Clarify that highlight is elongated along the tangent direction (#111)

index.html

@@ -65,8 +65,7 @@
 Slabs
 -------------------------------------
 
-To define our model unambiguously we use a simple formalism which describes an abstraction of a physical optical material consisting of "slabs" of dielectric or conductor, bounded by BSDFs. We work in the local space of a 2d surface, so horizontal refers to directions in the local plane, and vertical means aligned with the outward shading normal. The direction towards the top of the material is understood to align with the local shading normal at the current light transport vertex.
-The notion that the slabs are homogeneous in the horizontal extent is an idealization where, in order to make the light transport tractable, we consider that on the "mesoscopic" scale of the depth of the slabs in the structure, the slabs are locally homogeneous. On a larger macroscopic scale, the properties of the slabs, and the overall structure of the material, will vary over the surface.
+To define our model unambiguously we use a simple formalism which describes an abstraction of a physical optical material consisting of "slabs" of dielectric or conductor, bounded by BSDFs. We work in the local space of a 2d surface, so horizontal refers to directions in the local plane, and vertical means aligned with the outward shading normal. The direction towards the top of the material is understood to align with the local shading normal at the current light transport vertex. The notion that the slabs are homogeneous in the horizontal extent is an idealization where, in order to make the light transport tractable, we consider that on the "mesoscopic" scale of the depth of the slabs in the structure, the slabs are locally homogeneous. On a larger macroscopic scale, the properties of the slabs, and the overall structure of the material, will vary over the surface.
 
 Each slab consists of a homogeneous dielectric or conducting medium $V$ with conceptually unbounded extent horizontally and a finite vertical extent, bounded above by a surface with a given interface BSDF $f$. We denote a general slab therefore as follows:
 \begin{equation}
@@ -132,7 +131,7 @@
 Layering
 -------------------------------------
 
-The layer operation generates a composite material by depositing a slab $S_\mathrm{coat} = \mathrm{Slab}(f_\mathrm{coat}, V_\mathrm{coat})$ on top of another substrate slab $S_\mathrm{sub} = \mathrm{Slab}(f_\mathrm{sub}, V_\mathrm{sub})$ (or composite material), conceptually bonding the base of $S_\mathrm{coat}$'s dielectric medium to the surface of $S_\mathrm{sub}$, so that $V_\mathrm{coat}$ becomes the adjacent medium above the microsurface of $S_\mathrm{sub}$. 
+The layer operation generates a composite material by depositing a slab $S_\mathrm{coat} = \mathrm{Slab}(f_\mathrm{coat}, V_\mathrm{coat})$ on top of another substrate slab $S_\mathrm{sub} = \mathrm{Slab}(f_\mathrm{sub}, V_\mathrm{sub})$ (or composite material), conceptually bonding the base of $S_\mathrm{coat}$'s dielectric medium to the surface of $S_\mathrm{sub}$, so that $V_\mathrm{coat}$ becomes the adjacent medium to the top interface of $S_\mathrm{sub}$.
 
 The physical act of bringing the two independent slabs $S_\mathrm{sub}$ and $S_\mathrm{coat}$ together and bonding them produces a new composite material $L$ which is "vertically" heterogeneous on the mesoscopic scale, denoted by:
 \begin{equation}
@@ -391,7 +390,7 @@
 
 We give here some general assumptions about the form and parametrization of the BSDFs which describe the interfaces in the model outlined in the previous section.
 
-The BSDFs $f_\mathrm{conductor}$, $f_\mathrm{dielectric}$ and $f_\mathrm{coat}$ of the [metal](index.html#model/metallicbase), [dielectric](index.html#model/dielectricbase) and [coat](index.html#model/coat) slabs respectively, are each assumed to be described by a standard microfacet model. This is a widely used approximation in which the surface is assumed to be composed of a heightfield consisting of smooth microfacets of either metal or dielectric, where the statistical distribution of the normal of these facets (termed the "micronormal") determines the surface roughness characteristics at the macroscopic scale.
+The BSDFs $f_\mathrm{conductor}$, $f_\mathrm{dielectric}$ and $f_\mathrm{coat}$ of the [metal](index.html#model/metallicbase), [dielectric](index.html#model/dielectricbase) and [coat](index.html#model/coat) slabs respectively, are each assumed to be described by a standard microfacet model. This is a widely used approximation in which the surface at a microscopic scale (the _microsurface_) is assumed to be composed of a heightfield consisting of smooth microfacets of either metal or dielectric, where the statistical distribution of the normal of these facets, termed the _micronormal_, determines the surface roughness characteristics at the macroscopic scale.
 
