The relation between block size and building shape

Abstract

Block restructuring has been strongly emphasized in Japan for renovating cities. However, little is known about the relation between block size and building shape. Moreover, the shape of buildings designed on a block after restructuring is unclear. In this study, the relation between block size and building shape is analyzed quantitatively, and a three-dimensional building shape is estimated by a model using an urban planning GIS data set of Tokyo. Results show the quantitative relation between block size and building shape, and the building shape image on the blocks. Higher buildings and buildings with a basement tend to be built in larger blocks, leading to efficient use of the maximum volume permitted in the block. In addition, the region composed by larger blocks can be spacious, because the range of building setback will be long in larger blocks. Designation of a high floor area ratio may induce integration and enlargement of blocks. Blocks are less likely to be partitioned in zones when a high floor area ratio is designated.

Keywords

Block estimation floor area ratio GIS

Introduction

Recently, urban policies in Japan have shifted from the development of new cities to the renovation of existing cities. Particularly in Tokyo, the importance of block (land surrounded by road) restructuring has been strongly emphasized. Three reasons can be identified. First, in a small block, large buildings cannot be constructed because of urban planning regulations, which may fail to motivate developers to renovate. Second, large offices cannot be located in a small block. Therefore, the city may lose the opportunity to attract investment from global industries. Third, the existence of roads that are too narrow is an obstacle to disaster preparedness. For all these reasons, block restructuring has become necessary.

Although the importance of block restructuring has been emphasized, there is no research on block restructuring, and the relation between block size and building shape has never been analyzed quantitatively. The building shape and location after restructuring are not evident. If the spatial images of the city can be estimated before renovation, they are useful for planning. Many kinds of estimations are used for planning. The result of population estimation is used for policy for each spatial level, country, prefecture, and city. The growth ratio of the population is an important factor to determine the policy. Land-use simulation is used in urban planning. This simulation affects the planning of zoning and transportation. In addition, if the building shape and location could be predicted, the process of the restructuring plan would be considerably improved. If software that estimates the building shape by block size and shape is developed in the future, municipalities and developers could use it to understand the future image of the block and the city. Moreover, environmental influences, such as the energy consumption and wind-induced response of the buildings, can be estimated. Therefore, the predictability of building development in blocks is significant and crucial for city planning.

In this study, the relation between block size and building shape is analyzed quantitatively, and the three-dimensional building shape is estimated by using an urban planning GIS data set of Tokyo (March 2013). Finally, the effect of block restructuring is discussed.

Literature review

Location of buildings

Malcata-Rebelo and Pinho (2010) found that land use and office location are the relevant variables to the mechanisms of office supply, office demand, and market equilibrium. They concluded that these results support municipal decisions concerning office location and management. However, it is not obvious where buildings are located in the new shape of a block after block restructuring.

There are many studies about the land use of blocks. Makio et al. (2006) analyzed apartment house location on the block scale after a change of building use to apartments. Targeting a provincial city, Saito and Kato (2013) researched changing land use and the current status of each block. Nam et al. (2007, 2008) analyzed the location of parking lots for condominiums based on the relation between green space conservation and business balance. Nagatomi et al. (2007) examined land use in blocks adjacent to a highway. Nakao and Ito (2012) analyzed urban conditions in terms of building density and the building coverage ratio (BCR) in a block. Kawaguchi et al. (2015) conducted a quantitative study about the relation between scale and fluctuation of open space and scale and fluctuation of green space. Matsumiya et al. (2014) calculated the distribution of the open space ratio among buildings in a city. Some researchers tried to extract the building footprint (Sahar et al., 2010; Zhang et al., 2006) and three-dimensional outlines of buildings (Sahar and Krupnik, 1999) by aerial images. These studies searched and analyzed the current use of space in blocks, such as land use, parking lots, and building density. But for predicting the use of space in blocks in the future, further analysis and a developing model are needed.

Building shape

Various studies have focused on the analysis of building shape. Chau et al. (2007) analyzed the optimal building shape under building height regulatory restrictions. Moreover, several Japanese researchers examined the association between building shape and city planning regulations, such as floor area ratio (FAR) and slant plane restrictions (see online Appendix 1, Table A1), as can be seen in Katsuragi and Watanabe (2000), Kirita et al. (2007), Aoki et al. (2010), and Kawakami and Ohnishi (2013). Shpuza and Peponis (2008) measured the floorplate shape in two different ways and analyzed its influence on office layout. Some studies have proposed models to simulate the optimal office building shape using a genetic algorithm (Grierson and Khajehpour, 2002; Ouarghi and Krarti, 2006; Wang et al., 2006). These studies analyze the optimal building shape by model. But these analyses did not use actual building samples. In this study, building shape is estimated by using a GIS data set.

