Freeform Shading and Lighting Systems from Planar Quads

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Freeform Shadng and Lghtng Systems from Planar Quads Cagu Jang, Jun Wang Kng Abdullah Unversty of Scence and Technology, Thuwal, Saud Araba Phllppe Bompas Archtect, 70 rue du Temple, Pars, France

ths paper s optmzed shadng and lghtng systems whch consst of planar fns arranged along the edges of a quad-domnant base mesh, that mesh tself coverng a reference surface.

): A reference surface consstng of three vaults wth curved valleys n between s panelzed by a quad mesh whose faces are not planar; planar glass panels are mounted on a grd of quadrlateral fns whch

1 Freeform Shadng and Lghtng Systems from Planar Quads Cagu Jang, Jun Wang Kng Abdullah Unversty of Scence and Technology, Thuwal, Saud Araba Phllppe Bompas Archtect, 70 rue du Temple, Pars, France Helmut Pottmann Kng Abdullah Unversty of Scence and Technology, Thuwal, Saud Araba and Venna Unversty of Technology, Wen, Austra Johannes Wallner Graz Unversty of Technology, Graz, Austra The topc of ths paper s optmzed shadng and lghtng systems whch consst of planar fns arranged along the edges of a quad-domnant base mesh, that mesh tself coverng a reference surface. Such an arrangement can be observed e.g. n the Kogod courtyard roof desgned by Foster + Partners for the Natonal Portrat Gallery n Washngton DC (Fg. ): A reference surface consstng of three vaults wth curved valleys n between s panelzed by a quad mesh whose faces are not planar; planar glass panels are mounted on a grd of quadrlateral fns whch follow the edges of the quad mesh. In our paper we consder structures of exactly that type, whose geometry s herarchcally set up as follows: The frst element n the herarchy s the reference surface, whch may be any freeform shape. Secondly, the reference surface s panelzed by a quad-domnant mesh, whch s referred to as the base mesh. Thrdly, planar fns are arranged along the edges of the base mesh such that n each vertex the planes of fns ncely ntersect n a common node axs. For the purposes of ths paper, the prmary herarchy the reference shape s gven, whle the second and thrd herarches are to be determned by geometrc obectves lke maxmal shadng or the gudng of lght by reflecton. It turns out that these second and thrd levels (mesh and fns) are ntmately connected to each other and cannot be chosen ndvdually and ndependently. The total of all three levels together s called a torson-free support structure. It s an mportant pont we wsh to make that even n case the prmary surface s flat (lke an ordnary vertcal facade, or a flat roof), the secondary and tertary level are stll freeform. Here we brefly revew the geometrc background, demonstrate results of geometry optmzaton, and dscuss ts applcaton to the desgn of reflecton patterns. We contnue wth manufacturng ssues, and we conclude wth the stablty and rgdty/flexblty of support structures. Foto: Foster+Partners Foto: Foster+Partners Foto: Foster+Partners Fgure : Kogod Courtyard, Smthsonan Natonal Portrat Gallery, Washngton DC. The glass canopy, desgned by Foster + Partners, represents an arrangement of quadrlaterals whch follow the edges of a quad mesh and whch actually consttute a torson-free support structure.

PREVIOUS WORK AND GEOMETRIC BASICS OF TORSION-FREE SUPPORT STRUCTURES From the abstract vewpont of geometry, our obects of study are quad-domnant meshes wth addtonal decoraton: each vertex (resp.

In order to represent a torson-free support structure, these two tems must be consstent: If an edge contans a node, ts correspondng plane contans the node axs (see Fg. enter).

, 203) propose a computatonal approach for the modelng and optmzaton of such support structures whch satsfy constrants lke the blockng of lght, and whch comply wth desgners wshes lke boundary

