# Symmetric Free Form Building Structures Arranged Regularly on Smooth Surfaces with Polyhedral Nets

## Abstract

**:**

## 1. Introduction

## 2. State of the Art

_{0}and w

_{0}contained in two orthogonal principal planes (x,z) and (y,z) of symmetry of a reference ellipsoid ω

_{r}, Figure 13a. The method allows the investigated form of a polyhedral structure to be a regular network and precisely take into account by the designer the variable curvature of ω

_{r}. The method replaces a finite number of the selected straight lines t

_{i,j}normal to ω

_{r}with side edges k

_{i,j}of the sought-after reference tetrahedrons. A specific feature of the reference tetrahedrons is that two their subsequent straight side edges k

_{i,j}and k

_{i}

_{+1,j}belonging to the same side must intersect, while two corresponding straight lines t

_{i,j}and t

_{i}

_{+1,j}normal to ω

_{r}do not intersect to each other, Figure 13b. The positions of t

_{i,j}and t

_{i}

_{+1,j}have to be replaced by k

_{i,j}and k

_{i}

_{+1,j}, so that the positions of k

_{i,j}and k

_{i}

_{+1,j}have to be defined based on the geometric properties of ω

_{r}. An architectural study of a free form created with the help of the method is presented in Figure 14.

## 3. Aim

## 4. Concept of the Method

_{ij}of Γ is limited by four adjacent planes of the system defined by means of four vertices W

_{ABij}, W

_{CDij}, W

_{ADij}, and W

_{BCij}, Figure 16. Each single shell segment Ω

_{ij}and each complete free form Σ

_{ij}are located in one mesh Γ

_{ij}, so that façade walls, roof directrices and eaves segments are included in the aforementioned quadruple of planes of Γ

_{ij}.

_{ij}intersect in the side edges: a

_{ij}, b

_{ij}, c

_{ij}, d

_{ij}, and, every two opposite planes intersect in the axes u

_{ij}or v

_{ij}of Γ

_{ij}. The side edges and axes of Γ

_{ij}are defined by means of four vertices W

_{ABij}, W

_{CDij}, W

_{ADij}, W

_{BCij}, Figure 16a. Thus, each possible triple of these vertices determines one plane of Γ

_{ij}. On the basis of these vertices, four points S

_{Aij}, S

_{Bij}, S

_{Cij}and S

_{Dij}are constructed on four side edges a

_{ij}, b

_{ij}, c

_{ij}, d

_{ij}. These points are vertices of a spatial quadrangle S

_{Aij}S

_{Bij}S

_{Cij}S

_{Dij}determining a certain piece of a reference surface ω

_{r}, Figure 16b. In relation to ω

_{r}, four vertices A

_{ij}, B

_{ij}, C

_{ij}, D

_{ij}of single eaves B

_{vij}are determined to obtain mutually skew roof directrices. Vertices P

_{Aij}, P

_{Bij}, P

_{Cij}, P

_{Dij}, Figure 16a, belonging to a flat horizontal base of the sought-after free form Σ are constructed in relation to the aforementioned four vertices W

_{ABij}, W

_{CDij}, W

_{ADij}, W

_{BCij}. The complex free form Σ created on the basis of such a reference network Γ is a sum of all individual free forms Σ

_{ij}. Finally, a resultant z-axis symmetric free form structure can be achieved, Figure 17a,b.

## 5. Method’s Algorithm

_{11}of a reference network Γ is created so that the positions of its four vertices W

_{AB}

_{11}, W

_{CD}

_{11}, W

_{AD}

_{11}, and W

_{BC}

_{11}are defined in the three-dimensional space. For this purpose, a global coordinate system [x,y,z] must be taken, Figure 18a, where a point O is the origin of [x,y,z]. These vertices are arranged symmetrically in relation to the principal planes of [x,y,z], so the sought-after mesh must be symmetric. The first set of initial data is formed from the measures of the vectors and angles employed to determine all characteristic points of Γ

_{11}.

_{11}are determined as follows. The position of vertex W

_{CD}

_{11}is the result of the translation T

_{OCD}

_{11}of the point O by the vector OW

_{CD}

_{11}whose measure is defined by means of one element of the first initial set. In an analogous way, the position of vertex W

_{AB}

_{11}is defined by means of the translation T

_{OAB}

_{11}of the point O by the vector OW

_{AB}

_{11}so that its location is on opposite side of O on the x-axis.

_{AD}

_{11}is the result of a composition of the rotation O

_{CD}

_{11_}

_{AB}

_{11}of the z-axis about the x-axis by the angle α

_{AD}

_{11}and the translation T

_{OAD}

_{11}of O by the vector OW

_{AD}

_{11}, where the measures of α

_{AD}

_{11}and OW

_{AD}

_{11}are two elements of the first set of initial data. The position of vertex W

_{BC}

_{11}can be obtained in an analogous way, that is, by means of the rotation O

_{AB}

_{11_CD11}of the z-axis about the x-axis by the angle α

_{BC}

_{11}opposite to α

_{AD}

_{11}and the translation T

_{OBC}

_{11}of O. If we want to achieve a z-axis-symmetric reference tetrahedron, the absolute values of the above vectors and angles must be equal to each other, respectively.

