A Description Method of Spatial Complexity in Terms of Visibility

Transcription

1 IPR IPT IGU UCI CIG ACG Table of contents Table des matièes Authos inde Inde des auteus each Recheches Eit oti A Desciption Method of patial Compleit in Tems of Visibilit Hiotaka uzuki Assistant Pofesso, Gaphic cience, Depatment of Uban Engineeing, Gaduate chool of Engineeing, Osaka Cit Univesit, , ugimoto, umioshi-ku, Osaka Cit, , Phone&FAX , suzuki@ach.eng.osaka-cu.ac.jp Abstact The final goal of this eseach is gasping the elation between visual qualit and shape of a eal space. The visual qualit of a eal space means effects on human pscholog o difficult of evacuation. As the fist step, this pape agues a desciption method of spatial compleit in tems of visibilit. A simple concave figue can be used as an eamine space to calculate distibution of visible quantit. Fistl, we define fomulas to obtain eact value of visible quantit in the space. Then, visible quantit distibution of the space is visualized based on the fomulas. Finall, we t to descibe the compleit of the space. Aveage and standad deviation of the visible quantit distibution ae used to descibe the compleit and tansition of them is obseved with tansfoming the space. The elationship between the intuitive impession of space and visible quantit was suggested as a esult of eamination. We pesent the possibilit of evaluating visual qualit of a eal space b visible quantit of the space. Kewods: visibilit, visible quantit, spatial compleit, concave space Intoduction Vaious attempts have been made to link phsical quantit based on geometic shape of eal spaces and pschological esponses of humans. Visible quantit is one eample and is being studied fom vaious pespectives. Visible quantit is a fied quantit in espect to the position of the visual point, not affected b visual diection. It can theefoe be discussed in elation to pschological effects imposed b the geometical shape at that position, without being confined to the scope of esticted visual diection. Howeve, most studies on visible quantit until now have focused on discetel investigating and descibing visible and invisible elations in actual space, and do not claif the intinsic values and distibution of visible quantit. The autho is studing the possibilit of the elation between phsical quantit and mposium on Geospatial Theo, Pocessing and Applications, mposium su la théoie, les taitements et les applications des données Géospatiales, Ottawa 00

2 pschological esponse of numeous space shapes amounting to elation between visible quantit and pschological esponse in all diections. As phase of this stud, this pape poposes a desciption method of spatial compleit in tems of visibilit. Fig.. The evaluation of concave compleit in tems of visibilit A wide ange of eseach esults in the computational geomet field can be given fo visible quantit. In elation to methods fo dividing visible domains based on spaces fom given dawings, museum poblems, fotification poblems, lighting poblems, etc. have been pinpointed(fo eample Akiama et al(996)). High speed algoithms fo obtaining visibilit elation has been studied b man eseaches(fo eample Asano(990) o B. Chazelle et al (989)). In the field of achitectual engineeing and uban planning, studies such as the following have also been conducted: stud on aangement chaacteistics of Mosque focusing on visibilit of the mosque fom inside Islamic cities b Oikawa(000), analsis of uban factos which have a lage effect on visibilit b Kametani et al(997) and Lee et al(997), stud on elation between uban stuctue and shape estictions and visibilit b Hsiao et al(997). etc. This stud aimed at gasping the elation between visual qualit and shape of a eal space using visible quantit defined in continuous space. The visual qualit of a eal space means effects on human pscholog o difficult of evacuation. This epot does not go into the elation between visible quantit and pschological level. It will howeve stud the feasibilit of applications to the ealistic space b claifing that visible quantit occus due to the shape of space. The appoach emploed hee is coesponding to evaluate degee of concave instead of classif plane figues into onl two categoies, conve figue and concave figue as illustated in Figue. Definition of Visible Quantit and Calculation Method. Definition of Visible Quantit Taking into consideation the closed domain shown in Figue, when the line connecting the two points in the given does not intesect the bounda of

