Analytic Spherical Harmonic Coefficients for Polygonal Area Lights

Transcription

1 Anaytic Spherica Harmonic Coefficients for Poygona Area Lights JINGWEN WANG, University of Caifornia, San Diego RAVI RAMAMOORTHI, University of Caifornia, San Diego Spherica Harmonic (SH) ighting is widey used for rea-time rendering within Precomputed Radiance Transfer (PRT) systems. SH coefficients are precomputed and stored at object vertices, and combined interactivey with SH ighting coefficients to enabe effects ike soft shadows, interrefections, and gossy refection. However, the most common PRT techniques assume distant, ow-frequency environment ighting, for which SH ighting coefficients can easiy be computed once per frame. There is currenty imited support for near-fied iumination and area ights, since it is non-trivia to compute the SH coefficients for an area ight, and the incident ighting (SH coefficients) varies over the object geometry. We present an efficient cosedform soution for projection of uniform poygona area ights to spherica harmonic coefficients of arbitrary order, enabing easy adoption of accurate area ighting in PRT systems, with no modifications required to the core PRT framework. Our method ony requires computing zona harmonic (ZH) coefficients, for which we introduce a nove recurrence reation. In practice, ZH coefficients are buit up iterativey, with computation inear in the desired SH order. Genera SH coefficients can then be obtained by the recenty deveoped sparse zona harmonic rotation method. CCS Concepts: Computing Methodoogies Computer Graphics; Rendering; Additiona Key Words and Phrases: PRT, spherica harmonics, area ighting ACM Reference Format: Jingwen Wang and Ravi Ramamoorthi Anaytic Spherica Harmonic Coefficients for Poygona Area Lights. ACM Trans. Graph. 37, 4, Artice 54 (August 218), 11 pages. 1 INTRODUCTION Spherica Harmonic (SH) ighting is a popuar method in interactive rendering, within Precomputed Radiance Transfer (PRT) systems [Soan et a. 22]. Whie stricty accurate ony for owfrequency ighting, the method enabes reaistic visua effects ike soft shadows and gossy refections at rea-time frame rates with dynamic ighting. PRT has been the subject of a substantia body of work (see survey [Ramamoorthi 29]), and is widey used in rea-time appications ike video games. 1 In recent years, SH ighting and PRT have aso become mainstays of offine rendering, and have been widey used in movie production [Pantaeoni et a. 21]. 1 PRT was at one time part of the DirectX API; modern games use a variety of rea-time techniques, incuding SH irradiance, voumetric transport, and variants of PRT. Authors addresses: Jingwen Wang, University of Caifornia, San Diego, jiw524@eng. ucsd.edu; Ravi Ramamoorthi, University of Caifornia, San Diego, ravir@cs.ucsd.edu. Permission to make digita or hard copies of a or part of this work for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. 218 Copyright hed by the owner/author(s). Pubication rights icensed to the Association for Computing Machinery /218/8-ART54 $15. Fig. 1. Images rendered in a PRT system with mutipe poygona area ights, with anaytic SH ighting coefficients computed in rea-time with our method. This scene contains 388K trianges and is rendered at 28.4 frames per second on an NVIDIA GeForce GTX 18 Ti GPU. The core PRT framework precomputes a transfer function for each vertex on object geometry, taking into account visibiity and BRDF effects. This transfer function is then converted (projected) into spherica harmonics and stored on the object, possiby with compression [Soan et a. 23]. For soft shadows on a Lambertian surface, this woud simpy be an SH representation of the visibiity in each direction. At run-time, these SH coefficients are combined with the ighting (a simpe dot product of SH coefficients of ighting and transfer for visibiity). In this paper, we do not modify the precomputation or core PRT framework at a, but address computation of the SH ighting coefficients. Most PRT methods assume distant, ow-frequency environment map ighting. In this case, the incident ighting is the same everywhere in the scene, and SH ighting coefficients can easiy be computed once per frame using standard numerica integration, and re-used for each vertex on the object. On the other hand, there is currenty imited support for near-fied iumination from area ight sources. Indeed, near-fied effects impy that the anguar distribution of incident radiance is now different at each vertex (ocation in space), and numerica integration to compute SH ighting coefficients separatey for each vertex or pixe is too expensive. In this paper, we address this chaenge by deveoping an efficient cosed-form computation for spherica harmonic coefficients of genera orders, for an arbitrary poygona area ight. As in most previous work, we assume the ight source emission is ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

2 54:2 Wang and Ramamoorthi uniform across its surface (and do not expicity support textured area ights). Our method is exact, does not require numerica integration, is impemented in a few ines of code in a GPU shader, and is fast enough for rea-time PRT systems (Fig. 1). We are inspired by eary methods to compute integras of poynomias against area ights [Arvo 1995; Snyder 1996], athough their purpose was different, to enabe non-lambertian effects. Since SH are poynomias, it is possibe in principe to use these methods directy. However, there are three concerns. First, they derive recurrence formuae for poynomias, rather than for SH directy. Computing a reevant poynomias and then combining them to find SH coefficients is inefficient, since an SH of order 2 n may have O(n) poynomia terms. Second, the recursion can be inefficient for high orders, with the number of recursive steps growing with the order. Third, the simper Phong formuae of [Snyder 1996] are attractive, but appy ony to radiay symmetric poynomias, whie we need a the SH coefficients. Our contributions, summarized beow, address each of these three chaenges in order: Nove Zona Harmonic Recurrence Formua. Our key technica contribution is the derivation of a nove zona harmonic recurrence formua (Sec. 5) for computing the ZH about a given direction, for a poygona