_{AB}

_{11}, W

_{CD}

_{11}, W

_{AD}

_{11}and W

_{BC}

_{11}determine four straight side edges: a

_{11}, b

_{11}, c

_{11}and d

_{11}of Γ

_{11}. In order to obtain four points S

_{A}

_{11}, S

_{B}

_{11}, S

_{C}

_{11}and S

_{D}

_{11}of a reference surface and four vertices A

_{11}, B

_{11}, C

_{11}and D

_{11}of eaves B

_{v}

_{11}of a shell roof structure, four vectors have to be measured along the side edges a

_{11}, b

_{11}, c

_{11}and d

_{11}, Figure 19a,b, so that S

_{A}

_{11}= T

_{SA}

_{11}(W

_{AD}

_{11}), S

_{B}

_{11}= T

_{SB}

_{11}(W

_{BC}

_{11}), S

_{C}

_{11}= T

_{SC}

_{11}(W

_{BC}

_{11}), and = T

_{SD}

_{11}(W

_{AD}

_{11}), and A

_{11}= T

_{A11}(S

_{A}

_{11}), B

_{11}= T

_{B11}(S

_{B}

_{11}), C

_{11}= T

_{C11}(S

_{C}

_{11}), and D

_{11}= T

_{D11}(S

_{D}

_{11}). The measures of these vectors belong to the set of initial data.

_{11}, a point P

_{D}

_{11}= T

_{PD11}(W

_{CD}

_{11}) has to be defined on d

_{11}. The measure of the vector W

_{CD}

_{11}P

_{D}

_{11}must be one of the elements of the first set of initial data. Other points of this base can be calculated as the intersection of the horizontal base plane passing through the point P

_{D}

_{11}and the four side edges of Γ

_{11}.

_{12}, Figure 20a,b. At first, a second set of initial data must be adopted. The set is composed of the measures of the angles and vectors employed in the algorithm to define the positions of all characteristic points of Γ

_{12}.

_{12}are determined as follows. Vertex W

_{AB}

_{12}is identical with W

_{CD}

_{11}of Γ

_{11}introduced previously. Positions of the vertices W

_{BC}

_{12}= T

_{BC}

_{12}(W

_{BC}

_{11}), W

_{AD}

_{12}= T

_{AD}

_{12}(W

_{AD}

_{11}) are defined on two side edges b

_{12}= c

_{11}, a

_{12}= d

_{11}so that the measures of the vectors W

_{AD}

_{11}W

_{AD}

_{12}and W

_{BC}

_{11}W

_{BC}

_{12}are elements of the second set of initial data. The position of vertex W

_{CD}

_{12}is obtained as a result of a composition of the rotation O

_{WAD}

_{12_WBC12}of O

_{12}W

_{AB}

_{11}about the axis W

_{AD}

_{12}W

_{BC}

_{12}by the angle α

_{CD}

_{12}and the translation T

_{CD}

_{12}of O

_{12}by the vector O

_{12}W

_{CD}

_{12}, where O

_{12}is a point of the (W

_{AD}

_{12}, W

_{BC}

_{12}) straight line. If the reference tetrahedron Γ

_{12}is to be symmetrical about the z-axis, W

_{BC}

_{12}and W

_{AD}

_{12}have to be symmetric to each other towards the (x,z)-plane and O

_{12}has to be the middle point of the edge W

_{BC}

_{12}W

_{AD}

_{12}. Four vertices W

_{AB}

_{12}, W

_{CD}

_{12}, W

_{AD}

_{12}, and W

_{BC}

_{12}determine four straight side edges: a

_{12}, b

_{12}, c

_{12}, and d

_{12}of Γ

_{12}.

_{r}and Γ are constructed so that S

_{A}

_{12}= S

_{D}

_{11}, S

_{B}

_{12}= S

_{C}

_{11}, S

_{C}

_{12}= T

_{SC12}(W

_{BC}

_{12}), S

_{D}

_{12}= T

_{SD12}(W

_{AD}

_{12}), where the measures of the vectors W

_{BC}

_{12}S

_{C}

_{12}and W

_{AD}

_{12}S

_{D}

_{12}are two elements of the second set of initial data. On the basis of two other elements of the second set, four points A

_{12}= D

_{11}, B

_{12}= C

_{11}, C

_{12}= T

_{C12}(S

_{C}

_{12}), D

_{12}= T

_{D12}(S

_{D}

_{12}), Figure 20b, constituting the vertices of a closed spatial quadrilateral line modeling shell roof eaves are constructed.

_{21}, Figure 21a,b. The third set of initial data, composed of the measures of the respective angles and vectors employed to define all specific points of Γ

_{21}is adopted.

_{AB}

_{21}, W

_{CD}

_{21}, W

_{AD}

_{21}, and W

_{BC}

_{21}of Γ

_{21}are determined as follows, Figure 21a. The vertex W

_{AD}

_{21}= W

_{BC}

_{11}. The positions of vertices W

_{AB}

_{21}= T

_{AB}

_{21}(W

_{AB}

_{11}), W

_{CD}

_{21}= T

_{CD}

_{21}(W

_{CD}

_{11}) are defined on two side edges a

_{21}= b

_{11}, d

_{21}= c

_{11}, Figure 21b, so that the measures of the vectors W

_{AB}

_{11}W

_{AB}

_{21}and W

_{CD}

_{11}W

_{CD}

_{21}are two elements of the third set of initial data. The position of W

_{BC}

_{21}= T

_{BC}

_{21}O

_{WCD}

_{21_WAB21}(W

_{BC}

_{11}) is obtained as a result of a composition of the rotation O

_{WCD}

_{21_WAB21}of O

_{21}W

_{AD}

_{21}about the axis W

_{CD}

_{21}W

_{AB}

_{21}by the angle α

_{BC2}

_{1}and the translation T

_{BC}

_{21}of O

_{21}by the vector O

_{21}W

_{BC}

_{21}, where O

_{21}is a point of the straight line W

_{CD}

_{21}W

_{AB}

_{21}. If the reference tetrahedron Γ

_{21}is to be symmetrical towards the (y,z)-plane, the positions of points W

_{CD}

_{21}and W

_{AB}

_{21}have to be symmetric to each other towards the (y,z)-plane and O

_{21}has to be the middle point of the segment W

_{CD}

_{21}W

_{AB}

_{21}. Four vertices W

_{AB}

_{21}, W

_{CD}

_{21}, W

_{AD}

_{21}, and W

_{BC}

_{21}determine four straight side edges: a

_{21}, b

_{21}, c

_{21}and d

_{21}of Γ

_{21}.