 The other interfaces are not standard microfacet models, but strongly related:
 
@@ -406,39 +405,38 @@
 \end{equation}
 where $h = (\omega_i+\omega_o)/|\omega_i+\omega_o|$ is the half-vector, i.e. the micronormal which mirror reflects $\omega_i$ into $\omega_o$. For dielectrics there is also a BTDF, i.e. the portion of the BSDF where the input and output directions lie in opposite rather than the same hemispheres, which has a similar form to the BRDF (except with a modified half-vector and Fresnel factor) [#Walter2007].
 
-The Fresnel factor $F(\omega_i, h)$ is determined by the index of refraction of the reflecting material, and its form differs depending on whether the material is a dielectric or conductor [#Walter2007]. Its parametrization in each case is described in the Dielectric base section and Metal section. The "masking-shadowing" function $G(\omega_i, \omega_o)$ accounts for the probability that the input and output directions are occluded by the microsurface. It is usually derived using the Smith model which determines $G$ given the NDF, and for the GGX NDF (below) the masking-shadowing function then has a well-known form [#Heitz2014b].
+The Fresnel factor $F(\omega_i, h)$ is determined by the index of refraction of the reflecting material, and its form differs depending on whether the material is a dielectric or conductor [#Walter2007]. Its parametrization in each case is described in the Dielectric base section and Metal section. The _masking-shadowing_ function $G(\omega_i, \omega_o)$ accounts for the probability that the input and output directions are occluded by the microsurface. It is usually derived using the Smith model which determines $G$ given the NDF, and for the GGX NDF equation [GGX] the masking-shadowing function then has a well-known form [#Heitz2014b].
 
 The Normal Distribution Function (NDF) $D(m)$ describes the relative probability of occurrence of micronormal $m$ on the surface, and thus the roughness characteristics.
 A popular form of NDF which well-approximates the roughness of real materials is the so-called GGX distribution ([#Walter2007], [#Burley2012], [#Heitz2014b], [#Pharr2023]), which has the basic [^normalization] form:
-$$
+\begin{equation} \label{GGX}
 D_\mathrm{GGX}(m) \propto \left( 1 + \frac{\tan^2\theta_m}{\alpha^2} \right)^{-2}
-$$
-where $\theta_m$ is the angle between $m$ and the (macroscopic) surface normal, and the parameter $\alpha \in [0, \infty)$ controls the roughness of the surface. As $\alpha \rightarrow 0$, the distribution of normals becomes highly peaked around $\theta_m=0$ i.e. flat microfacets, while as $\alpha \rightarrow \infty$ microfacets of all orientations are equally prevalent.
-Following [#Burley2012], we set
-$$
-\alpha = r^2 \ ,
-$$
-where $r$ is termed the roughness, as this produces a more perceptually linear resulting change in apparent roughness as $r$ is varied.
-
-In the general case the roughness is anisotropic, that is the NDF is not circularly symmetric but stretched along some direction in the surface plane, producing an elongation of the specular highlight along the stretch direction. This simulates coherent microscale geometry due to processes like scratches or brushing.
-This is parametrized by two separate $\alpha$-roughnesses $\alpha_t$ and $\alpha_b$ along orthonormal tangent and bitangent vectors in the surface plane (where only the tangent vector direction needs to be specified, as an angle relative to some specified original tangent vector field). The form [^normalization] of the GGX distribution in this anisotropic case is (given the polar angle $\phi_m$ of $m$ relative to the tangent):
-$$
+\end{equation}
+where $\theta_m$ is the angle between $m$ and the (macroscopic) surface normal, and the parameter $\alpha \in [0, \infty)$ controls the roughness of the microsurface. As $\alpha \rightarrow 0$, the distribution of normals becomes highly peaked around $\theta_m=0$ i.e. flat microfacets, while as $\alpha \rightarrow \infty$ microfacets of all slopes are equally prevalent.
+Following [#Burley2012], we set (in the isotropic case)
+\begin{equation}
+\alpha = r^2
+\end{equation}
+where $r$ is termed the _roughness_, as this produces a more perceptually linear resulting change in apparent roughness as $r$ is varied.
+
+In the general case the roughness is anisotropic, that is the NDF is not circularly symmetric but stretched along some direction in the surface plane, producing an elongation of the specular highlight along that direction. This simulates coherent microscale groove geometry perpendicular to the stretch direction due to processes like scratches or brushing. It is assumed that a reference tangent vector field is defined (via **`geometry_tangent`**), and the surface tangent vector direction is then specified as a polar angle relative to it. The tangent direction corresponds to the direction in which the highlights are elongated (equivalently, the surface grooves lie along the orthogonal bitangent). The GGX distribution in the anisotropic case is then parametrized by two separate $\alpha$-roughnesses $\alpha_t$ and $\alpha_b$ along the tangent and bitangent vectors as follows, given the polar angle $\phi_m$ of the micronormal $m$ relative to the tangent (counterclockwise about the normal):
+\begin{equation}
 D_\mathrm{GGX}(m) \propto \left( 1 + \tan^2\theta_m \left(\frac{\cos^2\phi_m}{\alpha_t^2} + \frac{\sin^2\phi_m}{\alpha_b^2}\right) \right)^{-2}
-$$
+\end{equation}
 which reduces to the isotropic form when $\alpha_t = \alpha_b = \alpha$.
 