In office design terms, offices require a high rentable ratio (the rentable area divided by the area available for use in a building) and an efficient working environment. According to Kooijman (2000), office design has changed because of globalization and the needs of the local environment. The office layout is said to be associated with work practices.

These studies mentioned several factors that are relevant to building shape. In particular, city planning regulations have been focused on for analyzing building shape (e.g. Chau et al., 2007; Grierson and Khajehpour, 2002; Katsuragi and Watanabe, 2000). Developers design a building in order to maximize the benefit of the block under the restriction. In this study, we consider city planning regulation for developing an estimation model of building shape.

Urban renewal

Changes in building shape through urban renewal have been analyzed. Some studies have concluded that the initial block shape affects the future building shape or block shape after urban renewal. Siksna (1998) examined the influence of the initial block shape on the building shape after development. Ryan (2006, 2008) suggested that new residential patterns are established after the transformation of blocks. Processes of urban renewal have also been studied. Lin and Lin (2014) adopted game theory to analyze urban renewal processes based on the characteristics of landowners. In addition, some studies have focused on the process of change in blocks in the center of Tokyo (Matsukura and Miyawaki, 2006), the development process of traditional rectangular blocks in Kyoto (Hayami, 2009), and the development process of blocks used for office buildings in the Marunouchi area in Tokyo (Nomura, 2014). These researchers studied the changing building shape and process of urban renewal. But the building shapes in the future could not be estimated by these analyses.

Estimation

In this study, building location and shape are estimated. Some estimation methods for urban physical status, such as building footprint location, floor area, and land use, have been developed in previous research.

Taima et al. (2016b) developed a model to estimate the building footprint area by using GIS. The future image of the building footprint on various blocks is visualized. Similarly, Asami and Ohtaki (2000) developed a model to estimate detached house location. Orford (2010) developed a methodology for estimating the floor area of individual properties from digital infrastructure data. Shiravi et al. (2015) assessed the utility of some models for estimating floor area using three data sources: a geographic vector building footprint layer, a LiDAR data set, and field survey data for the south side of the city of Fredericton, Canada. They discussed the reliability and accuracy of each model. In other research, Brunner et al. (2009) extended a methodology for building height estimation and tried to improve its accuracy. Schmidt et al. (2010) presented an approach to the estimation of building density on the block scale. Land use (Debnath and Amin, 2016; Güneralp et al., 2012; Spyratos et al., 2017) and floor area (Orford, 2010) are popular topics and estimated in previous studies of the urban field, but estimation of building shape has seldom been a focus in the literature. Three-dimensional estimations of buildings cannot be found. If software to estimate building shape by block shape and other conditions was developed, it would be useful to determine urban planning, such as population estimation and land-use estimation. In this study, an estimation model is developed and applied to certain areas.

Method

Study area and data source

In this study, an urban planning GIS data set of Tokyo (March 2013) is used. This data set contains building types, the number of floors, land-use zones, the designated FAR, and the designated BCR (overview on urban planning regulations in Japan is shown in online Appendix 1, Table A2).

The relation between block size and building shape is first analyzed quantitatively. For analyzing the relation, we aggregate the number of blocks, lots, and buildings. Moreover, we calculate the area of blocks, lots, and building floors. The study area encompasses the special wards of Tokyo (626.70 km²). The population (9,256,625 people) and many functions are dense in this area (Tokyo Metropolitan Government, 2016). Moreover, urban renewal and block restructuring are strongly emphasized in this area.

All buildings, lots (parcel of land), and blocks in the special wards of Tokyo are used for analysis. When the relation between block size and building shape is analyzed, samples are distinguished by designated FAR, because the block size and building shape are expected to be much different in each designated FAR.

Next, the building shape is estimated using a model. The accuracy of the estimation will be higher when sample blocks with only similar conditions are used, such as land use and building type. Therefore, blocks in urgent urban renewal areas of commercial zones are chosen for analysis because the areas are designated for the most urgent block restructuring in Tokyo, where the government finances and promotes the renewal projects. In this estimation, office buildings are used in this area, because office supply is promoted in this area and several studies have focused on office shape (Grierson and Khajehpour, 2002; Kooijman, 2000; Ouarghi and Krarti, 2006; Shpuza and Peponis, 2008; Wang et al., 2006). For our analysis, two typical designated FARs are selected, namely 600% (termed group A: 27 blocks) and 700% (termed group B: 36 blocks). The FAR of 600% and 700% is found to be the mode value and the second mode value of all the blocks in the urgent urban renewal area, respectively. These blocks are used as reference blocks for estimation of building shape.