3 domain (when is a polgon, the sides), these two points ae visible. Hee, the visibilit vis (,) is defined as the aea of the domain made up b the aggegate of points that ae visible fom P(,) inside (see figue ). inv (,) is defined as the invisible aea of the domain fom P(,). If is a conve dawing, since an combination of points inside is visible, visible quantit is equivalent to aea inside at an point. In this pape, as the fist step in studing visible quantit in space, concave figues made b cutting out pat of a squae space with pimitive dawings (squaes, ectangles, cicles) ae used to stud visible quantit. And we obseve changes in visible quantit due to the cutting out method and diffeences in the shape of dawings cut out. Regading the effects of the shape of space on pschological aspects, since the size of space ma also pla a pat, calculation using visible quantit diectl is also possible. Howeve in this epot, based on the aim of investigating changes in visible quantit due to diffeences in shape, visible ate vis (,) was calculated b dividing visible quantit vis (,) b aea, and this value was compaed. In addition, the aveage of visibilit ates ove the whole domain and standad deviation wee calculated. is consideed to be coesponding to the degee of ease in gasping the whole space, while is consideed to be equivalent to the intensit of changes in visibilit quantit in space. Table. mbols and notation P(,). vis (,) inv (,) vis (,) (,) (,) closed domain point inside the domain the aea of the domain the visible aea of the domain the invisible aea of the domain the visible atio of the domain vis (,) / the aveage of visibilit ates ove the whole domain the standad deviation of visibilit ates ove the whole domain Hee, the elation with phsical quantit used in past studies is descibed. In the stud in Oikawa(000), the numbe of minaets which ae visible fom the measuing points at the cente of steets in actual space ae taken as numbe of visibilities. This is equivalent to the case of limiting visible subjects and discetel calculating in the visibilit ates of this pape. B distibuting measuing points at egula intevals in a domain and taking the subjects of visibilit evaluation as othe measuing points, the value obtained b dividing the numbe of visibilities b the numbe of measuements appoaches the visibilit ate in this pape when the distibution densit of measuing points is inceased. In the studies in Kametani et al(997) and Lee et al(997), the aanged two tpes of visible subjects, whole domain and inside steets, and also aanged two phsical quantities, elative visibilit ate and visibilit ate in the visible domain. The visible domain is also calculated discetel and the domain is consideed as the thee-dimensional space. in these studies. Despite diffeences, the idea is same as the visibilit ate used in this pape.

4 Fig.. Definition of visibilit quantit Fig. 3. Paametes and patial domain fo calculation of visible quantit fo L- shaped space The following shows how visibilit ate is calculated fo a L-shaped space, squae cut out space, and cicula cut out space. Calculation fo L-shaped space As shown in Figue 3, the domain made b cutting out a ectangula shape measuing fom a squae space( ) is called L-shaped space. In ode to

5 seek the visible quantit of such spaces, it is necessa to divide the into five sub domains fom A to E based on the etended lines of the sides of the domain bounda, and the lines joining vetices of the polgon. The following shows the visible quantit fo segment domains A to E shown in Figue 3. Hee,, and follow the smbols in Figue and Figue 3. Taking the visible quantit as vis (,) and invisible quantit as inv (,): vis + inv = = () inv is epessed in the following fo convenience. domain A: inv () B: inv (3) C: inv 0 (4) D: inv (5) E: inv (6) inv (,) in equations () to (6) ae continuous in domain. Thus both vis (,) and vis (,) ae also continuous in accoding to equation ()..3 Calculation fo squae cut out space As shown in Figue 4, the squae spaces( ) ae cut out using the squae with side aanged paallel to the squae making up the etenal cicumfeence of space, taking the oigin as the cente. In this case, it is necessa to stud onl the of the fist quadant fom the smmet. till as shown in Figue 4, thee is a need to divide it into the five sub domains fom A to E using the etension lines of the sides of the cut out squae and staight lines linking the vetices of two squaes. Because this calculation becomes slightl complicated than the L-shaped space, a sepaate calculation method is applied. As shown in Figue 5, taking P to be the fist point intesecting the squae shape cut out b otating the visual line diection stating fom the positive diection of the ais with P as the visual point, P to be the final point of intesection, to be the aea of the domain enclosed b staight line PP, staight line PO and the sides of the oute squae, and to be the aea of the domain enclosed b staight line PP, staight line PO,