 !!! WARNING
    GGX (originally "Ground Glass unknown" [#Walter2007]) is based on the Trowbridge-Reitz NDF. [#Burley2012] proposed Generalized-Trowbridge-Reitz (GTR) which has an exponent $\gamma$, where GGX is the special case $\gamma=2$. This gives more control over the tail of the NDF. It is an open question whether a more general NDF such as this would be a valuable addition to the model.
 
 The NDF terms $\alpha_t$ and $\alpha_b$ are more conveniently parametrized as the total roughness $r$ and an anisotropy $a \in [0, 1]$. We suggest the following mapping from $r, a$ to $\alpha_t, \alpha_b$, to preserve the average roughness regardless of the anisotropy. Figure [ndf_anisotropy] shows the resulting shape of the highlight (technically, these are contour lines of the NDF $D_\mathrm{GGX}(m)$ in the 2D slope space). A rationale is that if a renderer doesn't support anisotropy (or if the feature is turned off for performance considerations, such as level of detail), using only the roughness parameter should result in an isotropic specular highlight perceptually close to the original anisotropic one. This formulation verifies $\alpha_t^2 + \alpha_b^2 = 2\alpha^2$ regardless of anisotropy $a$:
-$$
+\begin{equation}
 \left\{
 \begin{aligned}
 \alpha_t &= r^2 \sqrt{\frac{2}{1 + (1 - a)^2}}  \; \ , \\[2ex]
 \alpha_b &= (1 - a) \, \alpha_t \ .
 \end{aligned}
 \right.
-$$
+\end{equation}
 
 ![Figure [ndf_anisotropy]: NDF shapes as a function of roughness $r$ and anisotropy $a$.](images/anisotropy.png width=60%)
 
@@ -474,7 +472,7 @@
 
 - **`specular_anisotropy`** for $f_\mathrm{dielectric}$ and $f_\mathrm{conductor}$, and **`coat_anisotropy`** for $f_\mathrm{coat}$, specify $a \in [0, 1]$ (the degree to which the specular highlight is stretched in the direction of anisotropic rotation).
 
-- **`specular_rotation`** for $f_\mathrm{dielectric}$ and $f_\mathrm{conductor}$, and **`coat_rotation`** for $f_\mathrm{coat}$, specify direction of the elongation of the specular highlight. The rotation is normalized to $[0, 1]$ representing angles from 0° to 360° counterclockwise, with 0° being the original tangent vector direction. Note that the appearance of the specular highlight is identical for rotations between $[0, 0.5]$ and $[0.5, 1]$; this allows the preservation of value when converting to/from other models that support directional anisotropy.
+- **`specular_rotation`** for $f_\mathrm{dielectric}$ and $f_\mathrm{conductor}$, and **`coat_rotation`** for $f_\mathrm{coat}$, specify direction of the elongation of the specular highlight. The rotation is normalized to $[0, 1]$ representing angles from 0° to 360° counterclockwise about the normal, with 0° being the original tangent vector direction. Note that the appearance of the specular highlight is identical for rotations between $[0, 0.5]$ and $[0.5, 1]$; this allows the preservation of value when converting to/from other models that support directional anisotropy.
 
 The single-scattering microfacet BRDF of equation [microfacet_brdf_ss] does not conserve energy, as it neglects to account for multiple scattering between the microfacets. An implementation should ideally account for this, via one of a number of schemes, otherwise the reflection from rough metals and dielectrics is dimmer and less saturated than it should be. A fully accurate approach is described in [#Heitz2016], where the multiple bounces are explicitly modelled via Monte Carlo. Simpler approximate models are presented in [#Kulla2017] (which functions by adding compensation lobes to account for the missing energy), and [#Turquin2019] (which scales the albedo of the lobe to maintain energy preservation at the expense of reciprocity).