The spatial resolution of 3D printing is nite. The necessary discretisation of an object before printing produces a step-like surface structure that in uences the appearance of the printed objects. To study the e ect of this discretisation on specular re ections, we print surfaces at various oblique angles. This enables us to observe the step-like structure and its in uence on re ected light. Based on the step-like surface structure, we develop a re ectance model describing the redistribution of the light scattered by the surface, and we study dispersion e ects due to the wavelength dependency of the refractive index of the material. We include preliminary veri cation by comparing model predictions to photographs for di erent angles of observation.
The appearance of objects with known optical properties can be modelled with
di erent rendering techniques [
In 3D printing, the surface requires discretisation. For example, in Fused
Deposition Modelling 3D printing (FDM), the object is created by applying
melted material in a layer-by-layer ow. Any oblique surface to be printed will
be a ected by this slicing. As a result, any oblique surface will be approximated
by a staircase-like shape. In this paper, we investigate plane surfaces of
FDMprinted wedges and the e ect of the surface discretisation on the appearance for
di erent wedge angles. For this, we consider wedges as schematically shown in
Fig. 1, a{c). We limit our study to surface re ection only. Once the scattering
of light by the surface has been modelled, it can be combined with subsurface
scattering using volumetric light transport simulation [
Re ectance models for ridges such as the symmetric V-groove cavity [
Angular redistribution of incident light after single or multiple re ections can
be quantitatively estimated with geometrical ray optics, together with the
Fresnel equations [
For the wedges described above, together with a given printing resolution, the unit cell considered for calculation is de ned as illustrated in Fig. 1, e). A unit cell is chosen as a concave right-angled structure with two slopes de ned by the vertical printing resolution. The length of one side corresponds to the printing layer thickness l1 (vertical printing resolution). The second side l2 is given by l1 and the wedge angle 2. The side corresponding to the layer thickness l1 forms the angle 1 = 90 2 with the wedged plane (Fig. 1, a,e). We thus have
Discretised surface (red) θi+2β2 θi-2β1 -θi
Model (side view) l1 3D Print b)
l2 e) a)
) d)
Wedge angle
It is su cient to consider a one-dimensional unit cell as the surface is constant
in the perpendicular lateral direction. The relation of the angle of incidence i
and the angle of observation o to 3D direction vectors describing the light-view
con guration and the surface geometry is covered by Luongo et al. [
Incident illumination is de ned through a number of rays hitting the section of the length l = pl12 + l22 between the tips of the slopes. By assuming that illumination consists of a number of rays homogeneously distributed with a certain density in the plane de ned by the unit cell, we can express radiance of incident light through the projection of any segment within the unit cell onto the rayfront. The radiance illuminating each slope is then expressed through their projections: r1 = l1 maxfsin( 2 + i); 0g + l2 minfcos( 2 + i); 0g r2 = l2 maxfcos( 2 + i); 0g + l1 minfsin( 2 + i); 0g : (2) (3) The max and min functions help us to avoid listing three di erent cases. When 2 i 1, the rst terms in the two expressions are nonzero, while the second terms are zero. On the other hand, when i < 2, we have r1 = 0 and both terms active for r2. The opposite is the case for i > 1. We also recall Lambert's cosine law: the more grazing the angle of incidence, the smaller the density of incident rays. The irradiance is thus proportional to the cosine of the angle of incidence, but this term is not part of the BRDF.
Light can leave the surface after being re ected once by any of the slopes or after a double re ection. Due to the mutually perpendicular slopes, each case where double re ection occurs is a retrore ection, with the scattered light direction coinciding with the direction of incidence (independently of the wedge angle). By letting our model account for this retrore ection, masking e ects do not need to be considered explicitly, but are intrinsically accounted for. One angle of incidence i can be associated with several scattering angles. Depending on the position of an incident ray along the slopes, light can be re ected with scattering angle o:
from left slope : o = i + 2 2 from right slope : o = i retrore ection : o = The convention regarding the sign of i and o still follows Fig. 1, d). For a given angle of incidence, we calculate the proportion of the rays being re ected once or twice. Those proportions are expressed through the projections of the corresponding sections onto the rayfront. We can use o from Eqs. (4{5) to get the direction of the incident light after the rst re ection. Retrore ection occurs for the part of the rst re ection being incident on the other slope. Insertion of the new directions in Eqs. (2-3) then provides the desired projections. For 2 < i 1 2, all rays hitting the right slope will be retrore ected. The rays hitting the left slope will be partially directly re ected and partially retrore ected. The direction of the light after re ection in the left slope is 180 i 2 2. Inserting this direction of incidence in Eq. (2), we get the projection of the part of the left slope from which retrore ection occurs: For angles of incidence satisfying i > 1 or i < 2, only direct re ection occurs, which we calculate using Eqs. (2{3). The proportion of scattered light in each direction can have a value between 0 and l cos i. We thus represent the proportions as re ectance ratios through division by l cos i.