Building shape estimation model

A building shape estimation (BSE) model was developed to estimate the building shape from a block shape. A similar model has been used to estimate the building footprint in blocks (Author’s previous article) and a detached house location (Asami and Ohtaki, 2000). However, the models in these previous studies estimate two dimensions of the shape, like a footprint. In this study, the model is extended to estimate three dimensions of the shape, including the height of the building.

Siksna (1998) explained that buildings and lots are developed depending on block forms. Moreover, several researchers have noted that building shape is strongly influenced by city planning regulations (e.g. Chau et al., 2007; Grierson and Khajehpour, 2002; Katsuragi and Watanabe, 2000; Kirita et al., 2007). This shows that building shape is determined by block shape and city planning regulations. Thus, we can expect that if two blocks are similar in shape and under similar city planning regulations, the building shapes in the blocks will be similar. This naive but simple assumption enables us to estimate the building location. Accordingly, a model was developed to estimate the probability of each point on every floor level covered by a building.

More specifically, the locations of buildings on reference blocks were overlaid so that the gravity center of the reference blocks matched that of a given block. A probability was assigned to each overlaid layer depending on the similarity of the block shape. The Lee–Sallee measure (Lee and Sallee, 1970) judges the similarity between two blocks by the quotient value—the intersection area divided by the union area of the two blocks.

An index expressing the similarity between two blocks, similarity index, s, is defined below. Generally, a block can be treated as a compact set in R². Let $x (\in X)$ be a point in block X and let $g (X)$ be a vector of the gravity center of block X and denote the area of block X as $A (X)$ . The similarity index, s, is calculated as the intersection area divided by the union area of the two blocks by matching the gravity center of the two blocks. Let $G (X)$ be the set that is given by moving the block X in parallel, so that the gravity center of the block coincides with the origin:

G (X) = {z : z = x - g (X), x \in X}

(1)

Based on the Lee–Sallee measure (Lee and Sallee, 1970), the provisional similarity index, $s'$ , between block X and block Y is defined as follows:

s' (X, Y) = \frac{A (G (X) \cap G (Y))}{A (G (X) \cup G (Y))}

(2)

The building location greatly depends on the direction of the adjacent roads. The direction of the road is different in each block. However, the influence of the adjacent road on the building location in blocks is not so different when a block revolves around a center of gravity within an angle up to $\pm π / 4$ . Allowing for such revolving, the final similarity index, s, is defined as follows. Let $R (X, θ)$ be the block revolved by angle θ around the center of gravity of the block X. The similarity index, s, is then defined as follows:

s (X, Y) = {max}_{- π / 4 \leq θ \leq π / 4} \frac{A (G (X) \cap R (G (Y), θ))}{A (G (X) \cup R (G (Y), θ))}

(3)

Reference blocks are used to calculate the probability of building location. Let I be an index set of reference blocks. Then, the set of reference blocks is denoted as $B_{i} (i \in I)$ . We assume that the actual building shape on reference blocks is known, and therefore this information is used to estimate a building shape on a new block.

If two blocks are similar in shape and other spatial features, we can expect that the building shapes and locations in the blocks will tend to be similar. Therefore, to estimate building shape in new block X, it is appropriate to use reference blocks that have a high similarity index value with the new block. It may be possible to use the similarity index, s, as a weight that determines the priority of the reference block. However, if this is the case, then all the reference blocks must be used for the BSE model. A better model can be developed by rejecting reference blocks of apparently different shape. To exclude differently shaped reference blocks, those weights should be zero. To this end, we define weight function, $f (s)$ , based on the similarity index, s, as follows:

f (s) = {\frac{s - t}{1 - t} s > t 0 s \leq t

(4)

where t is a parameter and its range is zero to one (

t \in (0, 1]

). When the similarity index, s, is less than parameter t, the weight function,

f (s)

, is zero.

The way to estimate building location can be described as follows. Let $B_{i}^{*}$ be the reference block by moving block $B_{i} (i \in I)$ in parallel so that its center of gravity coincides with the origin (i.e. $B_{i}^{*} = G (B_{i}), i \in I$ ). Let J be the set of the possible number of building floors, where a negative integer means a basement. In Japan, the first floor is the ground floor. Thus, “0” is not included in J. Let $F_{ij}$ be a set given by projection of the $j (\in J)$ th floor of the building on block B_i onto the block plain.

