Building height distribution under zoning regulations: Theoretical derivation based on allometric scaling analysis and application to harmonise building heights

Building height is one of the most important determinants of streetscapes. According to zoning regulations in countries such as Japan, building heights are indirectly controlled by the floor area ratio (FAR). The maximum FAR and the building coverage ratio (BCR) represent the main tools of zoning regulations, which legally regulate building shapes and volumes based on plot sizes at the district scale. If plot sizes and shapes are uniform due to zoning regulations, uniform building shapes (both their plane shapes and heights) may emerge. However, in Japan, plot sizes and shapes are not only non-uniform but irregular. Furthermore, the plot is a conceptually and analytically ambiguous concept and data regarding plot shapes and building heights are unavailable. Therefore, it is difficult to understand the relationship between variations in building heights and zoning regulations. These issues prompt the following research question: How can we estimate building height distribution under zoning regulations at the district scale? To answer this question, plot shapes are objectively defined as area Voronoi cells whose generators are building polygons and road networks. Then, the following two analyses are carried out in downtown districts of the Tokyo metropolitan region. First, the relationship between plot sizes, BCR and building heights is analysed. It is found that building heights scale sub-linearly to plot sizes. Second, following this relationship, the probability density function of building heights under zoning regulations is theoretically derived as a log-normal distribution, primarily estimated from building density, road network density, average road width, BCR and FAR. This is compared with the empirical distribution of discretised building heights and the difference is investigated. Furthermore, how to harmonise variations in building heights is discussed by considering their opportunities and external costs. These findings are expected to provide urban planners with a theoretical basis for creating harmonious streetscapes.