Minimum-area ellipse bounding an isovist: towards a 2D GIS-based efficient implementation

ABSTRACT

In geographic information science and technology, various methods and studies exist to characterize the linearity, rectangularity, convexity, circularity or compactness, sinuosity or tortuosity of a given spatial shape. Although there is much work on ellipticity in image processing, we do not address, in geomatics, the issue of matching to a reference elliptical shape. Regarding this issue, this article is a contribution to the qualification of urban open spaces. It provides an operating algorithm for determining a minimum-area bounding ellipse for any given polygonal shape. It also proposes an implementation of this algorithm in the context of a Geographic Information System and a Jupyter Notebook. As an application, it focuses on two real urban configurations on fields of about 300 isovists. The results from the application of this approach in two urban areas in France show that the ellipse is a better minimum bounding geometry than are circle or rectangles, at least for the half-dozen descriptors studied. The improvement relatively to the minimum bounding rectangle is particularly significant in terms of correlation concerning the orientation (+20%) and drift (+10%).

KEYWORDS:

• Minimum-area bounding ellipse
• isovist
• elliptic hull
• caliper
• diameter

Disclosure statement

The authors have no conflicts of interest to declare.

Data and codes availability statement

The data and codes that support the findings of this study are available with a DOI at http://doi.org/10.6084/m9.figshare.c.4964291. In particular, we provide a Jupyter notebook that explains step by step how to use the Python plugin (released under the GPL v3 license) that implements the method presented in this article.

Supplemental data

Supplemental data for this article can be accessed here.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1. We adopt Žunić et al. (2017)’s definition: ‘Notice that by the shape ellipticity we mean the similarity of a given shape to the planar region bounded by an ellipse’.

2. Let’s call chord length the distance in a straight line between two points of the shape contour. The diameter is defined as the greatest chord regardless of the selected pair of contour points.

3. The Feret diameter, also known as caliper diameter, is defined as the distance between two parallel tangents of the shape contour. It depends on the chosen direction (Toussaint 1983).

4. The shapes under study in this paper all derive from the isovist and as such are generated from a viewpoint. By extending the concept defined by (Conroy 2001), we call ‘drift’ the Euclidean distance between the generating viewpoint and the centroid of the corresponding shape.

5. For any non-circular polygon shape, orientation is defined as the measure of the azimuth of its diameter in the range of 0 to 180^°.

6. Indeed, the QgsEllipse class as defined in QGIS 3.0 core library is a container for elliptical geometries defined by a centre point, a semi-major axis, a semi-minor axis and an azimuth. This class does not in any way provide a solution for determining a minimum-area bounding ellipse.

7. The underlying idea is to solve linear systems as well-conditioned as possible.

8. In Appendix A, we have grouped together the few general equations related to ellipses that we use in this paper. All equations with labels beginning with the letter A refer to this appendix.

9. The proportionality factor being equal to the absolute value of its determinant.

10. Using the routine solve of the numpy.linalg package.

Additional information

Notes on contributors

Thomas Leduc

Dr. Thomas Leduc has a master degree in mathematical engineering. He holds a doctorate in computer science from the University of Paris VI, now renamed Sorbonne University. He was Deputy Director of the AAU laboratory (CNRS) at the School of Architecture in Nantes from 2014 to 2018. His research activities focus on the urban morphology characterization using geographic information science and technology. He is involved in several projects that aim to describe the morphology of open spaces using visibility-based methods.

Michel Leduc

Honorary Prof. Michel Leduc defended a state doctorate in mathematics in 1971. Appointed lecturer in 1964 and assistant professor in 1966 at the Faculty of Sciences of Orsay, he was successively appointed professor at the University of Rouen in 1971 and of the University of Le Havre in 1984. His research has focused on mathematical analysis, geometry, and optimization. He supervised five theses of modelling in the Laboratory of Ultrasonic Acoustics and Electronics at the University of Le Havre. He was President of the University of Le Havre from 1991 to 1994.