Geo-spatial Information Science 11(2):121-126 V olume 11, Issue 2 DOI 10.1007/s11806-008-0032-9 June 2008
Article ID: 1009-5020(2008)02-121-06 Document code: A Fractal Features of Urban Morphology and Simulation of Urban Boundary ZHANG Yi YU Jie FAN Wei
Abstract Using fractal dimension to reflect and simulate urban morphology are two applications of fractal theory in city geography. As the only consistent description of a fractal, fractal dimension plays an important role in describing the basic features of fractals. Just like other fractals, our cities have similar characteristics. Fractal dimension to some extent is regarded as an indicator of urban expansion, and it may change with urban morphology in different time and space. Bad on the Geographic Information System (GIS), taking Wuhan city as a test area, the fractal dimensions of different land u were calculated, and a linear regression equation was established to analyze the relationship between fractal dimension and residential areas. Then the author ud fractal dimension to simulate the urban boundary which is an important part of urban mor-phology. A mid-point subdivision fractal generator is needed in the simulation process, and the shape of the generator is determined by fractal dimension. According to the fractal theory, fractal boundaries in different scales have lf-similarity and they have the same fractal dimensions. Bad on this fact, the simulation method the aut
hor ud could quantitatively keep the similarity of configuration of the urban boundaries. Keywords fractal dimension; urban morphology; fractal simulation; fractal generator
CLC number P237.3
Introduction
Ever since the establishment of fractal theory in the 1970s[1], it has become more and more widely ud in city geography. Fractal theory provides both good mathematic tools of describing the urban geographic phenomena and the mathematical foundation for frac-tal simulation of urban spatial systems. Thereby, many scholars have done lots of rearches on urban morphologic structure and its expansion.
As a key quantity of fractals, fractal dimensions are ud as indicators of the complexity of curves and sur-faces, hence, rearching on the methods of calculating the fractal dimensions is an important part of the frac-tal city theory. There are some common fractal dimen-sions such as Houdoff dimension, Box-Counting dimension, etc. Scholars have propod some uful ways for deriving the fractal dimensions, for example, Structured Walk method, Equipaced Polygon method, Hybrid Walk method, Box-Counting method, etc. In this paper, Box-Counting method is adopted, and
the geographical significance of fractal dimensions calcu-lated with this method indicates the degree of similar-ity. Fractal dimensions of different geographic objects are obtained from the digital land u map, and changed with the urban expansion and evolution dur-ing different time periods. Then we establish a linear regression equation to analyze the relationship between fractal dimension and areas of residence.
Geo-spatial Information Science 11(2):121-126
122Fractal dimension is employed to simulate the ur-ban boundary which is an important part of urban morphology. In the experiment, the urban boundary is defined as a clod line with a fractal dimension of more than one but less than two. It’s hard to store data of every scale of urban boundaries as a linear part of the map. More details of the urban boundaries are ex-pected to be exhibited as the map scale is enlarged. Some conventional methods pay too much attention to the smoothness of linear element in this process. How-ever, they isolate the isotropy which is an obvious geo-graphic feature of the spatial change [2]. Simulation bad on fractal dimension can avoid this disadvantage.
1 Exploring fractal features
1.1 Mathematic model of fractal dimension
calculation According to Mandelbrot, supposing the fractal dimension is D , then
中日语在线翻译
()D N r cr −= (1)
Where ()N r is the number of measurements; r is the yardstick; c is a constant to be determined; D is the fractal dimension.
The equivalent equation is:
ln ()ln ln N r D r c =−+i (2)
In the rearch of city morphology, r stands for the yardstick of certain type of land u. ()N r stands for the measurement of this yardstick, if they have the law which do not change with the yardstick:
()D N r r ∝ (3)
It is considered that the configuration of city land
u has fractal quality: the spatial configuration fea-tures do not change with the measurement of the yardstick.