ight. This is a significant generaization of the recurrence for monomias z n in [Snyder 1996], and requires keeping track simutaneousy of 3 different surface and boundary integras. Whie integras of monomias (as needed for Phong obes in [Snyder 1996]) naturay end themseves to recursive formuae due to the repetition of factors in the integrand, such reations are more difficut to find for integras of poynomias. Our work makes use of the inherent recurrence properties of the Legendre poynomias. Computation of a ZH coefficients. Unike previous work, we seek to determine a ZH coefficients up to some order n (Sec. 4.1). Hence, instead of recursing for each term (which can invove O(n) compexity for each term), we iterativey buid up the ZH terms from constant to ower-order poynomias. Thus, each new ZH coefficient can be computed in O(1) time, and tota computationa time and storage is O(n). Whie a simiar iterative computation is certainy possibe for [Snyder 1996] to obtain poynomias up to order n, directy appying that method to compute ZH coefficients woud sti invove O(n 2 ) time, since each ZH of order n is a inear combination 3 of O(n) poynomia terms. 4 Sparse Zona Harmonic Rotation for SH coefficients. Finay, to compute a (n + 1) 2 SH coefficients at order n from ZH coefficients up to order n, we everage the recent sparse zona harmonic factorization method for efficient SH rotation [Nowrouzezahrai et a. 212] (Sec. 4.2). This is substantiay simper and more efficient than appying [Arvo 1995] to each poynomia term separatey. In particuar, we compute the ZH recurrence iterativey for each of the 2n + 1 obe directions in [Nowrouzezahrai et a. 212](shared among a 2 We use the terms order and degree of the poynomia interchangeaby. 3 Whie the addition/inear combination of poynomia coefficients to compute ZH coefficients is very fast in practice, it sti adds enough overhead to be 2-4 sower than our method, as discussed in Sec This anaysis is for a singe ZH obe direction. As discussed next, we actuay need 2n + 1 obe directions, so our approach has cost O(n 2 ), whie appying the recurrence in [Snyder 1996] woud take O(n 3 ) time. SH bands). This takes O(n 2 ) time, consistent with the tota number of SH coefficients. We then appy the fast rotation formua. In principe, this step invoves O(n 2 ) SH coefficients, each expressed as a inear combination of each ZH obe direction, and can be O(n 3 ). However, there is considerabe sparsity, and the rotation is very fast in practice [Nowrouzezahrai et a. 212], with minima overhead (Fig. 6). Hence, the agorithm s effective cost is O(n 2 ), inear in the number of SH coefficients. 2 RELATED WORK We are inspired by recent work on anaytic ighting [Heitz et a. 216], which can efficienty compute direct ighting from area sources. However, they cannot address shadows or other effects. In contrast, traditiona PRT systems enabe compex ight transport, but do not permit near-fied area ights. We bridge this gap by computing SH coefficients, thus enabing area ight sources with soft shadows and other compex ight transport effects. Note that we integrate zona harmonics, which are more compex functions than BRDF obes, so we cannot directy appy the methods or transformation matrices in [Dupuy et a. 217; Heitz et a. 216]. This paper focuses on SH ighting, which is widey used in practice. A-frequency methods using waveets [Ng et a. 23] or radia basis functions [Tsai and Shih 26] can be more accurate, and require fewer than 1 basis functions at run-time, comparabe to SH. However, the fu transport matrix, often with tens of thousands of basis functions, must sti be precomputed and stored, which has imited practica appications. Note that our method appies to any PRT system using SH ighting, and is orthogona to the core PRT framework, which need not be modified at a. Whie we demonstrate our resuts primariy for visibiity precomputation, invoving diffuse soft shadows in static scenes, the method can aso be incuded with PRT methods that support gossy refection and interrefections [Soan et a. 23], and dynamic scenes [Soan et a. 25]. Near-fied iumination in PRT has been incuded with source radiance fieds [Zhou et a. 25] for a-frequency reighting. However, this requires precomputing a fu 5D space-ange radiance fied, and imits the abiity to change the shape of the ight. The affine waveet method of [Sun and Ramamoorthi 29] reduces storage by propagating ighting in waveets, incuding for textured ights. However, the method is approximate, and ights cannot be rotated out of pane. Another approximate method, in the origina paper [Soan et a. 22], was to compute incident ighting at a few ocations on object geometry and interpoate. In contrast, we compute exact SH area ight coefficients at each vertex in a simpe GPU shader. Spherica harmonic exponentiation [Ren et a. 26] addresses rea-time soft shadows in dynamic ow-frequency environments by forming spherica approximations of object geometry and ighting, but provides anaytic ights ony for circuar/spherica ight sources. Spheres (with circuar cross-sections from a views) cannot easiy be packed to accuratey represent a panar poygon. Since we aow out-of-pane rotations and arbitrary shapes, this is especiay true for poygona ights that have thin/irreguar shapes or are at grazing anges to the receiver. Moreover, projection of arge numbers of spherica ights might require considering occusion between spheres, In contrast, our method computes exact SH coefficients. ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

