    manufacturing approaches.From an artistic point of view, the proposed surface moiré technology exhibits interesting visual properties.It could also be used for the authentication of products and for fighting counterfeits.We plan to streamline our current framework of curved surface moiré rendering in order to explore design variants at interactive rates.Although our low dimensional design space does not require quantifying the moiré quality, more general-purpose light field displays may need to look at this problem.Especially, quantifying the changes in light field quality when going from the original planar display (as a quality landmark) to the surface display could be very useful.Finally, our device is a rigid device which needs to be tilted in order to see the moiré effect.In order to extend the range of effects, we can imagine designing moiré mechanisms where, by clasping the device, the revealing surface moves relative to the base surface and changes the appearance of the moiré.

A. Level-line moiré
The theorem of level-line moiré [20], [ [1] pp.370-371] states that the level set of an elevation profile appears as moiré lines when superposing an unshifted line grating on a base line grating whose lines are shifted proportionally to the elevation profile.See Figure 1 for an example of a level-line moiré.The general equation of the level-line moiré can be determined by considering the indexed line families of the revealer, the base and the moiré ( [26], [ [1] sec.11.2 pp.353-360]).As shown in Figure 13, the transparent lines of the revealer are indexed with = 1, 2, 3, ... , the base lines are indexed with = 1, 2, 3, ... , and the moiré lines are indexed with = ..., −2, −1, 0, ... .The index characterizing the moiré fringe lines is equal to the index of the revealer minus that of the base: Let us now consider a revealer composed of a periodic grating of lines, which can be straight or curvilinear.Let us define such that the implicit equation of one of these lines is ( , ) = 0.The family of revealer lines is thus defined by: where is the index of the lines in the revealer and is the period of the lines in the revealer.For example, = • would represent a grating of horizontal lines of period .
Similarly, let us consider a base composed of a periodic grating of lines, which can be straight or curvilinear.Let us define such that the implicit equation of one of these lines is ( , ) = 0.The family of base lines is thus defined by: where is the index of the lines in the base and is the period of the lines in the base.The implicit equation of the moiré grating can then be obtained by combining Eq. ( 9), Eq. ( 10), and Eq. ( 11): ( , ) − ( , ) = As stated in the level-line moiré theorem suggested in [20] and formulated in [ [1], section 11.2, pp.353-360], the level lines of a surface defined by = ( , ) can be obtained by superposing two gratings.As first grating, we consider a revealing grating of horizontal lines and of period .We consider as the second grating a base layer grating that is a copy of the revealer in which we additionally translate vertically each point ( , ) by a distance equal to ( , ).The line equations of the revealer and the base are respectively: The resulting moiré is defined by the indicial equation Eq. ( 9).After replacing and thanks to Eq. ( 13) and Eq. ( 14), we obtain: This means that the moiré lines are indeed level lines of the surface = ( , ).

B. Parametric Distortion
We are interested in calculating how a 1D segment in the parameter domain (akin to ) would deform when mapped to a curve on a surface ( , ).The ratio / is the same ratio of the lens radius on the surface to its radius in the flat mode.Therefore, it gives us the varying radius along the surface and consequently the focal distance for any point on .An arbitrary curve on the parametric domain can be described as ( ( ), ( )).On a given parametric surface ( , ), the curve is mapped to ( ) = ( ( ), ( )).By means of the chain rule, the derivative (curve tangent) is given by where u = [ ( ) ( )] .On the surface, the square length of the tangent of the curve ( ) is then given by where the matrix represents the first fundamental form.
In our current moiré setting, we move only along the -term in the parametric domain, this induces ( ) = 0 and hence | | 2 = ( • ) ( ) 2 .Since the length of a curve is measured by integrating the squared norm of its derivative, the metric distortion in our setting is simply captured by the squared norm | | 2 .

Fig. 1 .
Fig. 1.(a) Elevation profile extracted from a gray-level image, (b) base layer shifted according to the elevation profile, (c) revealing layer, and (d) superposition of the base and revealing layers yielding the level-line moiré (taken with permission from [5]).

Fig. 2 .
Fig. 2. Top row: (a) Elevation profile formed by a gray-level image, (b) base layer shifted according to the elevation profile, (c) a closer view of the base layer.Bottom row: superposition of the base and revealing layers yielding the level-line moiré, shown at different tilt angles.

Fig. 3 .
Fig. 3. (a) A cylindrical lens array on top of a base layer.(b) Cross section of a single lenslet with : lens radius, : lens pitch (or width), ℎ: sag-height, : focal length, and : lens body thickness.

Fig. 4 .
Fig. 4. (a) Three steps in creating surface moiré.(b) Lenses with equal radius ( ) are fitted, resulting in different angular field of views (AFOV).(c) By adapting the radius of each lens, the angular field of view remains constant.

Fig. 5 .
Fig. 5. (a) Inspired by [19], we design a base surface (in green) that is both parallel to the individual lenses and in their focal plane.(b) The rendered simulation of such a design.The nonsmooth base-layer causes artifacts.(c) For a given lens surface, we can compute the offset surface (blue), the focal surface (red) and one of the interpolation surfaces (green).

Fig. 6 .
Fig. 6.Cross-section of a transparent device formed by two gratings of transparent cylindrical lenses with light rays oriented towards the eye of an observer having (a) an unshifted base and (b) a base shifted by quarter of the lens repetition period (schematic view).

Fig. 7 .
Fig. 7. Design space explored by varying the weighting parameter and the distance parameter , see formula 8.
13g.13.Superposition of a revealer (gray) and a base (black).The transparent lines of the revealer are indexed with .The black lines of the base are indexed with .The moiré fringe lines are indexed with .