Suppo ln ()N r equals Y , ln r equals X , ln c equals C , changing the yardstick r, veral groups of X , Y is measured, and the method of least squares is ud to fit the line which is described by Eq.(2), fitting equa-tion
1
1
21
1
1
N
N
i i
i i N
N
N
i i i i
i i i N C D X Y C X D X X Y =====+=+=∑∑∑∑∑i i i i (4)
confirmingThe fractal dimension D is calculated by Eq.(4). 1.2 Fractal dimensions and result analysis The data ud in this study cover approximately the same spatial extent at the same scale, and the differ-ent land parcels are extracted in the polygon, and the urban boundary is extracted in line according to the digital urban planning maps of different time periods with the aid of GIS.
Fig.1 The digital vegetation and urban boundary
Thereby, five layers of each map, namely, residen-tial area, industrial area, vegetation, water system and boundaries are available for calculating the fractal dimensions (Fig.1). Box-Counting method is carried out in our experiment which is widely ud in geog-raphy. Firstly, we transformed the vector layers of different land u into raster images, then fractal di-mensions of land parcels distributed in the images are evaluated with the help of GIS. The urban planning maps in our experiment reprent a main process of evolution of urban spatial pattern of Wuhan which includes three time periods: at the beginning of the establishment of new China, in the early part of Re-form and Opening, and the 1990s, a time at which urban expansion tide occurred.
Table 1 Fractal dimensions of different time periods
1950 1.325 3 1.350 2 1.462 6 1.542 3 1.201 21959 1.396 5 1.361 4 1.572 1 1.538 9 1.189 91980 1.413 5 1.372 2 1.525 4 1.550 1 1.210 01989 1.437 7 1.441 4 1.520 8 1.547 8 1.191 21997
1.449 6
1.487 2
1.521 0 1.548 5
1.232 5
As shown in Table 1, we list some cross-ctional years. The first stage (1949~1960), the entire urban spatial configuration extended little by little. Apart from the water system, fractal dimensions of different land u incread. The fractal dimension of indus-trial area incread 0.7% every year. The cond stage
ZHANG Yi, et al./Fractal Features of Urban Morphology and …
123
(1960~1980), spatial configuration became more and more complex. Increments of fractal dimensions were smaller than the previous stage, fractal dimension of industrial and residential area incread 0.1% and 0.11% every year, respectively. Vegetation area was reduced due to the expansion of industry and habita-tion, thus fractal dimension also decread. The frac-tal dimension of water system fluctuated within a very small range. At the third stage (1980~2000), the industry of Wuhan rushed into a high speed of devel-opment. Both the industrial and residential fractal dimensions apparently incread. With an increa of 11.5% every year of fractal dimension, the pattern of the residential area held a more complex spatial con-figuration.
There is an inherent constraint or effect on the rela-tionship between fractal indicator of urban morphol-ogy and human activities. Here we discuss the changes of fractal dimensions over a period of time to explore the relationship between fractal dimensions and residential area, for which a linear regression equation was established [3].
ln 9.8963 1.7274Areas D =− (5)
Table 2 Fractal dimensions and areas of residence
of different time
1959 116 442 11.665 1 1.361 4 1980
190 456 12.157 2 1.372 2 1989 285 423 12.561 7 1.441 4 1997
413 110
12.931 5
1.487 2
Fig.2 The relationship of fractal dimensions and areas
From the result of the linear regression equation shown in Fig.2, the correlation coefficient calculated is 0.952 7. It shows that the fractal dimensions of ur-ban morphology are great in relation to residential
areas.
2 Simulation with fractal features
Fractal dimension of spatial morphology has in-herent evolutional rule in different temporal sizes. Its change is greatly affected by human activity [4]. In the previous part of this paper, fractal dimensions
of dif-ferent land u in different time periods are derived using Box-Counting method. The quantitative re-arch on urban spatial characteristics has significant practical application for urban management and plan-ning [3], but just exploring the relationship between fractal dimension and urban morphology is not enough. Fractal theory should play a more important role in the rearch of geography, and fractal dimen-sion will be more widely ud in the area of spatial analysis. In this part of our paper we pay more atten-tion to the urban boundary, which is an important part of city configuration. As well as the coastline, cloud and mist, lightning and some other fractals, urban boundary has lf-similarity. It’s hard for the conven-tional Euclidean geometry to describe the phenom-ena. Since the establishment of fractal theory, some other subjects were promptly attracted by this theory [5] and some fractal generators are promoted to simulate the abnormal curves – for instance, mid-point subdi-vision fractal generator which is ud in our rearch. 2.1 Fractal generator and its dimension According to Eq.(1), fractal dimension D can be derived by
11ln(/)
免费人工翻译ln(/)
n n n n N N D r r ++=
(6)同义词典
Fig.3 shows the mid-point subdivision fractal gen-erator.