6 and the sides of the oute squae. Hee, when the coodinate of the point of intesection between staight line PP and staight line = is taken as X and that of the point of intesection between staight line PP and staight line = is X, then X, X Using these, will be as follows fo domains A and B o when, A,B: ( ) ( ) Fo domains C and D, o when and X C,D: ( ) ( ) ( ) Fo domain E, o when X, then, then E: 3 ( ) Likewise will be as follows fo domains B and C, X, B,C: ( ) (7) (8) (9) (0) Fo domains A, F, and E, o X, A,D,E: 3 As and include the visible domains as shown in Figue 5, it is impossible to simpl total these esults. Taking the aeas of the visible domains included in and espectivel as ' and ', then: A,B: ' 3 () () C,D,E: ' Whole domain: ' (3) (4)

7 Taking equations (7) to (4) into account, the invisible quantit is then inv (,) ' ' (5) 4 This is a continuous function in (). Both visible quantit and visibilit ate ae also continuous in domain. Fig. 4. Paametes and patial domain fo calculation of visible quantit fo squae cut out space Fig. 5. Calculation of squae cut out of space

8 .4 Calculation fo cicula cut out space Figue 6 shows the cutting off of a squae space( ) using a cicle with a adius aound the oigin. In the same wa as cutting out using a squae space, onl the ange of the fist quadant needs to be taken into consideation. Thee is a need to split the aea into the fou sub domains A to D shown in Figue 6 accoding to the tangent of the cicle passing though the vete of the oute squae. As shown in Figue 7, like.3, calculation is done using and. P and P ae tangent points ling on the tangent of the cicle passing though P. Taking P as the visual point, the fist tangent point appeaing when the visual line is otated stating fom the nomal diection of the ais is taken as P and the othe point as P. Hee, when the coodinate of the point of intesection between staight line PP and staight line = - is taken as Y and the coodinate of the point of intesection between staight line PP and staight line = - is taken as X, then Y X Using these equations, is as follows fo domains A and B, o when Y A,B: ) ( (6) It is as follows fo domains C and D, o when Y C,D: ) ( (7) Likewise, is as follows fo domain B, o when X B: ) ( (8) It is as follows fo domains A and C, o when X A,C: ) ( (9) It is as follows fo domain D, o when X D: ) ( (0)

9 Like.3, and include the visible domain 3 as shown in Figue 8. As shown in Figue 8, 3 is the quantit depending onl on the distance fom the oigin, and can be epessed b the following equation egadless of the domain. All domains: 3 cos The invisible quantit is as follows consideing equations (6) to () inv (,) 3 () () This is a continuous function inside (). Both the visible quantit and visible ate ae continuous in domain as well. Fig. 6. Paametes and patial domain fo calcuation of visisble quantit fo cicula cut out space

10 Fig. 7. Calculation of ciucla cut out space Fig. 8. Visible domain 3 included in and 3 Changes in Visible Quantit Accoding to Diffeence in hape of Given pace The pevious chapte discusses the methods of calculating the visible quantit when the cones of a squae space ae cut b a ectangle and when the cente is cut out b a squae and b a cicle. The following descibes the actual calculations

11 pefomed and evaluation of changes in the visible quantit while changing paametes. Also studied wee the diffeences in the changes in the visible quantit accoding to the method of aanging the cut dawing and the shape of the dawing with cut out cente. As shown in Figue 9 to, the distibution dawings on the visibilit ate in the following investigations plot vis (,) to obtain an isometic pojected dawing fom the nomal diections of,, and z. Contou lines ae dawn eve 0. (), and a domain composing is dawn on the plane. Othe than the distibution of the visibilit ate, the aveage visibilit ate and the standad deviation of visibilit ate in domain wee also calculated. These wee obtained numeicall b dividing the inside of into a mesh of , not obtained analticall. 3. When the size of the squae shape is changed fo a L-shaped space with = Figue 9 shows the distibution of the visibilit ate when (= )=.5,.0, and 0.5, and the aveage visibilit ate and standad deviation inside when (= ) is changed continuousl fom to 0. The maimum visibilit ate is.0. As the figue shows, the visibilit ate is constant (=.0) in the segment domain whee whole domain is visible. Howeve fo the domains at the two ends of L-shaped space, the visibilit ate deceases towads the ends. At the same time, the futhe awa is fom a squae, in othe wods, the geate is, the moe intense ae the fluctuations in the visibilit ate nea staight line = -, and staight line =. As the gaph on the aveage visibilit ate shows, the geate is, the moe the aveage visibilit ate appoaches 0.5, and the moe the two spaces become independent visuall. Regadless of whethe is small o lage, the standad deviation becomes small, and is a consistent space visuall. Fig. 9. Distibution of visible ates vis (,) when size of cut out squae shape is changed, aveage visible ate and standad deviation