Moreover, let $F_{ij}^{*}$ be given by parallel transformation of $F_{ij}$ so that the center of gravity of B_i coincides with the origin. That is,

F_{ij}^{*} = {z : z = x - g (B_{i}), x \in F_{ij}}

(5)

Let Z be a block to be estimated by the BSE model. Let $Z^{*}$ be the block by moving block Z in parallel so that its center of gravity coincides with the origin (i.e. $Z^{*} = G (Z)$ ). The building existing probability $p (x, Z^{*}, I, j)$ , i.e. the probability that point $x (\in Z^{*})$ is covered by buildings of block $Z^{*}$ , is estimated as follows. First, the similarity index, $s,$ of two blocks, $Z^{*}$ and B_i, is calculated as follows:

s (Z^{*}, B_{i}^{*}) = {max}_{- π / 4 \leq θ \leq π / 4} \frac{A (Z^{*} \cap R (B_{i}^{*}, θ_{i}))}{A (Z^{*} \cup R (B_{i}^{*}, θ_{i}))}

(6)

Let $θ_{i}^{*}$ be the solution of the maximization problem in equation (6):

θ_{i}^{*} = \arg {max}_{- π / 4 \leq θ \leq π / 4} \frac{A (Z^{*} \cap R (B_{i}^{*}, θ_{i}^{*}))}{A (Z^{*} \cup R (B_{i}^{*}, θ_{i}^{*}))}

(7)

If multiple solutions exist, then the solution closest to zero will be assigned to $θ_{i}^{*}$ . Moreover, if two such solutions exist, then the positive one is assigned.

Revolved block $B_{i}^{* *}$ and building floor $F_{ij}^{* *}$ are denoted as follows:

B_{i}^{* *} (Z^{*}) = R (B_{i}^{*}, θ_{i}^{*})

(8)

F_{ij}^{* *} (Z^{*}) = R (F_{ij}^{*}, θ_{i}^{*})

(9)

The indicator function expressing that point x is included in $F_{ij}^{* *}$ is defined as follows:

Then, the building existing probability $p (x, Z^{*}, I, j)$ at point x on block Z is defined as follows:

p (x, Z^{*}, I, j) = \frac{\sum_{i \in I} f (s (Z^{*}, B_{i}^{* *} (Z^{*}))) χ (x, F_{ij}^{* *} (Z^{*}))}{\sum_{i \in I} f (s (Z^{*}, B_{i}^{* *} (Z^{*})))}

(11)

Parameter t is decided so that the accuracy of the building location estimation is maximal. To this end, let $I_{- i}$ be the reference block set except for block B_i. Define estimation accuracy index, ρ, as follows:

ρ = \sum_{j \in J} \sum_{i \in I} \int_{x_{i} \in B_{i}^{*}} [p (x_{i}, B_{i}^{*}, I_{- i}, j) χ (x_{i}, F_{ij}^{* *} (B_{i}^{*})) - p (x_{i}, B_{i}^{*}, I_{- i}, j) (1 - χ (x_{i}, F_{ij}^{* *} (B_{i}^{*})))] d x_{i}

(12)

The estimation accuracy index, ρ, is given by the integral value of the building existing probability at the points covered by buildings minus the integral value of the building existing probability at the points not covered by buildings. The value of parameter t, which maximizes the estimation accuracy index, ρ, is used for the BSE model. Applying this method to the sample of group A and group B yielded t = 0.73 for group A and t = 0.74 for group B.

The method of visualization

The probability of building coverage for each point on every floor level can be used to predict the potential urban environment before the development. In particular, it is important to know office building shapes, because office supply is promoted and many offices will be developed in Tokyo’s central zones. Moreover, many researchers are interested in office shape (Grierson and Khajehpour, 2002; Kooijman, 2000; Ouarghi and Krarti, 2006; Shpuza and Peponis, 2008; Wang et al., 2006). Therefore, the estimation of the probability of building coverage for each point on each floor level needs to be visualized to obtain the spatial image.

The building existing probability is visualized as follows. First, hypothetical blocks (not actual blocks with a simple block shape) are set, and the building existing probability of the points on the block is visualized. Points are set every 1 m in a north–south direction and an east–west direction. Then, reference blocks are overlaid on a hypothetical block. Second, the building existing probability of each point is calculated by summing the building existing probability of the point overlaid by reference blocks, which is calculated based on the similarity between the reference blocks and the hypothetical block. The probability is expressed by brightness. In the white area, the probability is high. Conversely, in the black area, the probability is low.

Hypothetical blocks are set by changing their size, and the probability of building coverage for each point on every floor level is calculated. Hypothetical blocks are various sizes of squares. All sizes of hypothetical blocks estimated by the samples of group A and group B are visualized.

In addition to the visualization of building existing probability, the building area ratio (BAR) is calculated. The BAR is defined as the estimated area considering probability in each floor divided by block area. The result leads to a quantitative analysis of building shape. The BAR of the first floor is the same value as the BCR.