_{r}are determined on the side edges of Γ

_{21}so that S

_{A}

_{21}= S

_{B}

_{11}, S

_{D}

_{21}= S

_{C}

_{11}, S

_{C}

_{21}= T

_{SC21}(W

_{BC}

_{21}) and S

_{B}

_{21}= T

_{SB21}(W

_{BC}

_{21}), where the measures of the vectors W

_{BC}

_{21}S

_{C}

_{21}and W

_{BC}

_{21}S

_{B}

_{21}belong to the third set of initial data.

_{22}, Figure 22a,b. The fourth set composed of the measures of the vectors and angles employed in this step is adopted.

_{AD}

_{22}= W

_{BC}

_{12}, W

_{AB}

_{22}= W

_{CD}

_{21}, W

_{BC}

_{22}= T

_{BC}

_{22}(W

_{BC}

_{21}), W

_{CD}

_{22}= T

_{CD}

_{22}(W

_{CD}

_{12}) of Γ

_{12}, Figure 22a, are defined so that the measures of the vectors W

_{BC}

_{21}W

_{BC}

_{22}and W

_{CD}

_{21}W

_{CD}

_{22}are two elements of the fourth set of initial data. Four vertices W

_{AB}

_{22}, W

_{CD}

_{22}, W

_{AD}

_{22}, and W

_{BC}

_{22}determine four straight side edges: a

_{22}, b

_{22}, c

_{22}, and d

_{22}and two axes u

_{22}and v

_{22}of Γ

_{22}Figure 23a.

_{r}are constructed so that S

_{A}

_{22}= S

_{C}

_{11}, S

_{D}

_{22}= S

_{C}

_{12}, S

_{B}

_{22}= S

_{C}

_{21}, and S

_{C}

_{22}= T

_{C22}(W

_{BC}

_{22}), Figure 23b, where the measure of the vector W

_{BC}

_{22}S

_{C}

_{22}is a value belonging to the fourth set of the initial data. On the basis of these points of ω

_{r}, four points A

_{22}= C

_{11}, B

_{22}= C

_{21}, C

_{22}= T

_{C22}(S

_{C}

_{22}), D

_{22}= C

_{12}constituting the vertices of a closed spatial quadrilateral line modeling complete shell roof eaves are constructed.

_{ij}constructed above is a subnet Γ

_{1}constituting about one-fourth of the designed reference network Γ. The other three parts of Γ can be built using z-axis symmetry and two (x,z)- and (y,z)-plane symmetries called 3D-mirrors, Figure 24a,b, in the way described Section 6 with the help of a certain example. All obtained points S

_{Aij}, S

_{Bij}, S

_{Cij}, and S

_{Dij}, Figure 24a, and their images obtained as a result of the aforementioned symmetries are the selected vertices of a certain net defining ω

_{r}, Figure 24b. In relation to this net, a roof structure Ω composed of nine sectors Ω

_{ij}is positioned. Thus, the vertices of the eaves of each Ω

_{ij}segment of the roof structure Ω are defined on the basis of ω

_{r}, Figure 25.

_{ij}designate side edges a

_{ij}, b

_{ij}, c

_{ij}, and d

_{ij}and planes of Γ. Each new reference tetrahedron Γ

_{i}

_{+1j}or Γ

_{ij}

_{+1}is created as a spatial mesh having four sought-after vertices defined in the selected side edges of the previously constructed tetrahedrons Γ

_{ij}to obtain subsequent pairs of the adjacent meshes having common planes. In the aforementioned planes of Γ, the locations of roof directrices are determined.

_{11}, one its vertex is laid outside the side edges of the already created subnet of Γ. This vertex determines a new plane of Γ passing through the already constructed axis of this tetrahedron. However, in the case of two directions diagonal in relation to the first Γ

_{11}, each new reference tetrahedron has to have two vertices identical to two from the four previously constructed vertices of Γ and two other new vertices have to be determined on two side edges of the previously created subnet of Γ.

_{v}is given in Section 6, where certain division coefficients of the selected pairs of some adjacent vertices by other vertices of the reference network are employed. In addition, some points belonging to a reference surface and vertices of the eaves edge line of each individual roof shell segment are defined at each side edge of the network Γ. It is also advisable to use analogous division coefficients of pairs of the reference network’s vertices in determining these points of the reference surface and the eaves limiting the individual roof shell segments. An example of making such a parameterization is included in Section 6.

## 6. Parametric Shaping of an Example Reference Network and a Free Form Shell Roof

_{v}networks. These coefficients and proportions allow us to define the positions of (1) the sought-after vertices of Γ with respect to the adopted or calculated at one of the previous steps pairs of other vertices of Γ, (2) the subsequent planes of Γ, (3) the points S

_{Air}, S

_{Bij}, S

_{Cij}, and S

_{Dij}belonging to a reference surface w

_{r}, (4) the vertices A

_{ij}B

_{ij}, C

_{ij}and D

_{ij}of B

_{v}relative to the already determined vertices of Γ and these points of w

_{r}.