3 Anaytic Spherica Harmonic Coefficients for Poygona Area Lights 54:3 Tabe 1. Tabe of notation for key symbos. Y m The SH basis function of degree, index m L m The SH coefficient for the poygona ight P Legendre poynomia of degree The poygona ight (projected to unit sphere) ω e Projection of vertex e onto unit sphere γ e Ange between poygona vertices e and e + 1 ω Centra axis of zona harmonic obe α,d m Weight of d-th ZH obe used to reconstruct Y m S The surface integra of the -th Legendre poynomia P (ω ω i )dω i B,e The edge integra γ e P (z e )dγ C,e The intermediate edge integra γ e z e P (z e )dγ D,e The intermediate edge integra γ e P (z e )dγ As noted earier, we buid on [Arvo 1995; Snyder 1996]. Like them, we assume uniform poygona panar area ights. We do not expicity hande texture, but in some cases, can do so by breaking the ight into smaer poygons, each of which has a uniform coor (Fig. 1). In future, it may be possibe to everage [Chen and Arvo 21] for ineary-varying uminaires and anguary-varying ighting. Finay, the concurrent work of [Becour et a. 218] computes a singe integra of a particuar SH expansion against a poygon. We differ conceptuay in simutaneousy computing integras of a SH coefficients, enabing PRT with any SH expansion. In practice, the methods are quite simiar, and [Becour et a. 218] aso uses a zona harmonic decomposition and combines cosed-form monomia integras [Arvo 1995; Snyder 1996]. We differ in deriving a nove zona harmonic recurrence, rather than combining monomias, and we demonstrate rea-time performance. Athough our paper focuses on PRT, the various other appications mentioned by [Becour et a. 218] appy to our method as we, such as basis function projection, hierarchica sampe warping, and use in offine rendering. 3 BACKGROUND In this section, we introduce some reevant background. We first introduce the basic refection and PRT equations for direct ighting, focusing on Lambertian surfaces with soft shadowing for simpicity (more compex ight transport can aso be incuded, since our method is orthogona to the core PRT framework). We then briefy introduce the spherica harmonics and integration against an area ight, ending with some key SH and Legendre poynomia formuas. Tabe 1 summarizes notation for key symbos used in the paper. 3.1 Precomputed Radiance Transfer The refection equation for direct ighting can be written, L r (x, ω r ) = L i (x, ω i )V (x, ω i )ρ(ω i, ω r, n) max(ω i n, )dω i, Ω (1) where x is the spatia coordinate of the current vertex or pixe, (ω i, ω r ) are the goba incoming and outgoing directions, and n is the surface norma. L r is the refected radiance or image intensity, whie L i is the incident radiance. 5 V is the visibiity and ρ is the BRDF (which incudes the norma as an argument, since it is expressed in terms of incoming and outgoing directions in goba coordinates). For simpicity of notation, we combine the visibiity and BRDF terms into a transport function T (x, ω i ), which is precomputed for each spatia ocation x and incident ange ω i. We have dropped the dependence on ω r for simpicity, but that can aso be expressed in spherica harmonics [Soan et a. 22], and gossy refections are demonstrated in Fig. 1. We may now write, L r (x) = L i (x, ω i )T (x, ω i )dω i. (2) Ω To proceed further, we project the ighting and transport function into spherica harmonics Y m (ω i ) with the order, and m. We wi discuss the spherica harmonic basis function forms [MacRobert 1948] briefy at the end of this section. We use spherica harmonics up to degree n with (n + 1) 2 terms; we demonstrate resuts for fairy high order spherica harmonics with n = 8 or n = 14, and our resuts are genera for even higher orders if needed, L i (x, ω i ) = T (x, ω i ) = n L m (x)y m (ω i ) = m= n T m (x)y m (ω i ). (3) = m= Given these coefficients, by orthonormaity of the basis functions, L r (x) = n = m= L m (x)t m (x) = L(x) T (x), (4) where L and T are vector representations of L m and T m, and the integra can be reduced to a simpe dot product. 3.2 Area Light Integration Note that T m is precomputed, based on numerica integration in the standard way for PRT methods. The major contribution of this paper is to derive a cosed form formua for L m from poygona area ights. In particuar, we seek a soution to L m = Y m (ω i )dω i, (5) x where the integra is over the projection of the poygon onto the unit sphere. Note that we work with the rea (not compex) spherica harmonics. In addition, we have dropped the spatia coordinate x in L m, since it is easy to transate the poygon so x is at the origin; we denote this using x, but wi drop the subscript in Sec. 4, to simpify notation. We wi first seek to determine the zona harmonic coefficients (m = ) corresponding to a particuar centra direction ω, L (ω) = Y (ω ω i )dω i. (6) x 5 We use superscripts for L r and L i to avoid confusion with subscripts such as L m for spherica harmonic coefficients. ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

4 54:4 Wang and Ramamoorthi 3.3 Spherica Harmonic Formuas Finay, we briefy discuss the mathematica forms we use for the spherica harmonics and zona harmonics in this paper, aong with reevant identities. Consider the geographic anguar parameterization of the sphere (θ, ϕ) with Cartesian components, (x,y, z) = (sin θ cos ϕ, sin θ sin ϕ, cos θ). The rea spherica harmonics are, (2 + 1) ( m )! Y m (θ, ϕ) = 4π (+ m )! P m (cos θ) f ( m ϕ), (7) where Pm are the associated Legendre poynomias and the trigonometric function f ( m ϕ) is set to 1 whenm =, given by 2 cos(mϕ) when m > and 2 sin( m ϕ) when m <. In particuar, the zona harmonics (ZH) for m = are given by, Y (cos θ) = 4π P (cos θ), (8) where we parameterize by cos θ = z, and P (z) are simpy the Legendre poynomias of degree. For exampe P (z) = 1, P 1 (z) = z and P 2 (z) = 1 2 (3z2 1). Note that the ZH are radiay symmetric and independent of ϕ, depending ony on z in a Cartesian coordinate system. To consider ZH obes (as a function of incident direction ω i = (θ, ϕ)) about an arbitrary axis ω, we can simpy use a dot product with that axis, defining the ZH as Y (ω ω i ) as in equation 6. A key resut reating SH and ZH enabes sparse ZH factorization of SH, and aows for fast rotation of ZH into SH. [Nowrouzezahrai et a. 212] show 6 that for any m and any order (for consistency with equation 6 we use ω i = (θ, ϕ)), Y m (ω i ) = α,d m Y (ω,d ω i ). (9) d That is, an SH of degree can aways be written as a sum of rotated ZH basis functions. Whie d can in principe range from to +, invoving obes, the actua number of obes needed is usuay much smaer. Moreover, the same obes are used across a band, and in fact one can optimize to share directions ω,d across bands. As such, there are ony 2n + 1 unique obe directions ω across a bands, and the weights α,d are very sparse. [Nowrouzezahrai et a. 212] have aready precomputed the obe directions and weights. Finay, we wi make use of some we-known reations for Legendre poynomias in the subsequent derivations