2 PREVIOUS WORK AND GEOMETRIC BASICS OF TORSION-FREE SUPPORT STRUCTURES From the abstract vewpont of geometry, our obects of study are quad-domnant meshes wth addtonal decoraton: each vertex (resp., node) s equpped wth a node axs, and each edge s equpped wth a plane carryng the fns. In order to represent a torson-free support structure, these two tems must be consstent: If an edge contans a node, ts correspondng plane contans the node axs (see Fg. enter). The faces of the base mesh (realzed as glass panels n Fg., rght) are of no concern n ths paper. (Wang et al., 203) propose a computatonal approach for the modelng and optmzaton of such support structures whch satsfy constrants lke the blockng of lght, and whch comply wth desgners wshes lke boundary algnment or prescrbed lnes of sght. Ths modelng procedure entals two steps, whch we do not present n detal here. The frst s to optmze a feld of node axes for a dense sample of vertces placed on the reference surface (wthout knowng the fnal locaton of nodes and edges yet). It turns out that ndeed t s best from the computatonal vewpont to have the node axes as varables n optmzaton. A result of that optmzaton procedure can be seen n Fg. 2a. In a second step one extracts a quad mesh such that the prevously obtaned axes are usable for a support structure wth planar fns (one has to choose edges such that the node axes at ether end are co-planar). The second step s conceptually very smlar to fndng a mesh whose edges follow the prncpal drectons of curvature, see Fg. 2a and Fg. 2b. Fnally one has to apply global optmzaton to ensure planarty of fns, see Fg. 2c. (a) (b) (c) Fgure 2: Typcal example of optmzaton procedure, after node axes have already been determned for a dense sample of ponts on the reference surface. (a) shows a small selecton of node axes (red) and also a sample of the prncpal drectons (blue) whch correspond to ths system of node axes. They are determned by methods of dfferental geometry, cf. (Pottmann and Wallner, 200). (b) already shows the 2nd and 3rd geometry layers whch augment the reference surface: A quad mesh whose edges follow the prncpal drectons, and quadrlateral fns whch are spanned by the edges and node axes. The quads whch occur here are already approxmately planar, so the arrangement of quads s already close to a support structure. Subfgure (c) shows the support structure after global optmzaton such that fns become planar. The degree of planarty s shown by color codng: the red color corresponds to quads whch are off a plane by 2% of ther dameter. These fgures are taken from (Wang et al., 203). The present paper s not concerned wth detals of those methods of optmzaton, whch would anyway ft nether ams and scope of ths conference, nor the usual length of ts papers. One of the questons studed n ths paper, namely reflecton patterns, does requre optmzaton, but we smply refer to the paper by (Wang et al., 203) and use optmzaton as a black box. We brefly menton further prevous work n ths area. Controllng patterns created by lght s an mportant topc n varous felds, e.g. both Computer Graphcs and Archtecture. For computatonal approaches, we refer to (Papas et al., 20), (Kser et al., 202) for modelng caustcs, and to (Weyrch et al., 2009) for modelng mcrogeometry reflectons. As already mentoned, the man reference for the computatonal part of the present paper s (Wang et al., 203). The dfferental geometry aspects whch occur n modelng reflectons are treated by (Pottmann and Wallner, 200), as are the bascs of knematc geometry whch we need for stablty consderatons.

Fgure 3: Left: Reference surface wth sun path. Rght: Shadng system optmzed for blockng lght for a gven sun poston, for shallow shadng fns, and for reflectng lght onto the celng.

DESIGNING REFLECTION PATTERNS We here dscuss the desgn of a torson-free support structure whch functons as a system of fns whch reflect ncomng lght towards a gven lght pattern.

3 Fgure 3: Left: Reference surface wth sun path. Rght: Shadng system optmzed for blockng lght for a gven sun poston, for shallow shadng fns, and for reflectng lght onto the celng. Here the detaled shape of the reflecton patterns s not the subect of optmzaton, only the general drecton of outgong lght s. Ths example s taken from (Wang et al., 203). DESIGNING REFLECTION PATTERNS We here dscuss the desgn of a torson-free support structure whch functons as a system of fns whch reflect ncomng lght towards a gven lght pattern. Artfcal examples of such desred reflecton patterns can be seen n Fg. 4. As regards the three-level geometry herarchy mentoned above, we assume that a reference surface Φ s gven, and so s a doman Φ, whch typcally les n some plane. We wsh to determne the 2nd and 3rd geometry layer (base mesh and reflectng fns) such that lght httng Φ s reflected n the fns towards Φ. In the followng we descrbe the modelng procedure n more detal: Frst we fnd a correspondence whch maps ponts P Φ to ponts P = Φ. For some examples, e.g. f Φ s a curve-lke doman lke the sne wave of Fg. 4c, ths s easy: After choosng local coordnate systems for both Φ and Φ, ths mappng could read P = ( x, y, z) P = ( ax, bsn( cy),0),where abc,, are sutably chosen parameters. For other patterns we use e.g. a correspondence defned by mean value coordnatesf. (Floater, 2003). For each pont P we can compute a unt vector L representng ncomng lght and a unt vector L2 = ( P P)/ P P representng outgong lght. Then the vector N( P) = L L2s a normal vector of a plane whch can effect the desred reflecton (n case we eventually use P as the locaton of a reflectng fn). If a support structure s to effect the desred reflecton, then at least one half of the fns must be used for that purpose. The support structures llustrated by Fg. and Fg. 2c genercally have four fns adacent to each node axs, formng a cross. If P Φmarks the locaton of such a node axs, two of these fns must be placed approxmately orthogonal to the vector N( P ). (Wang et al., 203) has a method of fndng a feld of node axes where one half of fns, so to speak, s already determned. We apply that method as a black box, whch yelds a feld of node axes from whch a support structure could be derved (that would be the steps depcted n Fgs. 2b,c). Numercal experments show that support structures generated by ths method do not fathfully reproduce the desred reflecton pattern. Ths s not really to be wondered at snce the spatal poston of one half of the fns s well-determned by the orgnal correspondence Φ Φ alone, and we cannot expect that a crcutous optmzaton procedure (known normal vectors of half of the fns computng node axes computng fns agan) yelds the very fn dstrbuton we started from. Snce exactness of fn placement s very mportant when desgnng reflecton patterns, we reconsder the feld of node axes, whch s n the stage depcted by Fg. 2a. It comes wth a feld of suggested prncpal drectons, one half of whch corresponds to the reflectng fns. We smply apply correcton to the feld of prncpal drectons, replacng the wrong drectons wth the known drectons whch stem from the correctly placed fns. From now on the constructon of a support structure s the same as appled by (Wang et al., 203). We have chosen a sne wave reflecton pattern to not only compute, but actually buld, a support structure. Fg. 5