Fig.3 Mid-point subdivision fractal generator
Geo-spatial Information Science 11(2):121-126
124According to Eq.(6), the fractal dimension of this generator is:
1ln(2/1)
ln(/)
n n D r r +=network是什么意思
(7)
It can be changed into
ln 2
ln 2ln(cos )
D α=
+ (8)
So the relationship between fractal dimension and the shape of generator be related by Eq.(9)
ln 2
ln 2arccos D
e
α−= (9)
If fractal dimension D can be calculated by some methods, then the value of αwill be derived by Eq.(9), thereby fractal generators ground on different dimensions will be defined. Bad on this notion, abnormal curves with different dimensions could be simulated by the generators. As we can’t store data of every scale of urban boundary as a linear part of the map, how to generate the data from one scale to another without changing their configuration is an is-sue often mentioned in the rearch of automatic map generation [6]. Some conventional methods focus on maintaining the smoothness of linear elements. How-ever, they isolate the isotropy which is an obvious geographic feature of spatial change. Simulation bad on fractal dimension will take this factor into account. 2.2 Implementation of simulation
Simulation is ud to construct the enlarged bound-ary by applying the generator to the line gments, then it will yield a new boundary with more details. This process is continued towards the limit which is the minimum length of the line gments at a certain scale.
2.2.1 Simulate a simple curve
Firstly, the experiment was implemented from a simple curve, who fractal dimension has been cal-culated. According to some law, the shape of the curve alters with the iterative process, then every line gment of the curve is instituted by the fractal gen-erator. If the minimum threshold is the minimum length of a chord, this process will be terminated. As show in Fig.4, simulation results with different
length limits were obtained.
Fig.4 Simulation of a simple curve
The black curve is a source curve which is simu-lated using mid-point subdivision fractal generator;
the pink curve is its analog. It is clear that simulation results are mainly affected by the limitation length which is counted by pixels. That means if a line g-ment is longer than the threshold, it will not be re-placed by the generator. The maximum threshold in-creas as the details decrea, as shown in Fig.4.The threshold incread from 2 pixels to 40 pixels, and in the end, the source curve is overlapped by the simula-tion result. Actually, in a map, if a line gment of a boundary is a long o
欧洲降息
ne, it may be an edge of a regular land parcel which has no more fine features to display as the map scale is enlarged. So length is taken into consideration in the urban boundary simulation. 2.2.2 Define the generator and obtain the data of
coordinates Cities are usually depicted in the plane as two-dimensional phenomena thus their boundaries immediately imply some measure of area which compo the entire urban morphology [7]. In our ex-periment, the urban boundaries are defined as clod lines; their fractal dimensions during different time periods are derived and listed in Table 1. According to Eq.(9), the angle α can be calculated as the frac-tal dimension D is available. Every line gment of the boundary has its own generator which is deter-mined by the angle and the length of the line gment. Coordinates of every node are stored when digitizing the urban boundary so the length of every line g-ment can be calculated, as shown in Fig.5.
There is a curve with certain dimension D , and (X 0, Y 0), (X 1, Y 1)… (X 4, Y 4) are the coordinates of the
ZHANG Yi, et al./Fractal Features of Urban Morphology and …
汽车起步熄火
125
Fig.5 Determine the fractal generator
nodes. Line gment BC is taken as an example, as the fractal dimension, coordinates of node B , C are available, so angle α can be acquired from equation (9) , the length and the azimuth angle of line gment BC can be easily obtained, then the fractal generator will be defined by the factors. Becau the uncer-tainty of natural fractal objects and the mechanism of the spatial phenomena is not understood deeply and roundly, some random factors should be added into the simulation process that will make the results clor to reality, so the position of N has two possi-bilities: above or below the BC,
1212()/2sin ()/2cos N N X X X J L R Y Y Y J L R
ββ=+−××+=++××+ (10)
Where 1J = or 1−, it will decide whether node N will be above or below BC ; L is the distance per-pendicular to the midpoint of BC ; βis the azimuth angle of BC ; R is a random number which will make the location of N or N ′ randomly scattered in the dashed rectangle as show in Fig.5.
