Geometry of Roofs from the View Point of Graph Theory
Edwin Kozniewski · 2004
Roofs discussed in this article are deflned 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 deflned by its base. Namely, each ridge of a roof can be ob- tained 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 po- sition, 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.