(a) (a2) (b) (b2) (c) (c2) Fgure 4: Playng wth reflecton patterns.

The pattern s smulated by renderng the behavour of planar quads whch are unformly dstrbuted along the reference surface, each of whch reflects ncomng lght towards a certan pont n the llumnated regon.

(a) (b) (c) Fgure 5: Modelng of reflecton patterns. (a) Torson-free support structure whose fns are to reflect ncomng lght towards a prescrbed shape, n ths case a sne wave.

Snce the other half of fns does not agree on a common destnaton for reflected lght, the orgnally ntended lght dstrbuton s the only one whch s clearly vsble.

4 (a) (a2) (b) (b2) (c) (c2) Fgure 4: Playng wth reflecton patterns. The subfgures labelled wth show the desred reflecton pattern whch s to occur f lght falls n from a fxed drecton and hts an arrangement of fns. The pattern s smulated by renderng the behavour of planar quads whch are unformly dstrbuted along the reference surface, each of whch reflects ncomng lght towards a certan pont n the llumnated regon. The subfgures labelled wth 2 show the smulated reflecton pattern created by a complete arrangement of planar fns whch meet ther neghbours at node axes. (a) (b) (c) Fgure 5: Modelng of reflecton patterns. (a) Torson-free support structure whose fns are to reflect ncomng lght towards a prescrbed shape, n ths case a sne wave. (b) smulaton of reflecton of ncomng lght n all fns, not only that half of fns whch s desgned to reflect lght nto the wave. Snce the other half of fns does not agree on a common destnaton for reflected lght, the orgnally ntended lght dstrbuton s the only one whch s clearly vsble. (c) shows a photo of the actual lght dstrbuton obtaned by llumnatng the model by a strong flashlght. (d) shows a cutout pattern for strps whch can be bent along node axes and stacked together, thus manufacturng the support structure. llustrates the computed support structure, a smulaton of lght reflected n the support structure, a cutout pattern for manufacturng (antcpatng the next secton), and a photo of the actually occurrng reflectons whch were created wth a flashlght. Remark. Dfferental geometers wll recognze the node axes as the elements of a lne congruence; planar fns approxmate torsal planes. The procedure above mmcs the dfferental geometrc fact that one famly of torsal planes determnes the lnes of the congruence and n turn the other famly of torsal planes. (d) MANUFACTURING Torson-free support structures have two geometrc propertes whch are relevant for manufacturng, vz., the planarty of fns, and the fact that successve fns meet each other n a node axs. These propertes suggest at least two ways of manufacturng. One of them s shown by Fg. 6. Here shadng fns are connected to each node axs n a way whch does not transmt moments about that axs. Another one s depcted n Fg. 7 and conssts n collectng a sequence of successve fns n a strp whch can be developed nto the plane. After cuttng out these strps and cuttng slts nto the strps where the node axes are gong to be, one can bend the strps nto shape and assemble them as shown n Fg. 7.

(a) (b) (c) Fgure 6: A way of connectng shadng fns at node axes. Subfgures (a) (c) respectvely llustrate the regular case of 4 fns onng n the node axs, and the sngular cases of 3 and 5 fns.

For applcatons whose dmensons are close to the glass canopy of Fg. t s unlkely that a manufacturng soluton wll be anythng lke cuttng out strps and bendng them.