in Sec. 5 (beow, P (z) stands for the derivative d dz P (z)), z 2 1 P (z) = zp (z) P 1 (z) (1) P (z) = (2 1)zP 1 (z) ( 1)P 2 (z) (11) (2 + 1)P (z) = P +1 (z) P 1 (z) (12) 4 ALGORITHM In this section, we introduce our agorithm for computing the anaytic SH coefficients for a poygona area ight source, i.e., determining L m in equation 5. The key technica contribution is the nove zona harmonic recurrence, which is derived ater in Sec. 5. Whie the mathematics and derivation is somewhat invoved, the actua impementation is simpe, as discussed at the end of Sec. 4.1 and 6 Versions of equation 9 appear earier in other fieds, going back at east to [Stern 1965]. (b) (d) (a) Fig. 2. Iustration of the integras we keep track of (surface, boundary). In (a), we show the projection of the poygon to the unit sphere with support. In (b) we show the corresponding surface integra S, whie (c) shows the boundary integra B. Note that B can be broken into edges or arcs B,e as shown in (d) using a very simpe parameterization. We aso keep track of intermediate boundary integra D,e, and compute C,e on the fy. shown in the pseudocode in Agorithm 1. In practice, the agorithm is impemented in a simpe GPU vertex shader, and the source code is pubicy avaiabe at Iterative Computation of Zona Harmonic Coefficients We first deveop an iterative agorithm to compute the zona harmonics coefficients, L (equation 6). Doing so requires iterativey updating the surface integra, as we as two boundary integras (a third boundary integra is computed on the fy). Figure 2 shows the quantities we keep track of. We start by defining the integras we need and reevant notation and quantities, then discuss the base cases, and finay state the recurrence reation. Then, we summarize the iterative agorithm with reference to the pseudocode in Ag. 1. Surface and Boundary Integras. We assume the poygon has been transated by x, so the spatia ocation of the vertex is at the origin, as shown in Fig. 2(a), and omit expicit dependence on x in what foows. We are now primariy interested in the surface integra (Fig. 2(b)), S = P (ω ω i )dω i L = 4π S. (13) S simpy integrates the projection of the area ight on the unit sphere against Legendre poynomias P of the dot product ω ω i. We can mutipy in the (precomputed) normaization constant at the end to compute fina ZH coefficient L. Note that S is computed separatey for each obe direction ω,d (there may be 2n + 1 such directions). For simpicity of notation, we just use ω, as in equation 6, and omit the expicit dependence of S on ω. (c) ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

5 Anaytic Spherica Harmonic Coefficients for Poygona Area Lights 54:5 We wi be deriving a recurrence for S in terms of boundary integras over the contours of the poygon s projection, denoted by (Fig. 2(c)). Specificay, we keep track of B = P (ω ω i )( x i dỹ i ỹ i d x i ) = M c e B,e. (14) e=1 Note that B is a sum over sub-integras B,e for the M edges of the poygon. The coefficient c e corresponds to the differentia measure, which wi be shown to be a constant over an edge, and a formua for c e is given ater in equation 2. The differentia measure in the integra is expressed in terms of Cartesian components of ω i, with respect to ω. The tides on x and ỹ denote these are with respect to a coordinate frame where ω is the Z axis. 7 One can aso express the differentia measure in terms of a cross product between ω i and dω i, taking the component aong ω, P (ω ω i )( x i dỹ i ỹ i d x i ) = P (ω ω i ) [ω (ω i dω i )]. (15) In fact, the integrand above for B,e has a simpe form in the appropriate parameterization. Let us define ω e as the projection on the unit sphere of vertex e of the poygon, so that edge e connects ω e and ω e+1, as shown in Fig. 2(d). We seek to parameterize this edge, as the arc of a great circe on the sphere, to evauate B,e. If the (great-circe) arc were in the XZ pane with ω e = Z, a simpe parameterization woud be ω i (γ ) = Z cosγ +X sinγ whereγ ranges from to the ange between ω e and ω e+1. In the genera case, the vector Z is simpy repaced by ω e, whie the vector X is repaced by a new vector λ e, given through cross products by λ e = ω e ω e+1 ω e ω e+1 ω e = ω e+1 ω e (ω e ω e+1 ). (16) ω e ω e+1 We are simpy creating a vector of unit ength orthogona to ω e so that ω e+1, or any intermediate point aong the edge (arc), can be expressed as a inear combination of ω e and λ e, ω i (γ ) = ω e cosγ + λ e sinγ, (17) where γ ranges from for ω e to cos 1 (ω e ω e+1 ) for ω e+1. This parameterization for the integra is visuaized in Fig. 2(d). ω e and λ e are computed once, and reused for a obes ω. We can now express B,e from equation 14, using the integrand in equation 15, and the expression for ω i above, P (ω ω i ) = P ((ω ω e ) cosγ + (ω λ e ) sinγ ). (18) For the differentia measure, dω i = ω e sinγ dγ + λ e cosγ dγ, ω (ω i dω i ) = ω (ω e λ e ) dγ = ω µ e dγ, (19) where we define µ e = ω e λ e = ω e ω e+1 ω e ω e+1 to create a coordinate frame (ω e, λ e, µ e ). It is convenient to define a number of scaar quantities, corresponding to the dot-products above, with (a e,b e,c e ) 7 This form of differentia measure is common when appying the Stokes theorem to area ight integration, and foows from the test function in Sec. 5 (equation 37). the coordinates of ω with respect to this coordinaate frame, a e = ω ω e b e = ω λ e c e = ω µ e z e = a e cosγ + b e sinγ γ e = cos 1 (ω e ω e+1 ). (2) Finay, it is now possibe to write the edge integra B,e from equations 14 and 15 simpy as (using equations 18-2), γe B,e = P (z e )dγ. (21) Note that B in equation 14 invoves an additiona factor c e for each edge, corresponding to the differentia measure. The foowing quantities are intermediate edge integra computations used to find the boundary integra B,e, C,e = D,e = γe z e P (z e )dγ (22) γe P (z e )dγ. (23) Base Cases. For iterative computation, we need to estabish the base cases, starting with the surface integra. S is simpy the soid ange subtended by the poygon, which can be obtained from the anges between edges [Arvo 1995], S = ( M e=1 cos 1 ( (ω e 1 ω e ) (ω e ω e+1 )) ω e 1 ω e ω e ω e+1 ) (M 2)π (24) Note the standard convention that vertex indexing is cycic, so ω M+1 = ω 1 and ω = ω M. Next, the vaue of S 1 just corresponds to the computation of irradiance [Baum et a. 1989], which in this case is simpy given by the corresponding boundary integra, S 1 = 1 2 B = 1 M c e B,e = 1 M c e γ e. (25) 2 2 e=1 e=1 The base case for the boundary integra is a we known resut for irradiance, and foows directy from equation 21, B,e = γ e. (26) Simiary, noting that P 1 (z e ) = z e = a e cosγ + b e sinγ, B 1,e = a e sinγ e b e cosγ e + b e. (27) Athough easy to find, we do not actuay need a base case for C,e in equation 22, since we derive an expicit non-recursive formua for it in terms of the other boundary integras. For D,e in equation 23, since it invoves derivatives, base cases are reated to those for B,e, D,e = D 1,e = γ e D 2,e = 3B 1,e = 3 (a e sinγ e b e cosγ e + b e ). (28) ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