_{1}of a reference network Γ, Figure 26, and a quarter B

_{v}

_{1}of B

_{v}consisting of closed spatial quadrangles B

_{vij}, is presented below. It is based on some adopted proportions. All vertices of the other three quarters Γ

_{2L}, Γ

_{3p}, Γ

_{4r}of Γ, Figure 26, and B

_{v}

_{2L}, B

_{v}

_{3p}, B

_{v}

_{4r}of B

_{v}are determined using: (1) a z-axial symmetry, in the case of Γ

_{2L}, (2) a (x,z)-plane symmetry called 3D-mirror, where Γ

_{3p}is constructed, (3) a (y,z)―plane symmetry for Γ

_{4r.}

_{11}, Figure 27. The subsequent meshes Γ

_{ij}are determined in the order presented in the previous section, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25. In order to build the mesh Γ

_{11}, the following quantities and proportions are adopted. The values of these variables are given in Table 1.

_{AB}

_{11}, W

_{CD}

_{11}, W

_{BC}

_{11}) and (W

_{AB}

_{11}, W

_{CD}

_{11}, W

_{AD}

_{11}), Figure 27a–c, are inclined to each other at an angle α

_{11}= 2 α

_{B}

_{C11}= 2 α

_{AD}

_{11}, Figure 19. The length of the edge W

_{CD}

_{11}W

_{AB}

_{11}contained in the u

_{1}-axis was adopted in accordance with the values given in Table 1.

_{BC}

_{11}W

_{AD}

_{11}’s length results from the adopted values of the angle α

_{11}and the height OW

_{BC}

_{11}of the triangle <W

_{AB}

_{11}, W

_{CD}

_{11}, W

_{BC}

_{11}> as well as the proportion.

_{OW}

_{11}= OW

_{AD}

_{11}/OW

_{BC}

_{11}

_{A}

_{11}, S

_{V}

_{11}, S

_{C}

_{11}, and S

_{D}

_{11}are defined with the same constant d

_{S}

_{11}listed in Table 1 as follows

_{SB}

_{11}= d

_{SA}

_{11}= d

_{SC}

_{11}= d

_{SD}

_{11}= d

_{S}

_{11}

_{AB}

_{11}and ending at W

_{AD}

_{11}, m($\overrightarrow{{W}_{AB11}{W}_{AD11}}$) is the measure of $\overrightarrow{{W}_{AB11}{W}_{AD11}}$, $\overrightarrow{{W}_{AB11}{S}_{A11}}$ is the vector whose starting point is W

_{AB}

_{11}and ending point is S

_{A}

_{11}, etc. Thus, the location of points S

_{A}

_{11}, S

_{B}

_{11}, S

_{C}

_{11}and S

_{D}

_{11}is defined on the basis of the adopted division coefficients of all pairs {W

_{AB}

_{11}, W

_{AD}

_{11}}, {W

_{AB}

_{11}, W

_{BC}

_{11}}, {W

_{CD}

_{11}, W

_{BC}

_{11}} and {W

_{CD}

_{11}, W

_{AD}

_{11}} of the vertices of the Γ

_{11}mesh. The subsequent four points S

_{A}

_{11}, S

_{B}

_{11}, S

_{C}

_{11}and S

_{D}

_{11}usually form a flat rectangle determining the reference surface ω

_{r}in relation to which the shell segments of the designed multi-segment shell roof are arranged in the three-dimensional space.

_{11}, B

_{11}, C

_{11}and D

_{11}of Γ

_{11}, Figure 27a, are defined by means of the following proportions

_{AB}

_{11}and the end point at A

_{11}, etc. Points A

_{11}, B

_{11}, C

_{11}, and D

_{11}determine a spatial quadrangle B

_{v}

_{11}constituting the eaves of a single, smooth, shell segment Ω

_{11}modeling a single shell of a complex roof structure. It was adopted a constant dd

_{11}to calculate the values of four division coefficients d

_{A}

_{11}, d

_{B}

_{11}, d

_{C}

_{11}, and d

_{D}

_{11}. This constant is used with positive or negative sign depending on whether the points A

_{11}, B

_{11}, C

_{11}, and D

_{11}lie above or below ω

_{r}defined in the corresponding area by means of the quadrangle S

_{A}

_{11}S

_{B}

_{11}S

_{C}

_{11}S

_{D}

_{11}, Figure 27a. The ratios d

_{A}

_{11}, d

_{B}

_{11}, d

_{C}

_{11}, and d

_{D}

_{11}are calculated as follows

_{A}

_{11}= d

_{SA}

_{11}+ dd

_{A}

_{11}

_{B}

_{11}= d

_{SB}

_{11}+ dd

_{B}

_{11}

_{C}

_{11}= d

_{SC}

_{11}+ dd

_{C}

_{11}

_{D}

_{11}= d

_{SD}

_{11}+ dd

_{D}

_{11}

_{B}

_{11}= −dd

_{A}

_{11}= −dd

_{C}

_{11}= dd

_{D}

_{11}= dd

_{11}

_{CD}

_{12}is defined by means of the angle α

_{11}adopted in the previous step and the following formula.

_{CD}

_{12}= α

_{CD}

_{12}/α

_{11}

_{CD}

_{12}, it was also adopted the following relationship

_{12}is the middle of the segment W

_{AD}

_{12}W

_{BC}

_{12}. The aforementioned values are listed in Table 2. Two new division coefficients d

_{WBC}

_{12}and d

_{WAD}

_{12}are adopted as follows

_{CD}

_{11}and the end point at W

_{BC}

_{12}, and m($\overrightarrow{{W}_{CD11}{W}_{BC11}}$) is the measure of the vector $\overrightarrow{{W}_{CD11}{W}_{BC11}}$, etc.