To analyze the situation of a building set back from the road, the estimated BAR is calculated in each buffer from the surrounding road of a block. The width of the buffer is from 2 m to 20 m, changing every 2 m.

Results

Block size and designated FAR

Table 1 shows the number of blocks and the average block area of each designated FAR (60%, 80%, 100%, 150%, 200%, 300%, 400%, 500%, 600%, 700%, 800%, 900%, and 1000%) in the special wards of Tokyo. These blocks are used for an analysis of the relation between block size and building shape. The blocks are categorized, and the ranges of the category are 0–100 m², 100–1000 m², 1000–10,000 m², 10,000–100,000 m², and 100,000–1,000,000 m².

Table 1.

The number of blocks and average block area in each designated FAR zone.

Floor area ratio (%)	The number of block	Average block area (m²)
60	113	3775
80	1525	4013
100	9376	3564
150	7882	3309
200	25,177	3597
300	16,089	3096
400	3837	2324
500	3391	1979
600	2437	1911
700	1306	1818
800	591	2445
900	73	3214
1000	49	4952
Total	71,846	3077

FAR: floor area ratio.

Table 1 shows that the average block area is larger in lower designated FAR (60–300%) or higher designated FAR (900% and 1000%) in comparison with middle designated FAR (400–800%). We conducted a t-test on each combination, and both p values were less than 0.01. Moreover, in Table 2, the ratio of the blocks of 10,000–100,000 m² is higher in lower designated FAR (60–300%) or higher designated FAR (900% and 1000%). This is because in low designated FAR areas, detached houses and parks are the popular land uses, and they require a large block for creating open spaces. On the other hand, in high designated FAR areas, blocks are often used for offices (Ministry of Land, Infrastructure and Transport, 2003). For developing large offices where many employees are working, large sizes of blocks are required. Accordingly, results clearly showed that the total space in blocks is relevant to land-use type and population size.

Table 2.

The number of blocks and the ratio (%) in each block area category.

Floor area ratio (%)	Block area (m²)
	0–100		100–1000		1000–10,000		10,000–100,000		100,000–1,000,000		Total
	Count	Ratio	Count	Ratio	Count	Ratio	Count	Ratio	Count	Ratio	Count
60	33	29	50	44	20	18	10	9			113
80	6	0	259	17	1160	76	100	7			1525
100	34	0	1602	17	7302	78	437	5	1	0	9376
150	42	1	1256	16	6302	80	280	4	2	0	7882
200	222	1	5149	20	18,570	74	1219	5	17	0	25,177
300	160	1	4200	26	11,104	69	608	4	17	0	16,089
400	87	2	1254	33	2386	62	110	3			3837
500	35	1	1181	35	2128	63	45	1	2	0	3391
600	34	1	854	35	1520	62	28	1	1	0	2437
700	4	0	368	28	927	71	7	1			1306
800			116	20	463	78	12	2			591
900	4	5	25	34	41	56	3	4			73
1000	2	4	6	12	34	69	7	14			49

In Table 3, it is found that the designated FAR varies with land-use types. The ratio of the use of a lot as a park is 68.26% in a 60% designated FAR block. The ratio of the use of a lot as a detached house and condominium is 39.78% and 23.70% in an 80% designated FAR block. These are high ratios out of all uses of lots. On the other hand, in the high designated FAR areas, these areas belong to a commercial zone (Ministry of Land, Infrastructure and Transport, 2003). Therefore, large offices are developed in a high designated FAR. For example, the ratio of offices is 37.68% in a 1000% designated FAR block. For a reason why office lots are set by higher designated FAR, a large commercial lot with access to a wide road is supposed to achieve higher FAR in comparison to a residential lot with the same size and road access.

Table 3.

Ratio (%) of number of uses of lots in the blocks.

Floor area ratio (%)	Use of lot
Floor area ratio (%)	Detached house	Condominium	Park	Office	Complex of residence and commerce
60	8.38	1.80	68.26		2.40
80	39.78	23.70	2.97	0.85	4.80
100	39.16	27.23	1.78	0.97	5.95
150	36.68	27.50	1.40	1.20	8.70
200	31.80	23.48	1.88	2.18	10.37
300	29.26	22.40	1.28	3.68	15.59
400	21.37	18.58	0.91	7.44	20.46
500	16.69	15.03	0.67	13.41	23.82
600	11.77	12.78	0.73	20.60	22.83
700	7.69	9.51	0.61	24.53	21.74
800	2.90	2.93	0.65	30.59	16.08
900	0.41	0.41	0.83	16.12	8.68
1000	0.72		0.72	37.68	5.07

