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Journal for Geometry and Graphics 08 (2004), No. 1, 041--058
Copyright Heldermann Verlag 2004

Geometry of Roofs from the View Point of Graph Theory

Edwin Kozniewski
Inst. of Civil Engineering, Engineering Graphics & Computer Methods Division, University of Technology, Wiejska st. 45E, 15-351 Bialystok, Poland

Roofs discussed in this article are defined as polyhedral surfaces on the basis of two assumptions:
(1) all eaves of a roof form a planar (simply connected or k-connected) polygon called the base of the roof,
(2) every hipped roof end makes the same angle with the (horizontal) plane which contains the base.
Thus every roof, and equivalently the orthographic projection of this roof onto a plane, is uniquely defined by its base. Namely, each ridge of a roof can be obtained as a line segment of the bisectrix of the angle formed by two appropriate edges of the base; if these axes are parallel, then the ridge is the axis of symmetry. Disregarding the metric properties of a roof, we can treat such roofs as planar graphs. Usually, i.e., if the vertices of the base of a given roof are in general position, these are 3-regular graphs. For such graphs (with a simply connected or k-connected base of the roof) we formulate and prove a new Euler formula (Euler formula for regular roofs), and the so-called equations of a regular roof.

Keywords: Geometry of roofs, generalized polygon, k-connected generalized polygon, planar graphs, connected graphs, regular graphs, Euler formula for regular roofs.

MSC: 51N05; 52B05, 05C90, 68U05

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