6 54:6 Wang and Ramamoorthi Recurrence Reation. In Sec. 5, we derive the recurrence reations for surface and boundary integras. Combined with the base cases above, these enabe us to start with S, S 1, B, B 1 and D, D 1, D 2, iterating to buid up higher vaues of S and B unti we reach S n. Specificay, we derive that S = 2 1 ( + 1) B 1 + ( 2)( 1) S ( + 1) 2. (29) This is a direct O(1) iterative expression for S assuming B 1 and S 2 have aready been computed. However, to compute S +1, we wi aso need an iterative formua for the boundary integra, B, which is derived for each edge as, B,e = 2 1 C 1,e 1 B 2,e. (3) An expicit formua for the boundary integra C 1,e is derived in the appendix, C 1,e = 1 ((a e sinγ e b e cosγ e ) P 1 (a e cosγ e + b e sinγ e ) +b e P 1 (a e ) + (a 2 e + b 2 e 1) D 1,e + ( 1)B 2,e ) Finay, there is a simpe recurrence for D, (31) D,e = (2 1)B 1,e + D 2,e. (32) After we compute the base cases, we iterate 2 n, storing the vaues of S for increasing orders, whie aso updating B and D per the recurrence formuas above. Note that each iteration for increasing is constant time, and the whoe time and storage compexity is O(n). Further note that C 1,e is computed expicity on the fy and does not require storage. Moreover, we do not need to store a vaues of the boundary integras B and D. Indeed, we need ony keep the three intermediate vaues of D 2,e, D 1,e, D,e, and can discard oder vaues, with a simiar strategy empoyed for B. Iterative Computation. The pseudocode in Agorithm 1 detais the iterative computation, given spatia ocation x and a poygon with vertices v e of M sides. The obe weights α,d m for ZH rotation, aong with obe directions ω,d are aso inputs. We compute SH coefficients up to order n. First (Line 2), we transate the poygon vertices by x and project onto the unit sphere, obtaining the vertices ω e on the sphere. The quantities λ e, µ e = ω e λ e and γ e are aso pre-cacuated at this stage for each edge (Lines 3-7). Finay, the soid ange of the poygon S is computed (Line 8). We now enter the iterative computation, which is repeated for each obe direction ω (Line 9). We cacuate a e, b e and c e as per equation 2 (Line 12), and the base cases for B,e, B 1,e (Lines 14-15) and D 1,e, D 2,e (Line 16). For the base case S 1, we must sum over a edges (Lines 1,13). Then, we iterate 2 n (Line 18), and update the boundary integras for each edge (Lines 2-25). Specificay we compute C 1,e (Line 21), and appy the recurrence formuae for B,e (Line 22) and D,e (Line 24). The boundary integra B is updated by summing over edge integras B,e (Lines 19, 23). For carity, we do not incude a number of practica storage optimizations. Ony a singe variabe C needs to be used for C 1,e since it is computed expicity. As discussed earier, we ony need to maintain the three most recent vaues for B and D. Finay, we update the surface integra S (Line 26) Agorithm 1 Pseudocode of agorithm for SH coefficients. 1: function ComputeCoefficients(x, v[], M, n, α[], ω[]) 2: fora e: v e = v e x ; ω e = v e v e Project to Sphere 3: for e = 1 to M do Pre-Cacuate per edge 4: λ e = ω e ω e+1 ω e ω e+1 ω e Eq. 16 5: µ e = ω e λ e Frame (ω e, λ e, µ e ) 6: γ e = cos 1 (ω e ω e+1 ) Ange of edge e Eq. 2 7: end for 8: S = SoidAnge(ω e [], M) Eq. 24 9: for a ω in ω[] do Up to 2n+1 for a bands 1: S 1 = Initiaize S 1 11: for e = 1 to M do Base Cases 12: a e = ω ω e ; b e = ω λ e ; c e = ω µ e Eq. 2 13: S 1 = S c eγ e Eq : B,e = γ e Eq : B 1,e = a e sinγ e b e cosγ e + b e. Eq : D,e = ; D 1,e = γ e ; D 2,e = 3B 1,e Eq : end for 18: for = 2 to n do Each degree 2 n 19: B = Initiaize B 2: for e = 1 to M do Edge Integras 21: C 1,e =C(, a e,b e, B 2,e, D 1,e ) Eq : B,e = 2 1 C 1,e ( 1)B 2,e Eq. 3 23: B = B + c e B,e Eq : D,e = (2 1)B 1,e + D 2,e Eq : end for 26: S = (+1) ( 2)( 1) (+1) S 2 Eq : L (ω) = 2+1 4π S Eq : end for 29: end for 3: for each SH basis function (,m) do ZH rotation 31: L m = Initiaize L m 32: for each d d() do Sparse set of obes ω,d 33: L m = L m + α,d m L (ω,d ) Eq : end for 35: end for 36: end function and incude the normaization factor to compute the zona harmonic coefficients L (ω) for each obe (Line 27). The ast stage is to use the sparse zona harmonic rotation formua to get a spherica harmonic coefficients L m from ZH coefficients L (ω), as discussed in Sec. 4.2 (Lines 3-35). Discussion. We briefy compare our recurrence with that derived in [Snyder 1996] for Phong ighting. Ours is a recurrence directy on Legendre poynomias rather than monomias in [Snyder 1996]. Thus, we directy have the ZH coefficients. In contrast, if we had used the monomia recurrence from [Snyder 1996], we woud have had to separatey compute and sum O(n) monomia integras for each ZH coefficient for each obe, with a tota compexity O(n 3 ). Indeed, our direct Legendre poynomia recurrence is the key contribution of the paper. Note that the derivation and recursive formua is more compicated since we are deaing with poynomia rather than ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