12 3. When the aspect atio of a cut ectangula shape is changed b changing onl the size of, taking =.5 (constant) in a L-shaped space Figue 0 shows the distibution of the visibilit ate when is vaied to.5,.0, and 0.5, and the aveage visibilit ate and standad deviation inside when is changed continuousl fom to 0. As the gaph on the aveage visibilit ate shows, domain foms a ectangula shape when is small and lage, and the aveage visibilit ate is maimum at the two ends of and is minimum at between them. Looking at the space distibution, even when = 0.5 at which the aveage visibilit ate is lage, local visibilit ate changes intensel, indicating that it is difficult to pinpoint distibution onl fom the aveage value. The standad deviation of the visibilit ate when is lage is lage compaed to othe cases. Fig. 0. Distibution of visible ates vis (,) when aspect of cut out ectangula shape is changed, aveage visible ate and standad deviation 3.3 When is changes in the squae cut out space Figue shows the distibution of the visibilit ate when is 0., 0.5, and 0.8, and the aveage visibilit ate and standad deviation inside when is changed continuousl fom to 0. When is nea, the space is divided into fou conve domains, thus the aveage visibilit ate appoaches 0.5. Though the visibilit ate is lage at the cones of a squae space than the cente in most cases, if is consideabl small, o the B domain in Figue 4 eists, the B and C domains become opposite, whee the cente has a lage visibilit ate (see distibution diagam of = 0. in Figue ). The smalle gets, the high the visibilit ate becomes. The smalle and the lage gets, the moe consistent the space becomes.

13 Fig.. Distibution of visible ates vis (,) when of cut out squae shape is changed, aveage visible ate and standad deviation 3.4 When is changed in the cicula cut out space Figue shows the distibution of the visibilit ate when is 0., 0.5, and 0.8, and the aveage visibilit ate and standad deviation inside when is changed continuousl fom to 0. When is nea, the space is divided into fou domains. Howeve because each of the domains ae not pefectl conve, the aveage visibilit ate dops below 0.5. As appaent fom the distibution diagam of visibilit ates, thee ae no intense changes in the visibilit ate. Coesponding to this, the aveage deviation of the visibilit ate changes at a small value compaed to othe cases. Like the squae cut out aea, if is consideabl small, o if the B domain in Figue 6 eists, the B domain and A domain become opposite, and the visibilit ate (see distibution diagam of = 0. in Figue ) becomes lage at the cente.

14 Fig.. Distibution of visible ates vis (,) when of cut out cicula shape is changed, aveage visible ate and standad deviation 3.5 Compaison of L-shaped squae cut out space and squae cut out space In 3. and 3.3, an obstacle aea is set up in pat of the squae space using squaes in both. Focusing on the aea of the squae cut out fo both, we studied the aveage visibilit ate fo both. Figue 3 shows the aveage visibilit ate and standad deviation of both. As evident fom this diagam, when a squae with the same aea is used, cutting at cones have an ovewhelmingl lage visibilit ate than cutting out at the middle. This matches empiical findings that in a eal space, placing an object at the cones than the middle povides bette view on the whole. Judging fom the phsical quantit that is visibilit ate, placing at cones alwas povide lage view than placing at the cente. When the aea is above 0.36, placing at cones is a value moe than 50% of the visibilit value when placed at the cente. Fom when aea eceeds 0.9, the standad deviation stats to evese, and when the object becomes lage, findings suggest that asmmetical placement at cones esult in inconsistenc of space visuall compaed to the cente.