The two propertes mentoned above are helpful n any case. STABILITY An mportant queston to be asked s f support structures are stable.

5 (a) (b) (c) Fgure 6: A way of connectng shadng fns at node axes. Subfgures (a) (c) respectvely llustrate the regular case of 4 fns onng n the node axs, and the sngular cases of 3 and 5 fns. The second method requres that the materal used for manufacturng allows for the necessary bendng, whle stll dschargng all structural functons. For applcatons whose dmensons are close to the glass canopy of Fg. t s unlkely that a manufacturng soluton wll be anythng lke cuttng out strps and bendng them. Obvously for each realzaton of a support structure n practce the manner of manufacturng must be chosen n a way whch mnmzes costs, under the sde condtons specfc to that applcaton. The two propertes mentoned above are helpful n any case. STABILITY An mportant queston to be asked s f support structures are stable. We here present an approach to ths queston whch avods a full-blown fnte element analyss and nvestgaton of forces and stresses, but nstead nvestgates the possblty of the structure beng flexble and, n consequence, unstable. Ths approach works by assumng that the structure conssts of ndvdual rgd bodes whch are connected together by hnges. Those rgd bodes could be the ndvdual shadng fns of a support structure whch are connected accordng to Fg. 6; or they could be rgd developable strps consstng of successve fns lke the ones shown n Fg. 7, wth hnge connectons at each node axs. In the followng we descrbe n more detal an algorthm to determne the degree of flexblty or rgdty of any such system consstng of rgd bodes connected wth hnges. N Algorthm. We enumerate all rgd bodes and call them,,. If and are connected by a hnge, we choose two ponts A, B on the hnge and declare them to belong to both and. It s well known that for any rgd body moton of the system the velocty state of s governed by the vector C of angular velocty, and another vector C whch s the velocty experenced by the orgn of the coordnate system; the velocty experenced by any pont P connected to s then gven by V ( P) = C P+ C, (*) cf. (Pottmann and Wallner, 200). Bodes whch are connected cannot move ndependently; for each hnge we have the condtons C A + C = C A + C, C B + C = C B + C, (**) whch express the fact that the ponts A and B on the hnge must experence the same velocty, regardless of the system they belong to. Snce the entre arrangement of rgd bodes can always move as a sngle rgd body, the lnear system (**) has a 6- dmensonal space of trval solutons. In order to elmnate these we constran one system, to reman at rest. Assumng our arrangement of systems and hnges s stll flexble, there wll be a nonzero velocty state whch solves Equaton (**); otherwse the only soluton wll be zero veloctes. In order to determne how far our arrangement s from beng flexble, we solve for a best-possble soluton of (**) under the sde condton that each system must move. It s well

Fgure 7: Buldng a model. Ths arrangement of strps s part of the larger shadng and lghtng system shown by Fg. 3.

Top: Renderng of shadng system and photo of the acrylc glass model. known that the soluton s obtaned by the followng standard procedure:.

2 N N 2( N ) unknown vectors C, C,, C, C. 2. The system (**) represents two vector equatons per hnge, and a total of 6 scalar equatons per hnge.

Actually these 6 scalar equatons per hnge only count as 5 equatons, because they are lnearly dependent, but ths detal s not essental here. 2 2 2 N N N 3.

6 Fgure 7: Buldng a model. Ths arrangement of strps s part of the larger shadng and lghtng system shown by Fg. 3. Strps consstng of successve shadng fns have been developed nto the planeut out from acrylc glass, and stacked together. Left: Cutout pattern of strps. Top: Renderng of shadng system and photo of the acrylc glass model. known that the soluton s obtaned by the followng standard procedure:. We constran one system, say, to reman at rest by enforcng C = (0,0,0) and C = (0,0,0). There reman 2 2 N N 2( N ) unknown vectors C, C,, C, C. 2. The system (**) represents two vector equatons per hnge, and a total of 6 scalar equatons per hnge. If the number of hnges s denoted by H, we get a total of 6H scalar equatons for the 6N scalar components c 2, c ( =,..., N) of whch 6( N ) are unknown. Actually these 6 scalar equatons per hnge only count as 5 equatons, because they are lnearly dependent, but ths detal s not essental here N N N 3. We put the 6( N ) varables c 2 3, 2 c3 n a sngle column vector X, and wrte the lnear condtons(**) n the form AX = 0 wth a certan matrx A of 6H rows and 6( N ) columns. 4. The above-mentoned best possble soluton s more precsely defned as the mnmzer of AX under the condton X =, where the usual 2-norm for vectors s employed. 5. It s well known that the soluton X s gven as the frst row of the matrx V n a sngular value decomposton A=UDV, where U, V are orthogonal matrces and D s a rectangular-dagonal matrx whose nonnegatve dagonal entres are sorted n ascendng order. The resultng vector X represents velocty states of the ndvdual rgd bodes whch are as consstent as possble n the sense that the devaton of veloctes n the hnges s mnmal (n the least squares sense). In case ths mechansm s truly flexble, the devaton of veloctes wll vansh, and (**) wll be fulflled exactly for all hnges.