The disposals are the embodiment of isotropy which is an obvious geographic feature of spatial change, and may well reflect the complex and fine features of fractals.
2.2.3 Simulation of urban boundaries
At the beginning of simulation, veral reference points are needed which approximately determine the configuration of urban boundaries, so when digitizing the urban boundaries we may as well choo points in the inflexion that will help to exhibit more details of the linear elements. The iterative process is con-fined by two thresholds——the maximum and mini-mum length which will be ud to terminate the itera-tive operation. If a line gment is shorter than the扬州新东方
minimum threshold there is no need for it to be estab-lished by the generator, and if a line gment is longer than the maximum threshold, it maybe the edge of a regular land parcel which form the entire urban boundary. For example, a wall of a steel fac-tory is not suited for the iterative process. Therefore, another point should be recognized when digitizing the urban boundaries. That is, if there are no regular land parcels at the edge of the city, we intend to u small line gments to digitize the urban boundaries, and the maximum threshold is decided by the average chord length d avg , and the length of every chord is computed using the equation:
221/2,111[()()],1,...1,
i i i i i i d x x y y i n +++=++−=− (11)
辅料英语
Where n is the number of total nodes, then the length of the boundary can be summed as:
1
,11.n i i i s d −+==∑ (12)
The average chord length is therefore:
avg 1
s
d n =
− (13) The minimum threshold is decided by the change of map scale. The algorithm is summarized as:
1) enter the original digital geographic curve, which stores as an array of node coordinates; 2) calculate the fractal dimension D ;
3) compute the average chord length of the curve and determine the minimum and maximum thresh
old; 4) identify whether each chord is between the mini-mum and maximum threshold, deriving the fractal generator using Eq.(10). This process will be contin-ued towards the minimum threshold; 5) export the new curve.
Fig.6 shows a part of the simulation results of ur-ban boundaries. From the result, two chords are not
taken into the iterative process, as they are out of the
Fig.6 A part of simulation results of urban boundaries
Geo-spatial Information Science 11(2):121-126 126
maximum threshold.
3 Conclusions
In this paper, urban spatial characteristics of Wu-han city are analyzed quantitatively using GIS. The spatial configuration characteristics and expansion of urban land u are discusd using the fractals, and the change of fractal dimension of the city fractal system is compared with the actual situation. The re-lationship between residential area and dimension are analyzed, which implies that fractal dimension can be ud to show the rule of urban spatial morphology change, then an experiment of simulating the urban boundary with fractal dimension is performed in this paper.
References
[1] Mandelbrot B (1967) How long is the coast of Britain?
Statistical lf-similarity and fractional dimension[J]. Sci-ence, 156(3775): 636-638.
waga[2] Jin Yiwen, Lu Shijie (1998) Fractal geometry theory and
application[M]. Hangzhou: Zhejiang University Press (in Chine)
[3] Li Jiang (2005) Fractal dimension of urban spatial mor-
phology and its application[J]. Engineering Journal of Wuhan University, 38(3): 99-103 (in Chine)
[4] Li Jiang (2004) Evolvement of spatial morphology in out
areas of cities[J]. Resource and Environment in the Yang-tzi Basin, 38(3): 208-211 (in Chine)
[5] Hu Rian, Hu Jiyang, Xu Shugong (1995) Computer im-
ages of fractals and its application[M]. Beijing: China Railway Publishing Hou (in Chine)
[6] Longley P A, Batty M (1989) Fractal measurement and
line generalization[J]. Computers & Geoscience, 15(2): 167-183
[7] Batty M, Longley P A (1994) Fractal citie: a geometry of
form and function[M]. London: Academic Press