Block size and number of lots

In Table 4, the number of lots in each categorized block is shown. Generally, the number of lots increases as the block becomes large. However, compared with lots in lower designated FAR (60–300%) blocks, lots in higher designated FAR (600–1000%) blocks do not increase as much as the block becomes larger. In the large blocks in the high designated FAR blocks (600–1000%), the number of lots is fewer than the smaller blocks. For example, in a 600% designated FAR block, the number of lots in 10,000–100,000 m² is 10.82 and decreases to 5.00 in a larger block, 100,000–1,000,000 m². We examined the Pearson product–moment correlation coefficient between block area and the number of lots in both lower designated FAR (60–300%) blocks and higher designated FAR (600–1000%) blocks. The correlation coefficients were 0.312 and 0.112, and both p values were less than 0.01. Each result was significant, but the relation between block area and the number of lots was stronger in lower designated FAR (60–300%) blocks. These results show that blocks tend to be split by many lots in low designated FAR zones as the block becomes larger. On the other hand, the blocks are less likely to be partitioned in high designated FAR zones.

Table 4.

The average number of lots in a block.

Floor area ratio (%)	Average	Block area (m²)
Floor area ratio (%)	Average	0–100	100–1000	1000–10,000	10,000–100,000	100,000–1,000,000
60	1.48	1.09	1.10	1.75	4.10
80	5.06	1.17	1.86	4.97	14.68
100	5.09	1.03	2.01	5.21	14.57	68.00
150	6.21	1.07	2.27	6.57	16.18	25.50
200	6.50	1.09	2.61	7.24	12.49	11.18
300	7.32	1.11	3.30	8.52	14.05	30.00
400	6.61	1.05	3.64	8.20	9.32
500	6.86	1.17	4.44	8.25	8.98	13.00
600	6.09	1.03	4.22	7.17	10.82	5.00
700	5.56	1.00	4.13	6.15	4.00
800	4.44		3.03	4.82	3.50
900	3.32	1.25	2.60	4.02	2.33
1000	2.82	1.00	1.50	3.47	1.29

Block size and number of building floors

In Tables 5 and 6, the average number of building floors above ground and underground in each block category is calculated. The results in Tables 5 and 6 showed similar tendencies. The number of floors above the ground and underground of the high designated FAR blocks (700–1000%) increases as the block becomes larger. On the other hand, the floors do not increase in low designated FAR blocks as the block becomes large.

Table 5.

The average number of floors above ground in each block area category.

Floor area ratio (%)	Average	Block area (m²)
Floor area ratio (%)	Average	0–100	100–1000	1000–10,000	10,000–100,000	100,000–1,000,000
60	1.53	1.00	2.00	1.90	1.31
80	1.99	2.50	1.99	2.00	1.94
100	2.01	1.90	2.06	2.01	1.95	1.83
150	2.10	1.78	2.15	2.10	2.03	1.88
200	2.21	2.00	2.21	2.22	2.19	1.55
300	2.45	2.14	2.42	2.46	2.50	2.22
400	3.14	2.23	2.90	3.15	3.86
500	3.79	2.56	3.42	3.87	5.09	2.13
600	4.59	1.54	4.03	4.75	5.01	5.00
700	5.40	1.50	4.71	5.55	8.94
800	6.18		5.12	6.25	9.89
900	5.63	1.40	4.65	5.59	13.67
1000	7.33	1.00	3.08	7.28	10.59

Table 6.

The average number of floors underground in each block area category.

Floor area ratio (%)	Average	Block area (m²)
Floor area ratio (%)	Average	0–100	100–1000	1000–10,000	10,000–100,000	100,000–1,000,000
60
80	0.0126		0.0091	0.0126	0.0133
100	0.0127		0.0141	0.0133	0.0086	0.0225
150	0.0114		0.0091	0.0118	0.0097	0.0064
200	0.0057		0.0045	0.0055	0.0078	0.0143
300	0.0077	0.0102	0.0051	0.0074	0.0128	0.0248
400	0.0168		0.0087	0.0149	0.0754
500	0.0433		0.0285	0.0429	0.2259	0.0132
600	0.0896		0.0683	0.0931	0.1782	0.1429
700	0.1707		0.1142	0.1800	0.9583
800	0.3975		0.3205	0.3996	0.7708
900	0.6736		0.5909	0.6493	1.8333
1,000	0.6452			0.7243	0.6410

In addition, the relation of the number of floors above ground and those underground probably has a positive correlation. Using all of the samples of buildings (1,043,634 buildings), the Pearson product–moment correlation coefficient between the number of floors above ground and those underground is 0.2653 (p < 0.01). There is a strong positive correlation between them. This is because higher buildings need to develop the basement for the reinforcement of the foundation as well as the requirement of the service area, such as parking. It must be noted that people cannot possess a basement that is deeper than 40 m from the ground or 10 m from the foundation of the building in Tokyo because it is prevented by the Act on Special Measures Concerning Public Use of Deep Underground. The law regulates private use of a basement for the promotion of public use, such as lifeline utilities, and efficient construction.