7 Anaytic Spherica Harmonic Coefficients for Poygona Area Lights * * 1 + * * 2 Fig. 3. Sparse zona harmonic rotation. The SH and equivaenty integra for SH coefficients can be viewed as a sparse sum of integras for ZH coefficients. Thus, we may sum a sma number of ZH coefficients (here, iustrated as ony two obes) for each SH coefficient. monomia terms, requiring us to simutaneousy keep track of the surface integra S and the boundary integras B and D, whie a third boundary integra C can be evauated expicity. Nevertheess, the actua impementation is quite straightforward, invoving ony very simpe formuae, as shown in the equations above, and the reativey simpe pseudocode in Agorithm Sparse Zona Harmonic Rotation We now proceed to compute a the SH coefficients Lm given the ZH coefficients L from the recurrence reation (reca that L is easiy obtained from S using equation 13). In fact, the same inearity reation in equation 9 aso appies to the coefficients, since they invove inear integras of the spherica harmonics. Therefore, as shown in Fig. 3, we simpy compute, Õ m Lm = α,d L ω,d. (33) d d ( ) Whie d can take vaues in the genera case (and there are a tota of 2n + 1 obe directions ω shared among a bands), there is considerabe sparsity in practice for the set of obes d() associated m are known beforehand, with SH order. Moreover, the weights α,d as a basic property of spherica harmonics. Therefore, the rotation in equation 33 is very efficient. In terms of computation and storage, computing a the ZH takes O(n) time, and is run for each of O(n) obe directions ω to compute L (ω), taking O(n 2 ) time and space, inear in the tota number of SH coefficients Lm, which is aso O(n2 ). (Of course, there is aso a O(M) factor for the number of edges of the poygon.) Since the m are sparse, the fina step in equation 33 is rotation coefficients α,d very fast, with overa compexity essentiay inear in the number of SH coefficients. 5 DERIVATION OF ZONAL HARMONIC RECURRENCE In this section, we derive the zona harmonic recurrence in equations Note that whie this section is mathematicay invoved, the actua agorithm is quite simpe, as shown in Agorithm 1. Readers more interested in impementation may want to skip ahead to the resuts in Sec. 6 on a first reading. 54:7 Main Recurrence We begin with the derivation of equation 29. Consider the Legendre poynomia identity in equation 11. Integrating over (i.e., with each term now as an integrand), we obtain P (z) dωi = zp 1 (z) dωi P 2 (z) dωi. (34) We now need to convert this to the notation of the previous section, and in particuar the surface integra S in equation 13. The main issue is that the canonica form above uses z for the argument of P, whie S uses ω ωi. For our derivation, we can assume, without oss of generaity, that we have rotated the coordinate system so ω is aigned with the Z axis, whie expicity preserving ω for the actua agorithm in Sec. 4. We can now view equation 34 as a reation for the surface integra, 2 1 ( 1) S = zp 1 (z) dωi S 2. (35) It is cear that the expression on the right is a previousy-computed vaue. Now we show how to compute zp 1 (z) dωi. We use Stokes Theorem to convert this into a boundary integra. Whie this approach has been used in many previous works, the form of our integras and recurrence is more compicated, requiring a much more invoved cacuation. Stokes theorem states that for a suitabe vector V, N ( V ) dωi = V dr, (36) where as usua, N is simpy the norma at ωi on the region of the sphere, with Cartesian components (x, y, z)t, and dr is a differentia curve segment on the boundary. To convert the integra expression in equation 35 to a boundary integra, we need to find the appropriate vector V, and further cacuations are sti required. Specificay, we use yp 1 (z) ª V = xp 1 (z) «(37) We take the cur, as in the eft-hand side of equation 36, d d P (z) y 2 P 1 (z) + 2zP 1 (z) dz 1 dz = (z 2 1)P 1 (z) + 2zP 1 (z), (38) N ( V ) = x 2 where we use P for dp/dz. We now appy equation 1 to rewrite P in the right-hand side of the above expression, N ( V ) = ( + 1)zP 1 (z) ( 1)P 2 (z). (39) We can now appy Stokes Theorem, converting the surface integra for the quantity above to a boundary integra. For simpicity of notation, we drop the argument z to the Legendre poynomias in the remainder of this subsection, ( + 1) zp 1dωi ( 1) P 2dωi = P 1 (xdy ydx). (4) ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

8 54:8 Wang and Ramamoorthi We can now to evauate the surface integra zp 1dω i in equation 35. Incuding the normaization 2 1 in equation 35 and rearranging terms in equation 4, 2 1 zp 1 dω i = (41) 2 1 (2 1)( 1) P ( + 1) 1 (xdy ydx) + P ( + 1) 2 dω i. Substituting this resut back into equation 35 gives S = 2 1 ( + 1) B 1 + S = 2 1 ( + 1) B 1 + (2 1)( 1) ( 1) S ( + 1) 2 S 2 ( 2)( 1) S ( + 1) 2, (42) which is exacty the main recurrence in equation Boundary Integras Now we derive a recurrence for the boundary integra B, which is simpy a sum of edge integras B,e, as in equation 21. We are interested in B,e = γ e P (z e )dγ, where as usua we have that z e = a e cosγ + b e sinγ. We first rewrite B,e using equation 11, γe P (z e ) dγ = 21 1 γe z e P 1 (z e ) dγ 1 γe P 2 (z e ) dγ. (43) The integras correspond respectivey to B,e on the eft-hand side, and C 1,e and B 2,e on the right-hand side. Considering the normaization factors, this is exacty the recurrence reation for B,e in equation 3. It is cear that the second term on the right has been previousy computed. The first term on the right corresponds to C 1,e = z e P 1 (z e )dγ. In the appendix, we derive an expicit formua for C,e, using integration by parts, deriving equation 31. Finay, to compute D,e, we can use equation 12, which we rewrite as (2 1)P 1 (z) = P (z) P 2 (z). Considering the corresponding integras and rearranging, γe γe γe P (z e )dγ = (2 1) P 1 (z e )dγ + P 2 (z e )dγ. (44) This can be written as D,e = (2 1)B 1,e + D 2,e, which gives the recurrence in equation IMPLEMENTATION AND RESULTS The agorithm is impemented in a simpe GPU vertex shader, which computes SH coefficients separatey at each vertex, foowing the pseudocode in Agorithm 1. We store pre-tabuated vaues of Legendre poynomias in 1D textures, which are interpoated in hardware and fetched within the shader program. This effectivey aows the Legendre poynomia evauations in equation 31 to be evauated in constant time. Once the SH ighting coefficients are computed, they can directy be used in the PRT system. Moreover, the SH ighting coefficients can aso be added to those from an environment map or from other ights to render images with mutipe area ight sources and environment ighting (Figs. 