_{A}

_{12}and S

_{B}

_{12}are similar to the positions of S

_{D}

_{11}and S

_{C}

_{11}determined previously for Γ

_{11}. The positions of S

_{C}

_{12}and S

_{D}

_{12}result from the following proportions

_{SC}

_{12}= d

_{SD}

_{12}= d

_{S}

_{11}

_{12}= D

_{11}, B

_{12}= C

_{11}are calculated previously for Γ

_{11}. The positions of points C

_{12}and D

_{12}can determined by means of the following proportions

_{C}

_{12}= d

_{SC}

_{12}+ dd

_{C}

_{12}

_{D}

_{12}= d

_{SD}

_{12}+ dd

_{D}

_{12}

_{C}

_{12}= −dd

_{D}

_{12}= dd

_{12}= dd

_{11}

_{AB}

_{13}, W

_{AD}

_{13}, W

_{CD}

_{13}, and W

_{BC}

_{13}of Γ

_{13}constituting one mesh of Γ

_{1}, Figure 28a, the points S

_{A}

_{13}, S

_{B}

_{13}, S

_{C}

_{13}, and S

_{D}

_{13}of ω

_{r}and the vertices A

_{13}, B

_{13}, C

_{13}, and D

_{13}of the closed eaves quadrangle B

_{v}

_{13}, Figure 28b, can be defined analogously as for Γ

_{12}and B

_{v}

_{12}using the following formula

_{CD}

_{13}= α

_{CD}

_{13}/α

_{11}

_{SC}

_{13}= d

_{SD}

_{13}= d

_{S}

_{11}

_{C}

_{13}= d

_{SC}

_{13}+ dd

_{C}

_{13}

_{D}

_{13}= d

_{SD}

_{13}+ dd

_{D}

_{13}

_{D}

_{13}= −dd

_{C}

_{13}= dd

_{13}= dd

_{11}

_{13}is the middle point of W

_{AD}

_{13}W

_{BC}

_{13}.

_{v}

_{12}and B

_{v}

_{13}results from the relationships adopted for all meshes of B

_{v}, which are as follows S

_{A}

_{13}= S

_{D}

_{12}, S

_{B}

_{13}= S

_{C}

_{12}, A

_{13}= D

_{12}, B

_{13}= C

_{12}, Figure 28a. The values of the adopted new proportions are given in Table 3.

_{1j}and B

_{v}

_{1j}meshes (for j = 1 to 3) of the first orthogonal direction in Γ, the manner of determining the subsequent Γ

_{i}

_{1}and B

_{v}

_{i}

_{1}meshes (for i = 1 to 3) of the second orthogonal direction in Γ, Figure 29a, needs a slight modification.

_{BC}

_{21}is a function of the angle α

_{11}, defined with the help of the following formula

_{BC}

_{21}= α

_{BC}

_{21}/α

_{11}

_{21}is the middle point of the edge W

_{CD}

_{21}W

_{AB}

_{21}, Figure 29b. The positions of points S

_{A}

_{21}and S

_{D}

_{21}are similar to the positions of points S

_{B}

_{11}and S

_{C}

_{11}obtained previously for Γ

_{11}. The positions of points S

_{B}

_{21}and S

_{C}

_{21}, Figure 29a, result from adopting of the following proportions

_{SB}

_{21}= d

_{SC}

_{21}= d

_{S}

_{11}

_{21}= B

_{11}and D

_{21}= C

_{11}. The positions of the points B

_{21}and C

_{21}result from adopting of the following proportions

_{B2}

_{1}= d

_{SB}

_{21}+ dd

_{B}

_{21}

_{C}

_{21}= d

_{SC}

_{21}+ dd

_{C}

_{21}

_{C}

_{21}= −dd

_{B}

_{21}= dd

_{21}= dd

_{11}

_{AB}

_{31}, W

_{AD}

_{31}, W

_{CD}

_{31}and W

_{BC}

_{31}of Γ

_{31}, four points S

_{A}

_{31}, S

_{B}

_{31}, S

_{C}

_{31}, and S

_{D}

_{31}of ω

_{r}and all vertices A

_{31}, B

_{31}, C

_{31}, and D

_{31}of B

_{v}

_{31}are determined like for Γ

_{21}, Figure 30a,b. For this purpose, the following proportions are defined

_{BC31}= α

_{BC31}/α

_{BC21}

_{SB}

_{31}= d

_{SC}

_{31}= d

_{S}

_{11}

_{B}

_{31}= d

_{SB}

_{31}+ dd

_{B}

_{31}

_{C}

_{31}= d

_{SC}

_{31}+ dd

_{C}

_{31}

_{B}

_{31}= −dd

_{C}

_{31}= dd

_{31}= dd

_{21}

_{31}is the middle point of W

_{CD}

_{31}W

_{AB}

_{31}. The locations of points S

_{A}

_{31}= S

_{B}

_{21}, S

_{D}

_{31}= S

_{C}

_{21}, A

_{31}= B

_{21}, and D

_{31}= C

_{21}. All values of the proportions defined above are shown in Table 5.