Block size and volume sufficiency ratio (VSR)

The VSR measures the ratio of the total floor area to the maximum legally allowable floor area under the designated FAR regulation. The VSR is calculated as the whole building floor area divided by the maximum volume of the block. The maximum volume of the block is calculated as the block area multiplied by the designated FAR.

The range of the VSR should be from 0% to 100%. However, some results of the VSR are over 100%. This is partly due to the fact that buildings constructed before the legislation of FAR regulation was introduced are regarded as legally allowable, particularly in small blocks, even if the floor area of the building exceeds the current allowable floor area. Moreover, the application of some special measures relaxing the designated FAR can be pointed out. Particularly in large blocks, the measures are often applied to promote development.

The results are shown in Table 7. The VSR increases gradually as the block becomes larger up to 10,000–100,000 m², excepting a lower designated FAR (60–300%). In the low designated FAR (60–300%) block, the VSR decreases as the block becomes larger. We calculated the correlation coefficient between block area and VSR in both blocks of lower designated FAR (60–300%) and that of other designated FAR (400–1000%). The results were −0.032 and 0.125 (both p values are less than 0.01). Accordingly, these results support the discussion above. However, in blocks that are too large, such as 100,000–1,000,000 m², the VSR is low. This result indicates that blocks that are too large sometimes lead to inefficient development.

Table 7.

VSR (%) in each block area category.

Floor area ratio (%)	Average	Block area (m²)
Floor area ratio (%)	Average	0–100	100–1000	1000–10,000	10,000–100,000	100,000–1,000,000
60	0.36	0.29	2.41	0.29	0.10
80	1.09	1.56	1.24	1.08	0.79
100	0.99	1.01	1.12	0.97	0.75	0.36
150	0.74	0.65	0.83	0.73	0.64	0.16
200	0.65	0.58	0.65	0.65	0.66	0.46
300	0.55	0.45	0.53	0.56	0.66	0.40
400	0.61	0.37	0.54	0.64	0.85
500	0.67	0.34	0.59	0.70	0.94	0.27
600	0.72	0.16	0.62	0.78	1.16	0.20
700	0.77	0.15	0.65	0.82	1.22
800	0.83		0.66	0.86	1.55
900	0.68	0.12	0.48	0.79	1.55
1000	0.95	0.06	0.61	0.90	1.65

VSR: volume sufficiency ratio.

Estimation of office shape

The building existing probability on the estimated hypothetical blocks is visualized in Figures 1 and 2. From these figures, the image of the building location can be grasped.

Figure 1.

Visualization of building existing probability in 600% designated FAR zones.

Figure 2.

Visualization of building existing probability in 700% designated FAR zones.

According to Figures 1 and 2, on the upper floors, the estimated BAR is higher in large blocks. In group A, the estimated BAR is almost the same under the seventh floor in all blocks, but for the eighth floor, in the smaller blocks, the range of each side is 25 m and 30 m; the estimated building ratio decreases rapidly compared with the seventh floor (25 m: 78.81% to 32.92%; 30 m: 61.07% to 23.10%). On the other hand, for the larger blocks, the range of each side is 40–60 m, and the estimated BAR does not decrease much. In group B, similar tendencies can be seen. In the smaller block, the range of each side is 25 m, and the estimated BAR decreases rapidly from the fourth floor to the fifth floor as 79.04% to 35.88%. In addition, similarly to the smaller block, the range of each side is 30 m, and the estimated BAR decreases rapidly from the third floor to the fourth floor as 67.23% to 33.67%. In the larger blocks, the range of each side is 40–60 m, and the estimated BAR does not decrease in such lower floors. These results show that the efficiency of using upper floors is high in large blocks. In small blocks, the efficiency of using upper floors is low. If higher buildings and efficient use of upper floors are desired, the blocks should be larger by restructuring.

According to the analysis of the BAR in the buffer area, the larger the blocks, the farther the buildings are set back from the roads. The estimated BAR in the buffer area decreases gradually from the small block to the large block. For example, in Figure 1, when the width of the buffer from the roads is 6 m, the BAR is 82.79% in the block of the range of each side is 25 m, and is 40.55% in the block of the range is 30 m. The BAR increases as the block becomes large, and the value is 15.59% in the block of the range is 50 m, and is 1.29% in the block of the range is 60 m. Similar tendencies can be observed in Figure 2. This is probably because high buildings can be built in the large blocks and do not need to be built near roads. If a city is composed of large blocks, it becomes spacious because much open space exists around buildings. Conversely, if a city is composed of small blocks, it is dense and narrow. In the city, buildings cannot be set back from roads, and open spaces tend not to be created.

