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Analysis and Classification of Public Spaces Using Convex and Solid-Void Models

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Part of the Springer Optimization and Its Applications book series (SOIA,volume 102)


Urban planning and design are increasingly often supported by analytical models of urban space. We present a method of representation for analysis and classification of open urban spaces based on physical measures including three-dimensional data to overcome some observed limitations of two-dimensional methods. Beginning with “convex voids” constructed from 2D plan information and 3D data including topography and building facade heights, we proceed to “solid voids” constructed by aggregation of convex voids. We describe rules for construction of both convex voids and solid voids, including basic forms and their adjustment for perception. For analysis we develop descriptive characteristic values such as enclosure, openness, granularity and connectivity, derived from more basic geometric properties of the void representations. We also show how combinations of these values can be correlated with urban open space typologies, including commonly accepted traditional ones as well as previously unnamed classes of space. Concluding with discussion of some future planned developments in this work, we also propose that such methods can contribute to better understanding of the relations between urban forms and their perception and use, so as to guide urban transformations for improved urban quality.


  • Urban open public space
  • Urban spatial analysis

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  1. 1.

    Hillier and Hanson propose a notation identifying steps in territorial depth from the most global or public as Y to the most local or private as X. Steps in public space towards the most global are denoted as y steps in y space; and steps in private space towards the most private are x steps in x space.

  2. 2.

    Hillier and Hanson describe an algorithmic procedure for drawing the convex map as: «(…) find the largest convex space and draw it in, then the next largest, and so on until all the space is accounted for.» (page 98).

  3. 3.

    Strictly objectively speaking, of course, all contiguous spaces are one, and the mere opening or closing of a portal potentially reconfigures a vast space/network of potential movement.

  4. 4.

    Portuguese medieval towns usually contain a long winding street crossing the grid called “rua direita” (ironically meaning straight street). Lisbon centre is accessed by very old streets with similar characteristics which can still be seen entirely or partially within its complex superposition of old and new grids. “Rua das Portas de St Antão” and “Rua dos Anjos” are two examples of such streets and many others can be identified.

  5. 5.

    Note that non-metric criteria can also be used, but in such cases only sorting, not ranking, is possible on that/those dimension(s).


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Ljiljana Cavic is a PhD scholarship holder funded by FCT (Fundação para a Ciência e Tecnologia Portugal) with a reference SFRH/BD/76730/2011

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Correspondence to José Nuno Beirão .

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Appendix: Summary of Measures Extracted from Convex- and Solid-Voids’ Model

Appendix: Summary of Measures Extracted from Convex- and Solid-Voids’ Model

Convex-voids’ generation
Convex-voids’ height cv H \( {}_{\mathrm{cv}}H=\left(\Sigma {A}_{\mathrm{f}}\right){\!/\!}{P}_{\mathrm{s}} \) Convex-voids’ height is derived from the average height of surrounding buildings
Corrected convex-voids’ height cv H c \( {}_{\mathrm{c}\mathrm{v}}{H}_{\mathrm{c}}={}_{\mathrm{c}\mathrm{v}}H+{}_{\mathrm{c}\mathrm{v}}H_{\mathrm{adj}} \) Corrected convex-voids’ height is sum of convex-void height with the height adjustment value (cv H adj) which can be positive or negative depending on the characteristics of the surrounding spaces
Convex-voids’ characteristics
Area cv A   
Volume cv V   
Aspect ratio    
Convex-voids properties
Spaciousness/containment \({{}_{\mathrm{cv}}S} {{}_{\mathrm{cv}}C}\) \( \begin{array}{lll}{{}_{\mathrm{cv}}S={A}_{\mathrm{g}}/{A}_{\mathrm{fv}}} \\ {{}_{\mathrm{cv}}C={A}_{\mathrm{fv}}/{A}_{\mathrm{g}}}\end{array}\) Spaciousness/containment represents the relation between the area of the convex-void façades and its floor area
Enclosure/openness to vision cvEV cvOV \( \begin{array}{lll}{_{\mathrm{cv}}\mathrm{E}\mathrm{V}={P}_{\mathrm{e}}{\!/\!}{P_{\mathrm{s}}}_{\mathrm{cv}}}\\ {\mathrm{O}\mathrm{V=1{\hbox{--}}} _{\mathrm{cv}}\mathrm{E}\mathrm{V}}\\ {\left({P}_{\mathrm{ev}}+{P}_{\mathrm{ov}}={P}_{\mathrm{s}}\right)}\end{array}\) Enclosure/openness encodes the proportion of open or closed vs total perimeter
Perceived enclosure/openness to vision cvEV pcvOVp \( \begin{array}{lll}{{}_{\mathrm{cv}}{\mathrm{OV}}_{\mathrm{p}}=}\\{\left(\Sigma {A}_{\mathrm{fv}}\hbox{--}\ \Sigma {A}_{\mathrm{fv}\mathrm{adj}}\right){\!/\!}\Sigma {A}_{\mathrm{fv}}}\end{array}\) Perceived enclosure/openness is a percentage of a convex-void’s faces that is overlapped by surrounding ones
Enclosure/openness to movement cvEM cvOM \( \begin{array}{lll}{{}_{\mathrm{cv}}\mathrm{E}\mathrm{M}={P}_{\mathrm{em}}/{P}_{\mathrm{s}}}\\{{}_{\mathrm{cv}}\mathrm{O}\mathrm{M}=1\hbox{--} {}_{\mathrm{cv}}\mathrm{E}\mathrm{M}=}\\{\left({P}_{\mathrm{em}}\hbox{--} {P}_{\mathrm{s}}\right){\!/\!}{P}_{\mathrm{s}}}\end{array}\) Enclosure/openness to movement takes into account obstacles to movement
Solid-voids properties
Number of convex-void particles sv N   Number of convex-voids that are forming the solid-void
Length of solid-void sv L   Sum of the lengths of all street segments forming the solid-void
Number of façades within a solid-void sv N f   
Granulation of built structure of solid-void sv G   Number of façades per hundred metres
Connectivity svCON   Number of physically permeable routes leading from one solid-void toward other spaces
Perceived connectivity svCONp   Number of all routes, physically permeable and not permeable, opening from one solid-void toward other spaces
Void connectivity svVCONp   Number of other solid-voids connecting (or crossing) a solid-void

A g: Ground area of space

A f: Areas of buildings’ façades

A fv: Area of vertical faces of convex-voids

A fn: Areas of neighbourhoods’ façades

A fvadj: Area of vertical faces of adjoining convex-voids

cv H adj: Height adjustment (=G H × (3/Sqrt(9 + A g)))

G H: Gross height correction value ((A f − A fn)/P s)

L f: Length (on ground) of façades (excl. 0-height)

P s: Total perimeter of space

P ev: Length of perimeter closed to vision (=ΣL f)

P ov: Length of perimeter opened to vision

P em: Length of perimeter closed to movement

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Beirão, J.N., Chaszar, A., Čavić, L. (2015). Analysis and Classification of Public Spaces Using Convex and Solid-Void Models. In: Rassia, S., Pardalos, P. (eds) Future City Architecture for Optimal Living. Springer Optimization and Its Applications, vol 102. Springer, Cham.

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