In addition to geometric considerations, the material refractive index in
uences the re ections of the individual slope surfaces. We calculate bidirectional
re ectances using the Fresnel equations with tabulated wavelength-dependent
refractive indices for the 3D printing materials obtained from spectroscopic
ellipsometry. P- and S-polarised light is considered separately. For our nal results,
we use unpolarised light incident in the plane depicted in Fig. 1. The re ectance
Similar considerations apply for 1 2 < i < 1. Here, all rays hitting the left
slope will be retrore ected. The part of the rays falling on the right slope and
being retrore ected is given by
r21 = l1 sin( 2 + i) :
r12 = l2 cos( 2 + i) :
is then half of each type of polarisation (also in the case of retrore ection as the
plane of incidence is the same for both re ections). The amount of light entering
the material is given by one minus the Fresnel re ectance. This refracted light
can be used with a model for volumetric light transport [
The described approach was used to predict re ectance properties of wedges printed with a fused deposition modelling (FDM) printer. Black polylactide (PLA) lament was used. Slope angles of the wedges were set to the values 10 , 20 , 30 , 40 , 45 , and 60 . Examples of printed wedges with 10 and 45 (side view) are shown in Fig. 2. The footprint of each slope was 2 2 cm2. The numbers of layers needed for each print were obtained after slicing the models with de ned layer thickness (resolution in z direction). Larger slope angles resulted in a larger number of layers needed. Thus the step in y direction was smaller for the higher slope angles. Table 1 summarises used and calculated parameters.
We computed the re ectance by combining the pure geometrical e ect with the Fresnel angular re ectance for each angle of incidence and re ection. Results at 550 nm are shown in Fig. 3. For the refractive index, a representative value of n = 1:46 for PLA has been chosen. The wavelength-dependent values of the wedge angle, number of layers y-step, mm refractive index were obtained by spectroscopic ellipsometry and are shown in Fig. 4. As the re ectance values have been calculated for a one-dimensional surface cross-section, i.e. one plane of incidence and re ection, they represent an in-plane BRDF of the printed surface.
Characteristic patterns can be observed in the diagrams: The straight line o = i corresponds to retrore ection. The other two lines represent direct re ections from the two slopes, l1 and l2. Retrore ection is weaker because it involves two re ections. Retrore ection is not possible for i < 2 or i > 1. The top-left line represents direct re ection from the left slope and the bottomright line corresponds to direct re ection from the right slope. The intensity of the direct re ectances becomes weaker when also retrore ection is possible. Otherwise, the radiance distribution along the lines is a result of both the Fresnel re ectances and the geometrically determined proportions.