Conclusion

In this study, the relation between block size and building shape is analyzed quantitatively. In addition, blocks used for offices are chosen, and the building shape is estimated by a developed BSE model.

Results show that the number of floors above the ground and underground of the higher designated FAR blocks increases as the block becomes larger. On the other hand, the floors do not increase in low designated FAR blocks as the block becomes large. Moreover, there is a strong positive correlation between the number of floors above ground and those underground.

The VSR increases gradually as the block becomes large, excepting blocks of a low designated FAR. However, in blocks that are too large—more than 100,000 m²—the VSR is low. In the low designated FAR block, the VSR decreases as the block becomes larger.

On the upper floors, the estimated BAR is higher in large blocks. Moreover, the larger the blocks, the farther the buildings are set back from the roads. Therefore, the significance of enlarging blocks throughout the block restructuring can be discussed by these results. In large blocks, high buildings can be built, and the volume in the block can be used efficiently. In addition, in the area composed of large blocks, many open spaces will be created because the buildings are set back from the surrounding roads. It is possible that large blocks create a spacious city. But it is necessary to designate high FAR of more than 600% on the blocks.

The average block area is large in low designated FAR zones and high designated FAR zones. Accordingly, high designated FAR may lead to enlarging blocks by promoting block restructuring. By high designated FAR, the blocks are less likely to be split by lots, even in large blocks. The BSE model, which estimates the building shape and location, was developed. The estimated results are shown as targeting offices. This technique can be applied to other buildings and cities.

If software to estimate building shape by block shape and other conditions was developed, it would be useful for determining urban planning, as well as population estimation and land-use estimation. In the future, it is possible to estimate the urban image by using this method, particularly in the city, where development and urban restructuring are active. By the estimation, planners can understand the building shapes and locations of the future and use the results for planning, and citizens can easily understand the future image of the block. They can judge whether the urban renewal plan is better. In addition, the estimation of building location and shape can be applied to many fields. Many studies have been conducted related to the building location and shape, the relation between the building shape and energy efficiency of office buildings (AlAnzi et al., 2009), the energy consumption and buildings (Ko and Radke, 2014; Mortimer et al., 2000; O’Brien et al., 2010; Ourghi et al., 2007; Rode et al., 2014), and the wind-induced response of buildings in a block (Kuang et al., 2011; Kwok, 1988; Merrick and Bitsuamlak, 2009). Combined with this knowledge, it is possible that the estimation of wind, energy efficiency, and consumption can be done using the estimation of buildings.

Limitations and future research

In this research, we aimed to examine the relation between block size and building shape, and we showed the method to estimate building shape from block size (BSE model). We tried to enhance the model’s accuracy by limiting blocks that are in similar situations as follows: designated FAR is 600% and 700%, land-use zoning is commercial zone, and building type is office. However, it is unclear how this model is influenced by each factor (designated FAR, land-use zoning, and building types) and what the most important factor for accurate estimation is. Moreover, we did not consider other factors (e.g. energy consumption, construction cost, width of roads adjacent to blocks) that may impact the building shape. For example, average construction cost per square meter varies for certain height ranges, and the developer decides the building shape within the regulations. For the examination of verification and improvement of the model, we desire to use more samples and apply the model to other regions, including other countries.

Footnotes

Acknowledgements

The urban planning GIS data set of Tokyo (March 2013) was provided by the Tokyo Metropolitan Government. Part of this article is based on the Proceedings of the 17th International Symposium on Spatial Data Handling (Taima et al., 2016a). We are thankful for the valuable comments from attendees of the symposium. In addition, we would like to thank the two referees for their useful comments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partly supported by JSPS KAKENHI Grant Numbers JP 26590036, JP16H01830.

Masahiro Taima is a graduate student at the Department of Urban Engineering, Graduate School of Engineering, The University of Tokyo. His current research interests include urban planning, spatial modelling, and urban simulation.

Yasushi Asami, PhD, is a professor at the Department of Urban Engineering, Graduate School of Engineering, The University of Tokyo. He received his PhD from the Department of Regional Science, University of Pennsylvania. His current research interests include residential environment, housing policy, and urban planning.

Kimihiro Hino, PhD, is an associate professor at the Department of Urban Engineering, Graduate school of Engineering, The University of Tokyo. He received his PhD from the Department of Urban Engineering, Graduate school of Engineering, The University of Tokyo. His current research interests include crime prevention and health promotion through environmental design.

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