_{22}and the quadrangle B

_{v}

_{22}, Figure 31a, the values listed in Table 6 are adopted. To determine the position of W

_{CD}

_{22}on the straight line c

_{22}, Figure 31a,c, a coefficient d

_{WCD}

_{22}defining the division of the edge W

_{BC}

_{12}W

_{CD}

_{12}by the point W

_{CD}

_{22}is defined as follows

_{BC}

_{22}on b

_{22}= c

_{21}, a coefficient d

_{WBC}

_{22}defining the division of the edge W

_{CD}

_{21}W

_{BC}

_{21}by the point W

_{BC}

_{22}is assumed, Figure 31c, so that

_{AB}

_{22}= W

_{CD}

_{21}, W

_{AD}

_{22}= W

_{BC}

_{12}. Similarly, the values of two coefficients d

_{SC}

_{22}and dd

_{C}

_{22}are adopted. The first value defines a division ratio of the edge W

_{CD}

_{22}W

_{BC}

_{22}by S

_{C}

_{22}, Figure 31c,

_{CD}

_{22}W

_{BC}

_{22}by C

_{22}

_{C}

_{22}= d

_{SC}

_{22}+ dd

_{SC}

_{22}

_{22}positioned diagonally towards Γ

_{11}are defined for: (1) Γ

_{23}and Γ

_{32}, located diagonally in relation to Γ

_{12}and Γ

_{22}. (2) Γ

_{33}located diagonally towards Γ

_{22}. Values of these proportions are listed in Table 7. A sum of all Γ

_{ij}(for i = 1–3) achieved so far is a subnet Γ

_{1}constituting about one quarter of Γ. It is contained in the dihedral angle limited by the planes (x,z) and (y,z) containing the positive y-axis and negative x-axis, Figure 32.

_{1}

_{j}and Γ

_{i}

_{1}for (for i, j = 1 to 3) of the subnet Γ

_{1}were arranged in two orthogonal directions along the principal planes (x,z) and (y,z) of [x,y,z]. However, other tetrahedrons Γ

_{ij}(for i, j = 2 to 3) are arranged in diagonal directions towards the Γ

_{11}mesh. To construct these tetrahedrons, a relatively small number of the respective proportions is employed.

_{1}and B

_{v}

_{1}are given in Table A1, Table A2 and Table A3 posted in Appendix A. Table A1 applies to all vertices W

_{ABij}, W

_{CDij}, W

_{ADij}, and W

_{BCij}of Γ

_{1}(for i, j = 1 to 3). Table A2 relates to the vertices S

_{Aij}, S

_{Bij}, S

_{Cij}, and S

_{Dij}of ω

_{r}. Table A3 concerns all vertices A

_{ij}, B

_{ij}, C

_{ij}, and D

_{ij}of B

_{v}

_{1}.

_{2}

_{L}and B

_{v}

_{2L}of Γ and B

_{v}, the z-axis-symmetry of all characteristic points W

_{ABij}, W

_{CDij}, W

_{ADij}, W

_{BCij}, A

_{ij}, B

_{ij}, C

_{ij}, D

_{ij}, S

_{Aij}, S

_{Bij}, S

_{Cij}, and S

_{Dij}of the previously constructed subnets Γ

_{1}and B

_{v}

_{1}is used. As a result of this transformation, the vertices W

_{ABijL}, W

_{CDijL}, W

_{ADijL}and W

_{BCijL}of Γ

_{2}

_{L}, the vertices A

_{ijL}, B

_{ijL}, C

_{ijL}, and D

_{ijL}of B

_{vijL}as well as the points S

_{AijL}, S

_{BijL}, S

_{CijL}, and S

_{DijL}of ω

_{r}are determined so that S

_{AijL}, S

_{BijL}, S

_{CijL}, S

_{DijL}, A

_{ijL}, B

_{ijL}, C

_{ijL}, and D

_{ijL}belong to the dihedral angle located between the (x,z)-plane and (y,z)-plane and including the positive x-half-axis and the negative y-half-axis, Figure 33. Examples of a single reference tetrahedron Γ

_{13L}of the subnet Γ

_{2}

_{L}and a mesh B

_{v}

_{13L}of the subnet B

_{v}

_{2L}are shown in Figure 33.

_{2}

_{L}and B

_{v}

_{2L}, there are many proportions between the lengths of their side edges and axes and the measures of their angles, identical to those obtained for Γ

_{1}and B

_{v}

_{1}. Some selected relations resulting from the z-axis-symmetry of the vertices of Γ

_{2}

_{L}and B

_{v}

_{2L}and the corresponding vertices of Γ

_{1}and B

_{v}

_{1}are listed in Table A4, Table A5 and Table A6 posted in Appendix A. Table A4 applies to the vertices W

_{ABijL}, W

_{CDijL}, W

_{ADijL}, and W

_{BCijL}of Γ

_{2}

_{L}. Table A5 relates to the vertices S

_{AijL}, S

_{BijL}, S

_{CijL}, and S

_{DijL}of ω

_{r}. Table A6 consists of the coordinates of the vertices A

_{ijL}, B

_{ijL}, C

_{ijL}, and D

_{ijL}belonging to B

_{v}

_{2L}.

_{3p}and B

_{v}

_{3p}of Γ and B

_{v}, a (x,z)-plane-symmetry, of the previously constructed nets Γ

_{1}and B

_{v}

_{1}is used. Based on this symmetry, the following are transformed: (1) all vertices W

_{ABij}, W

_{CDij}, W

_{ADij}, and W

_{BCij}of Γ

_{1}, (2) all vertices A

_{ij}, B

_{ij}, C

_{ij}, and D

_{ij}of B

_{v}

_{1}, (3) the points S

_{Aij}, S

_{Bij}, S

_{Cij}, and S

_{Dij}of ω

_{r}. As a result of this transformation, the vertices W

_{ABijp}, W

_{CDijp}, W

_{ADijp}, and W

_{BCijp}of Γ

_{3p}, the vertices A

_{ijp}, B

_{ijp}, C

_{ijp}, and D

_{ijp}of B

_{vijp}and the points S

_{Aijp}, S

_{Bijp}, S

_{Cijp}, and S

_{Dijp}of ω

_{r}are determined, Figure 34. The obtained points S

_{Aijp}, S

_{Bijp}, S

_{Cijp}, S

_{Dijp}, A

_{ijp}, B

_{ijp}, C

_{ijp}, and D

_{ijp}belong to the subspace contained between the planes (x,z) and (y,z), so that both the negative x-half-axis and the negative y-half-axis are included in this subspace. Examples of the reference tetrahedron Γ

_{23p}of Γ

_{3p}and the mesh B

_{v2}

_{3p}of B

_{v}

_{3p}are shown in Figure 34.