1,9). Anaysis of Accuracy. The spherica harmonic formua in Sec. 4 is exact. To verify the correctness of the formua and its derivation, we compare with numerica integration with 146k sampes per vertex Fig. 4. We show image comparisons using our anaytic SH computation, numerica integration (sow), and the pixe-wise difference. Note that the differences between the two resuts are at most one grey eve, and dispayed here at 1 times the intensity. For further verification, we aso pot the intensity aong the centra image scanine, where the anaytic and numerica curves agree amost perfecty. using stratified Monte Caro samping computed offine in Fig. 4. We have aso conducted a number of other simiar tests with different scenes, ights and SH orders. Figure 4 uses SH up to degree n = 8 (81 spherica harmonic terms) to show that we get high accuracy for genera degree SH. A comparison of anaytic and numericay computed images shows they are identica to within one gray eve. To expore further, we consider the centra scanine, potting overa intensity, which is again essentiay identica; the red numerica integration curve is amost indistinguishabe (and therefore invisibe) from the bue anaytic computation. Our method aows us to compute the spherica harmonics up to any order. In Fig. 5 we show the effects of increasing the order, obtaining sharper, more accurate shadows. We aso show one exampe with gossy refections ater in Fig. 1. Timings. We now discuss the time required to compute the SH ighting coefficients for increasing SH orders, shown in Fig. 6. For our method, we aso break out the time for iterative zona harmonic computation (Sec. 4.1) and the sparse zona harmonic rotation (Sec. 4.2). We profied the GPU time for a singe draw ca using NVIDIA s Nsight, performing our ighting method on a scene with 4k vertices. To obtain timings for the different steps of our method and accuratey observe performance impact of increasing orders, we disabed compier optimizations for these measurements. 8 Because of this, as we as the sight additiona performance overhead from using the profier, we show normaized time in Fig. 6. It can be seen that the iterative ZH computation, which is inear in the tota number of SH coefficients (Sec. 4.1), is the major time compexity of the method. Sparse zona harmonic rotation adds ony a sma overhead, which is negigibe at ower orders (not noticeabe at a unti order n = 12 or 169 SH terms). It is a maximum of 15% of 8 Optimized compies aso give roughy (but not exacty) inear performance with increasing coefficients. Differences are due to e.g. oad baancing at various GPU stages. ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

9 Anaytic Spherica Harmonic Coefficients for Poygona Area Lights 54:9 Fig. 7. Pot of timings for increasing the number of edges (vertices) in the poygona area ight, showing the method behaves ineary. Tabe 2. Running time for our method in frames per second. We aso incude a comparison to using the monomia recurrence in [Snyder 1996]. Fig. 5. Area ighting with SH orders 2, 4, 6, and 8. Note increasing accuracy in terms of sharpness of shadows with higher orders. Scene Tris. Lights Speed (Us) Monomia Pant (Fig. 5) 163k fps 91. fps Buddha (Fig. 1) 388k fps 8.2 fps Dragon (Fig. 9) 1M fps 15.3 fps Face (Fig. 8) 98k fps 15. fps Fig. 6. Pot of timings for increasing numbers of spherica harmonic coefficients. Note that the rotation step is negigibe at ower orders, and the overa method is therefore inear in the number of SH coefficients. tota runtime at order n = 14 (225 SH terms). Thus, in practice, the time compexity of our method is O(n 2 ) or inear in the number of SH coefficients, as desired. Finay, Fig. 7 pots normaized runtime vs. the number of ight poygon edges M. As expected the time compexity is inear in the number of edges (or vertices) of the poygona area ight. We now discuss actua wa-cock running times, using an NVIDIA GeForce GTX 18 Ti GPU, with compier optimizations enabed. For the pant scene in Fig. 5 with 163K trianges and one rectanguar area ight, at SH order n = 8, our method runs at 35 frames per second. We aso compared with using the monomia recurrence in [Snyder 1996] (the direct computation per [Arvo 1995] woud be even sower). To do so, we used our iterative method to compute the monomias instead of the ZH, foowed by combining them to compute the ZH coefficients, and then using our sparse SH rotation. Note that this is not the method of [Snyder 1996], since we ony use their recurrence, and ony one part differs from ours (computation of monomias rather than directy using the ZH recurrence). That method ran about 4 sower, at 91fps because of the time compexity in ineary combining monomias to obtain ZH. Whie the compier can optimize the required additions to make them very fast, the extra O(n) compexity for inear combinations of monomias sti adds significant overhead. Tabe 2 shows additiona runtime comparisons on a variety of scenes. The compexity depends on poygon count and the number of ights, achieving 3+fps for modes of about 1K trianges and one ight (Figs. 5, 8), and sti being rea-time at 28-39fps for the 1M poygon dragon scene in Fig. 9 as we as the 388K poygon Buddha scene in Fig. 1 with three poygona ights. In a cases, our agorithm is 2 4 faster than using the monomia recurrence of [Snyder 1996]. Appications. A number of appications and extensions enabed by our approach can be seen in the accompanying video and are briefy described here. Our method supports genera poygona area ights, and enabes editing of the ight source, rotations, or transations as shown in Fig. 8. (The scene can aso be rotated/transated, corresponding to an inverse transformation of the ights). Moreover, mutipe area ights can be combined, simpy by adding their SH coefficients (Fig. 1). The time compexity for SH ighting cacuation wi grow ineary with the number of ights, but the cost of ighting-visibiity computation in the PRT framework is not affected, since it need ony be done once at the end. We can aso incude environment maps (Fig. 9), with their SH coefficients computed in the conventiona way by numerica integration once per frame, and then simpy added to the anaytic area ight SH coefficients. ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.

10 54:1 Wang and Ramamoorthi Fig. 9. Our method can support traditiona reighting with environment maps in addition to area ights. Both the area ight and environment can be rotated/changed interactivey at rea-time framerates. Fig. 8. Our method supports transation, in/out of pane rotations, and editing of the shape of the ight source. Insets show area ight positions. In this exampe, we show SH order n = 8. Finay, whie we do not support textured ights (since the derivation and formuae assume constant emission over the poygon), we can in some cases break the area ight into smaer poygons, each with a different coor. Since we ony use ow-order spherica harmonics, detaied textures are not needed. An exampe is shown in Fig. 1. This resut aso uses a gossy ground refector and Buddha, eading to appearance that switches from reddish to buish, in accordance with the ight texture, from one end of the ground pane to the other. Simiary, the Buddha has buish as we as yeowish highights (combining red and green parts of the ight). Gossy refection is computed using spherica harmonic tripe products (Cebsch-Gordan coefficients) to mutipy the precomputed visibiity and area ighting computed with our method at each vertex, before convoving with a Phong BRDF [Soan et a. 22]. 7 CONCLUSIONS AND FUTURE WORK Spherica Harmonic ighting is a popuar technique today for reatime appications based on the precomputed radiance transfer method. We enabe the use of near-fied poygona area ights, amost as easiy as distant environment map ighting, by deriving anaytic resuts for integrating spherica harmonics against panar poygons. The key technica contribution is a nove recurrence directy for zona harmonic coefficients, which enabes a coefficients up to a given degree to be computed iterativey in inear time, foowed by fast spherica harmonic rotation. The agorithm is impemented in a simpe GPU shader for area ighting within any spherica harmonic ighting method. In the future, we woud ike to generaize the resuts to non-uniform or directiona emission. So far, we have not expoited the smoothness of area ighting. This can be done in at east two ways. First, the SH coefficients are ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218. Fig. 1. More compex ight sources with different patterns and coors can be broken into constituent poygons and used within our system. This exampe aso shows gossy refections. smooth spatiay. One coud compute the SH projections at a much ower-frequency coarse mesh (or even a sparse voume grid for the whoe scene) and interpoate, whie sti representing visibiity and performing PRT at vertices of the dense mesh. Moreover, at each vertex, the integra to compute SH coefficients from the area ight is smooth, usuay without the discontinuities typicay present in graphics integras. As such, an optimized samping-based numerica integration or quadrature scheme may have ow error, whie being competitive in speed with our exact anaytic approach.

11 Anaytic Spherica Harmonic Coefficients for Poygona Area Lights 54:11 In summary, we have shown that anaytic ighting approaches can be combined with methods for compex ight transport such as soft shadows, enabing wider avaiabiity of near-fied effects in future rea-time rendering appications. ACKNOWLEDGMENTS We thank the reviewers for many hepfu suggestions, James Ha for discussion regarding the recurrence and Weiun Sun for impementation suggestions. The Happy Buddha and Dragon modes are from the Stanford 3D Scanning Repository. The pant mode was provided by Xfrog through TurboSquid. This work was supported in part by NSF grant IIS , Samsung, the Ronad L. Graham endowed Chair, and the UC San Diego Center for Visua Computing. REFERENCES J Arvo Appications of irradiance tensors to the simuation of non-lambertian phenomena. In SIGGRAPH D. Baum, H. Rushmeier, and J. Winget Improving Radiosity Soutions through the use of anayticay determined form-factors. In SIGGRAPH L. Becour, G. Xie, C. 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We start with the formua for C,e in equation 22, and integrate by parts, ( ) γ e ( ) zp (z) dγ = P (z) zdγ zdγ dγ P dz (z) dγ = (a sin γ b cos γ )P (z) γe + ( a sin γ + b cos γ ) 2 P (z)dγ. (45) In the ast ine, we use z = a cos γ + b sin γ, making it easy to find z dγ and dz/dγ. It is now convenient to write C,e = C,e 1 + C2. The first term,e above is simpe to evauate directy, eaving us with C 1,e = (a e sin γ e b e cos γ e ) P (a e cos γ e + b e sin γ e ) + b e P (a e ). (46) The second term C,e 2 is found by integrating by parts again, ( ) ( a sin γ + b cos γ ) 2 P (z)dγ = ( a sin γ + b cos γ ) 2 P γ e (z)dγ ( ) P (z)dγ ( 2(b cos γ a sin γ )) (a cos γ + b sin γ ) dγ ) = (( a sin γ + b cos γ ) 2 P γ e ( ) (z)dγ P (z)dγ ( 2zdz). (47) We now perform another integration by parts for the second term, ( ) ) γe (( a sin γ + b cos γ ) 2 P (z)dγ + P (z)dγ z 2 z 2 P (z)dγ = ( (( a sin γ + b cos γ ) 2 + z 2) ( P (z)dγ )) γe ( = (a 2 + b 2 ) ) P (z)dγ z 2 P (z)dγ z 2 P (z)dγ. (48) Note that we substitute for z = a cos γ + b sin γ to obtain the ast ine. In the process, we aso removed the evauation from to γ e, since a 2 + b 2 is a constant. We now use the identity in equation 1, rearranging to obtain z 2 P (z) = (zp (z) P 1 (z)) + P (z). Substituting inside the second integra, the above expression becomes, ( ) (a 2 + b 2 ) P (z)dγ ( + ) ( P 1 (z)dγ ( = (a 2 + b 2 1) ( P (z)dγ ) ) ( P (z)dγ ) zp (z)dγ ) ( zp (z)dγ + Pugging back into equation 45 gives us ( + 1) zp (z)dγ = (a sin γ b cos γ )P (z) γe + ( ) ( ) (a 2 + b 2 1) P (z)dγ + P 1 (z)dγ (49) ) P 1 (z)dγ. We are now ready to obtain a formua for C,e, which corresponds to the integra on the eft-hand side. The right-hand side integras are simpy D,e and B 1,e, whie equation 46 can be used for the evauation at and γ e. Putting this together, and re-introducing the index and subscript e for the edge, ( + 1)C,e = (a e sin γ e b e cos γ e )P (a e cos γ e + b e sin γ e ) + b e P (a e ) + (a 2 e + b 2 e 1)D,e + B 1,e. (51) If we now consider the corresponding formua for C 1,e, we obtain, C 1,e = (a e sin γ e b e cos γ e )P 1 (a e cos γ e + b e sin γ e ) + b e P 1 (a e ) (5) + (a 2 e + b 2 e 1)D 1,e + ( 1)B 2,e. (52) Upon dividing through by, we obtain equation 31. A APPENDIX: BOUNDARY INTEGRAL C In this appendix, we derive the formua for C,e. For simpicity of notation, we drop the subscript and index e for the edge, and omit the imits of the ACM Transactions on Graphics, Vo. 37, No. 4, Artice 54. Pubication date: August 218.