15 Fig. 3. Changes in visible quantit accoding to aea of cut out squae shape 3.6 Compaison of squae cut out space and cicula cut out space With both 3.3 and 3.4, obstacle aea is placed at the cente. The elation between the aea of the laout dawings and visibilit ate was investigated and the diffeences in the visibilit ate accoding to the shape of the dawings wee studied. Figue 4 shows the aveage visibilit ate and standad deviation in each case. Though the aveage visibilit ate was moe o less the same, in the ange of spatial shapes which ma eist, in othe wods, ange whee paametes ae not eteme values, the squae cut out aea has lowe aveage visibilit ate. Howeve, the standad deviation is alwas small with the cicula cut out space, indicating that an even space with no sudden changes in the visible quantit is ealized b cicula shapes without cones. Fig. 4. Changes in visible quantit accoding to cut out aea of cut out space

16 4. umma and Futue Tasks Focusing on visible quantit in space, the autho claified that the distibution of visible quantit changes divesel accoding to the shape of space using pimitive dawings. The effects of diffeences in shape on visible quantit wee also studied, and esults suggested the feasibilit of using visible quantit in actual space. Though the stage of linking visible quantit with specific pschological quantit has not et been eached, pospects of elation between the impession of space obtained intuitivel and shape of distibution of visible quantit have been achieved, and effots will be stated fom the job of epessing intuitive impessions quantitativel. Also planned ae investigations on elations with othe phsical quantities obtained b gaphic sciences appoach such as (peipheal length) / (aea), (peipheal length) /(aea), visibilit ate of oute cicumfeence sides, taking into account mathematical backgound. The visible quantit can also be applied fo complicated cases b a discete appoach using values of sampling points. Paticulal fom a viewpoint of visual envionment field, it would be inteesting to stud how phsical quantities elated to visible quantit change when pat of the bounda has specula eflecting chaacteistic. B etending visible gaphs investigated in the field of calculation geomet to the mati composed b the value at all points in space, applications to disaste pevention is also possible b investigating ease of visual ecognition of the space and studing the visibilit of evacuation lamps. This epot discussed two-dimensional investigations, howeve the definition of visible quantit itself can easil be etended to thee dimensions as seen in pevious studies. It should be possible to link the attaction of thee-dimensional comple uban space and visible quantit, and evaluate the attaction of given space quantitativel. In the futue, we hope to continue studies on the natue of visible quantit and feasibilit of applications to establish means to claif the impessions of spatial shapes on people. Acknowledgements Finall, the autho would like to epess thanks to all the pominent efeees who povided valuable comments and views. Notes () Punched squae domains and cicula domains ae not counted as visible domains no non-visible domains. () As the potion obtained b the intesection of the cuved suface dawing of the thee-dimensional gaph epessing distibution of visibilit ate and domain enclosed b the two sides and is black, the width of the black aea diffes accoding to the cosine of the angle between the nomal vecto of the cuved suface and z ais.

17 Refeences Akiama, J. and Gaham, R. (996) Visibilit: Fist book of Discete Mathematics, Revised, Toko, Asakua-shoten (in Japanese, the title of the chapte and the book had been tanslated into English b uzuki, H) Asano, T. (990) Visibilit: Computational Geomet, Toko, Asakua-hoten. (in Japanese, the title of the chapte and the book had been tanslated into English b uzuki, H) B. Chazelle and L. J. Guibas (989) Visibilit and intesection poblem in plane geomet, Discete Computational Geomet, Vol.4, 989, Oikawa, K. (000) An Analsis on the Visibilit of Minaet in an a Old Cit of Yemen, ummaies of technical papes of annual meeting F-, Achitectual Institute of Japan, 8-0 eptembe, 000, Koiama, Japan, (in Japanese) Kametani, K., Ohgai, A. and Lee, M. (997) 3-D imulation Modeling fo Evaluating teet and teet Block Foms and tation Visibilit:Pat, ummaies of technical papes of annual meeting F-, Achitectual Institute of Japan, 3-5 eptembe, 997, Funabashi, Japan, (in Japanese) Lee, M., Ohgai, A. and Kametani, K. (997) 3-D imulation Modeling fo Evaluating teet and teet Block Foms and tation Visibilit:Pat, ummaies of technical papes of annual meeting F-, Achitectual Institute of Japan, 3-5 eptembe, 997, Funabashi, Japan, -. (in Japanese) Hsiao, N. et al. (997) Analsis on the Visible Volume of the Opennes of Town teet Analsis of Oita Cit-, ummaies of technical papes of annual meeting F-, Achitectual Institute of Japan, 3-5 eptembe, 997, Funabashi, Japan, (in Japanese)