_{3p}and B

_{v}

_{3p}, many specific proportions between their side edge and axis lengths and angle measures similar to those obtained for Γ

_{1}and B

_{v}

_{1}can be found. Some relations resulting from the (x,z)-plane-symmetry of the vertices of Γ

_{3p}and B

_{v}

_{3p}and the corresponding vertices of Γ

_{1}and B

_{v}

_{1}created previously are listed in Table A7, Table A8 and Table A9 posted in Appendix A. Table A7 relates to the vertices W

_{ABijp}, W

_{CDijp}, W

_{ADijp}, and W

_{BCijp}of Γ

_{3p}. Table A8 concerns the points S

_{Aijp}, S

_{Bijp}, S

_{Cijp}, and S

_{Dijp}of ω

_{r}. Table A9 applies to the vertices A

_{ijp}, B

_{ijp}, C

_{ijp}, and D

_{ijp}of B

_{v}

_{3p}.

_{4r}of Γ and subnet B

_{v}

_{4r}of B

_{v}, a (y,z)-plane-symmetry of Γ

_{1}and B

_{v}

_{1}is used. The positions of the vertices W

_{ABijr}, W

_{CDijr}, W

_{ADijr}, and W

_{BCijr}of Γ

_{4r}, the vertices A

_{ijr}, B

_{ijr}, C

_{ijr}, and D

_{ijr}of B

_{vijr}and the points S

_{Aij}, S

_{Bij}, S

_{Cij}, and S

_{Dij}of ω

_{r}are determined as a result of the transformations of (1) the vertices W

_{ABij}, W

_{CDij}, W

_{ADij}, and W

_{BCij}of Γ

_{1}, (2) the vertices A

_{ij}, B

_{ij}, C

_{ij}, and D

_{ij}of B

_{v}

_{1}, (3) the points S

_{Aij}, S

_{Bij}, S

_{Cij}, and S

_{Dij}of ω

_{r}, so that S

_{Aijr}, S

_{Bijr}, S

_{Cijr}, S

_{Dijr}, A

_{ijr}, B

_{ijr}, C

_{ijr}, and D

_{ijr}belong to the dihedral angle limited by the (x,z)-half-plane and (y,z)-half-plane containing the positive x-half-axis and the positive y-half-axis, Figure 26.

_{4r}and B

_{v}

_{4r}and the corresponding vertices of Γ

_{1}and B

_{v}

_{1}, resulting from the (y,z)-plane-symmetry of Γ and B

_{v}are given in Table A10, Table A11 and Table A12 included in Appendix A. Table A10 relates to the W

_{ABijr}, W

_{CDijr}, W

_{ADijr}, and W

_{BCijr}vertices of Γ

_{4r}. Table A11 concerns the points S

_{Aijr}, S

_{Bijr}, S

_{Cijr}, and S

_{Dijr}of ω

_{r}. Table A12 applies to the vertices A

_{ijr}, B

_{ijr}, C

_{ijr}, and D

_{ijr}of B

_{v}

_{4r}. Finally, the sought-after nets Γ and B

_{v}are created as the sums of the respective symmetric subnets Γ

_{1}, B

_{v}

_{1}, Γ

_{2}

_{L}, B

_{v}

_{2L}, Γ

_{3p}, B

_{v}

_{3p}, Γ

_{4r}, and B

_{v}

_{4r}have already been constructed. These nets have to be supplemented with roof shell sectors and a plain base to obtain complete building free form model.

## 7. Discussion

_{Aijk}, S

_{Bijk}, S

_{Cijk}, and S

_{Dijk}(for i, j = 1–3, k = ϕ, L, p, r, where ϕ is the empty set) used in the example presented in the previous section designate a regular double-curved surface ω

_{r}characterized by the positive Gaussian curvature, Figure 26 and Figure 35, because the values of the proportions d

_{S}

_{Aijk}, d

_{S}

_{Bijk}, d

_{S}

_{Cijk}, and d

_{S}

_{Dijk}are bigger than 1.0. Double-curved surfaces having the negative Gaussian curvature can be obtained when the coefficients range from 0.0 to 1.0, Figure 36. The investigated method allows one to enter certain points to determine such networks Γ and B

_{v}for which the resultant reference surface is a single-curved surface having the zero Gaussian curvature. For this case, selected groups of the axes of some reference tetrahedrons Γ

_{ij}have to be contained in the same straight lines. This problem is going to be presented in further publications.