0 00 0 00 0 Fgure 8: Illustraton of stablty of a shadng system va ts rgdty. In (a) the long strps of Fg. 7 are consdered rgd, whle n (b) and (b2) each shadng fn s a rgd body of ts own.

The veloctes of an almost consstent flexon are computed as descrbed n the text; veloctes are normalzed by fxng one system and lettng the velocty of a dstance node equal unt/sec.

The computatons descrbed n the prevous paragraph yeld a velocty state ( C, C ) for each rgd component of a support structure.

7 Fgure 8: Illustraton of stablty of a shadng system va ts rgdty. In (a) the long strps of Fg. 7 are consdered rgd, whle n (b) and (b2) each shadng fn s a rgd body of ts own. In all cases the node axes act as hnges. The obect s scaled such that the average dstance of nodes from the barycenter s 000 unts of length. The veloctes of an almost consstent flexon are computed as descrbed n the text; veloctes are normalzed by fxng one system and lettng the velocty of a dstance node equal unt/sec. The colors llustrate the dscrepancy experenced by hnge vertces after travelng for sec. Interpretaton of Results. The computatons descrbed n the prevous paragraph yeld a velocty state ( C, C ) for each rgd component of a support structure. In partcular, each pont P belongng to system s assgned a velocty V ( P) = C P+ C. We now let each pont move for a tme nterval of second, so that P moves to the new poston P+ V ( P). For a hnge pont we may compute the updated postons w.r.t. both systems t belongs to. The dscrepancy ( ) ( ) P+ V ( P) P+ V ( P) = V ( P) V ( P) (a) (b) (b2) experenced by that hnge s llustrated by Fg. 8. One can clearly see that n case the entre strps are consdered rgd, the devaton at hnges s way too hgh to be accommodated by the usual leeway present n connectons; whereas n case each ndvdual fn s rgd, ths devaton s rather small, so that for all practcal purposes the arrangement must be consdered flexble n consequence, the fn arrangement of Fg. 7onnected va hnges as shown by Fg. 6, s unstable and cannot be expected to bear loads (ncludng the dead load) unless the connecton s made rgd so as to transmt moments also. CONCLUSION After reportng on motvaton we revewed the capabltes of a computatonal approach to shadng and lghtng systems publshed by (Wang et al., 203). Ths paper extends the applcatons gven there, n partcular we gven examples where reflecton patterns are desgned. A man focus of ths paper s manufacturng: We not only demonstrate cutout patterns correspondng to shadng and lghtng systems, but we also study the stablty of such systems. The methods whch are employed n ths analyss are knematc n nature and are founded on the prncple that flexblty and stablty exclude each other. In concluson we beleve that ths topc of controllng lght and shade s an attractve one whch nvolves dfferent dscplnes, n partcular geometry processng, and whch can beneft from each other. REFERENCES Floater, M. S., Mean value coordnates. Comp. Aded Geom. Desgn, 20:9 27. Kser T.; Egensatz M.; Nguyen M. M.; Bompas P.; Pauly M., 202. Archtectural caustcs controllng lght wth geometry. In L. Hesselgren et al., edtors, Advances n Archtectural Geometry 202, pages Sprnger. Papas M.; JaroszW.; JakobW.; Rusnkewcz S.; MatuskW.;Weyrch T., 20. Goal-based caustcs. Computer Graphcs Forum, 30(2):503 5, Proc. Eurographcs. Papas M.; Hout T.; Nowrouzezahra D.; Gross M.; Jarosz W., 202. The magc lens: refractve steganography. ACM Trans. Graph., 3(6):86: 86:0, Proc. SIGGRAPH Asa. Pottmann H.; Wallner J., 200. Computatonal Lne Geometry. Sprnger. Wang J.; Jang C.; Bompas P.; Wallner J.; Pottmann H., 203. Dscrete lne congruences for shadng and lghtng. Computer Graphcs Forum, 32(5):53 62, Proc. Symposum Geometry Processng. Weyrch T.; Peers P.; MatuskW.; Rusnkewcz S., Fabrcatng mcrogeometry for custom surface reflectance. ACM Trans. Graph., 28(3):32: 32:6, Proc. SIGGRAPH.

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges