*2.1. Definition of the Geometric Model of the Canopy*

Canopy roofs of hyperbolic paraboloid shape were chosen as a case study. It was assumed that the roofs covered a square plan of an area of 100 square meters. Each roof was supported by four columns placed symmetrically, whereas each column was joined with the grid by four branches, Figure 2. The positions of the columns have been set as parametric variables, as well as the locations of the branches' nodes.

**Figure 2.** The view of the considered structure.

The hyperbolic paraboloid roof surface as the ruled surface was established by two skew lines—the directrix lines and a director plane to which all surface's rulings are parallel, Figure 3. This surface constituted the base surface to form a grid of bars.

**Figure 3.** A hyperbolic paraboloid with the directrix lines expressed.

The Grasshopper's algorithm, composed of the connected block components, was created in such a way that two skew lines defined parametrically by two pairs of various points were distinguished as its input. Each of the lines were next divided into the same number of elements to establish a series of points on them. These series of points were next joined by lines to define a hyperbolic paraboloid, which constituted a base surface for structural grid creation. Therefore, the obtained surface was discretized by dividing it into the same number of parts in two directions. Thanks to this a three-dimensional quadrate grid was obtained, whose vertices lay on the base surface. The Grasshopper's block script for roof's base surface creation is presented in Appendix A, Figure A1.

Next, each spatial polygon of the obtained grid was divided into two triangles to form a triangular bar grid. Depending on the division direction of each of the quadrangles, which can be done along shorter or longer diagonals, and depending on the number of subdivisions of the base surface as well as its type, various patterns of bar grids can be obtained. In Figure 4, the examples of grid patterns obtained due to eight-fold division of the hyperbolic paraboloid surface are shown. The structure with the grid pattern split along a short diagonal is further called the structure of type a in the paper, whereas the structure with the grid pattern split along a long diagonal is called the structure of type b.

**Figure 4.** Rectangular projections of considered grid patterns: (**a**) the pattern—split along a short diagonal; (**b**) the pattern—split along a long diagonal.

The geometry of each considered structure was determined using a block algorithm with variable parameters. However, during the simulations carried out, the following variables were adopted:


**Figure 5.** Presentation of the allowed area of supports' positions: (**a**) a horizontal projection; (**b**) a perspective view.

Moreover, it was assumed that each branch node was placed at the column's end point, whereas the column length was equal to a distance of between sixty and eighty percent of the distance of the ground support from the roof surface. However, the columns were assumed to be located within the rectangular plan, but no further than one meter from the place's border (offset from the edge of the square in both *x* and *y* directions of 0.0–1.0 m), Figure 5.