Considering light incident from all directions (di use lighting), the re ected contributions towards a given viewing angle can be accumulated (by integrating over all angles of incidence). An example is shown in Fig. 5. Here, characteristic peaks for certain angles of observation of each wedge can be observed. The peaks correspond to the situation when the longer slope is being illuminated under shallow angles. For example, for a 10 wedge with 2 = 10 , 1 = 80 , this occurs when i is close to 90 and hence, o approaches 90 + 2 2 (here 70 ). From Fresnel equations, larger angles of incidence (with respect to the slope's surface normal) lead to the higher re ectance values (for angles greater than Brewster's angle). However, there is no contribution if i = 90 . This causes a minimum at o = 90 + 2 2. For wedge angles larger than 45 ( 1 < 45 ), there is a strong contribution of light re ected at shallow angles directly o the right slope. This happens for i approaching +90 and o approaching 90 2 1. Comparing di erent wedge angles in Fig. 5, the longest printing step in y-direction was made in the case of the 10 wedge angle. Therefore the contribution to the peak value is the highest in this case, while the peak is least pronounced for 45 . Furthermore, as expected, the BRDF for the wedge angle 45 is symmetric with respect to o = 0 , as the unit cell is mirror symmetric. Similarly, the BRDFs of 30 and 60 wedges are mutually symmetric. For comparison, we calculated also the spectral re ectance for a plane surface which is shown in Fig. 5. 3.1
In order to be able to interpret calculated light quantities as appearance attributes, the above described calculation has to be carried out for each wavelength of the visible spectrum (380 to 780 nm) separately because the refractive index is dispersive. Accumulated re ectance spectra for a given viewing direction were calculated after integrating over all incidence angles. An example of spectral re ectance for viewing angle 0 is shown in Fig. 6. Assuming a certain illuminant (e.g. D65), integrated re ected (scattered) radiance spectra towards a given angle of observation can be calculated and interpreted as surface contribution to the overall appearance. Interpretation of the colour under standard illuminants D65 and A is demonstrated in Fig. 7. The radiance changes with varying viewing angle. Nevertheless, the redistributed specular re ections represent mostly a colour of the light source. As can be seen in Fig. 6, the re ectance is relatively uniform over the visible spectrum. 3.2
Real surfaces contain rough details. In the case of the studied printed wedges, each individual slope surface is rough, also the edges of the layers are not perfectly sharp, but are to a certain degree round. Those two factors lead to statistically uctuating angular deviation of scattered radiation, which introduces the di usion and blurring of specular re ection. Additionally, the nite size of specimen and light source induce an angular spread. Furthermore, angular deviations can be observed for light that is incident from outside the plane of interest depicted in Fig. 1. All these factors can be represented through applying a convolution with Gaussian broadening to the computed BRDF. Fig. 8 shows the resulting scattered light angular distribution after applying convolution representing additional angular spread of re ection caused by surface roughness.
As a rst attempt at verifying our approach, we compare the computed predictions to real visual impressions. For this purpose, a series of photographs was prepared. A small point-like LED light source was placed in 50 cm distance from the specimen. Photographs were taken at di erent angles at 90 cm distance with 135 mm objective and the following setting: ISO 100, exposure time 1 second and aperture 5.6. Figure 9 compares theoretical expectations with real appearance. The main quality that varies with changing viewing angle is the brightness which is casued by the redistribution of specular re ections. Despite of large scattering between the pictures, the general trend can be observed. Apparently, the real structures reveal large angular broadening. Remarkably, the 10 wedge has a generally darker appearance than the other ones. Also, the re ectances of 30 and 60 wedges di er from each other while the model predicts symmetric behaviour. The unit cells for both con gurations have the same geometry, however the size of the unit cell decreases with the wedge angle. The unit cell of the 30 wedge is approximately 1.7 times bigger than that of the 60 one. As a conclusion, the absolute unit size plays a role in the appearance of the oblique 3D printed surfaces. A possible explanation is the impact of the real (rounded) edges of the printed layers (cf. Fig. 2). If the unit cell size approaches the curvature radius, those edges will dominate the scattering characteristics. Another consequence of large angular broadening for structures with smaller unit cells is the bright intensity scattered towards 80 for wedge angles around 40 as seen in the photographs. For example, for the 45 wedge, we would expect a direct re ection toward 90 which would not be observable. The brightness at 80 may result from the discussed angular broadening by imperfections like the round edges. In this work, the e ects of oblique surfaces created with 3D printing on the object's appearance were studied. Due to the limited printing resolution, these surfaces are discretised according to the surface inclination. Using geometrical optics, shadowing e ects and angular redistribution of specular re ections were modelled. Geometrical considerations were complemented by calculating wavelength dependent re ectance. The so calculated re ectances allowed us to predict the surface contribution to the object appearance, especially for highly absorbing materials. The approach was used to compute the colour tone of black material under a certain illuminant for di erent viewing angles. A series of wedges was 3D printed using FDM technique and black PLA lament. A comparison of modelpredicted scattering with photographs taken at di erent viewing angles indicates reasonable correspondance in particular for those wedges with coarser discretisation. Such imperfections as rounded edges are not discribed by the model but have strong impact on surfaces with ner discretisation. For future work, we suggest a study of other aspects such as di use scattering by surface microfacets and subsurface scattering in a translucent print material.
Funded by the Horizon 2020 programme of the European Union. Grant # 814158.