_{v}determining an unconventional shell roof structure can be built. All Γ

_{ijk}(for i, j = 1–3, k = ϕ, L, p, r) of Γ are tetrahedrons whose vertices, side edges, planes and axes take specific mutual positions influencing diversified types of the created reference surfaces ω

_{r}and eaves roof networks B

_{v}. Meshes Γ

_{ijk}affect the rationality of the designed building free form structures due to the specific mutual positions of the side edges a

_{ijk}, b

_{ijk}, c

_{ijk}and d

_{ijk}of Γ. B

_{vijk}(for i, j = 1–3, k = ϕ, L, p, r) of B

_{v}are closed spatial quadrangles whose two opposite sides can be taken as roof directrices. The parametric shapes and mutual positions of the directrices may positively affect a process of designing the attractive building structures by automatically changing the proportions d

_{Aij}, d

_{Bij}, d

_{Cij}, and d

_{Dij}defining the positions of the B

_{vijk}‘s vertices on a

_{ijk}, b

_{ijk}, c

_{ijk}and d

_{ijk}in accordance with ω

_{r}.

_{v}introduced in Section 6, Figure 26, and the network shown in Figure 36 are characterized by the fact that each pair of their adjacent quadrilateral meshes B

_{vijk}and B

_{vmns}arranged orthogonally in relation to the mesh B

_{v}

_{11}(for i = 1 or j = 1 and m = 1 or n = 1 and k, s = ϕ, L, p, r) has one common edge, including their directrix, and two common vertices. In addition, each tetrad of adjacent quadrilaterals B

_{vijk}has one common vertex. In contrast, the B

_{v}network shown in Figure 35 was created so that each of the two adjacent quadrangles B

_{vijk}and B

_{vmns}arranged in any of the orthogonal directions compatible with the principal planes (x,z) and (y,z) do not have a common edge, but their corresponding edges are inclined to each other and intersect in one point. In this case, each two adjacent meshes-quadrangles arranged diagonally in B

_{v}have only one common vertex.

_{ijk}limited by eaves quadrangles B

_{vijk}(for i = 1 or j = 1 and m = 1 or n = 1 and k = ϕ, L, p, r). The edges model a set of ribs between the complete transformed shells of a roof structure. In the parametric description of Γ implemented to computer applications, it is possible to easily change the positions of all vertices A

_{ijk}, B

_{ijk}, C

_{ijk}, D

_{ijk}of the meshes B

_{vijk}along the side edges of Γ, relative to reference surface ω

_{r}by modifying the division coefficients of some specific pairs of—the vertices of Γ. The change may cause the resultant structure Ω to become discontinuous, Figure 35. The structure Ω can contain many empty flat areas dividing the roof shell sectors Ω

_{ijk}. These openings should be built by windows illuminating the interior of the designed building with the sunlight. This problem is also going to be analyzed in the further publications.

## 8. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**The coordinates of the vertices W

_{ABij}, W

_{CDij}, W

_{ADij}, W

_{BCij}(for i, j = 1, 2, 3) of the polyhedral reference network Γ

_{1}shown in Figure 27.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

W_{AB}_{11} | 4500.0 | −4793.6 | 54,790.7 |

W_{CD}_{11} | 4500.0 | 4880.7 | 55,786.9 |

W_{AD}_{11} | −4500.0 | 4793.6 | 54,790.7 |

W_{BC}_{11} | −4500.0 | −4880.7 | 55,786.9 |

W_{AB}_{12} | xW_{CD}_{11} | yW_{CD}_{11} | zW_{CD}_{11} |

W_{CD}_{12} | 3254.3 | 0.0 | 468.2 |

W_{AD}_{12} | −100.0 | −958.7 | 10,958.1 |

W_{BC}_{12} | −100.0 | 958.7 | 10,958.1 |

W_{AB}_{13} | xW_{CD}_{12} | yW_{CD}_{12} | zW_{CD}_{12} |

W_{CD}_{13} | 5572.7 | 0.0 | zS_{C}_{12} |

W_{AD}_{13} | −435.4 | −1054.6 | 12,007.1 |

W_{BC}_{13} | −435.4 | 1054.6 | 12,007.1 |

W_{AB}_{21} | −1100.0 | −87.2 | −996.2 |

W_{CD}_{21} | 1100.0 | −87.2 | −996.2 |

W_{AD}_{21} | xW_{BC}_{11} | yW_{BC}_{11} | zW_{BC}_{11} |

W_{BC}_{21} | 0.0 | 2574.0 | 9677.1 |

W_{AB}_{31} | −1210.0 | −353.3 | −2063.5 |

W_{CD}_{31} | 1210.0 | 353.3 | −2063.5 |

W_{AD}_{31} | −xW_{BC}_{21} | yW_{BC}_{21} | zW_{BC}_{21} |

W_{BC}_{31} | 0.0 | 4374.6 | 9074.6 |

W_{AB}_{22} | xW_{CD}_{21} | yW_{CD}_{21} | zW_{CD}_{21} |

W_{CD}_{22} | 3589.7 | 0.0 | −580.8 |

W_{AD}_{22} | xW_{BC}_{12} | yW_{BC}_{12} | zW_{BC}_{12} |

W_{BC}_{22} | −110.0 | 2840.1 | 10,744.4 |

W_{AB}_{23} | xW_{CD}_{22} | yW_{CD}_{22} | zW_{CD}_{22} |

W_{CD}_{23} | 6173.6 | −105.5 | 435.5 |

W_{AD}_{23} | xW_{BC}_{13} | yW_{BC}_{13} | zW_{BC}_{13} |

W_{BC}_{23} | −480.0 | 3133.7 | 11,876.9 |

W_{AB}_{32} | xW_{CD}_{31} | yW_{CD}_{31} | zW_{CD}_{31} |

W_{CD}_{32} | 3959.7 | −389.5 | −1713.3 |

W_{AD}_{32} | xW_{BC}_{22} | yW_{BC}_{22} | zW_{BC}_{22} |

W_{BC}_{32} | −121.0 | 4847.4 | 10,188.4 |

W_{AB}_{33} | xW_{CD}_{32} | yW_{CD}_{32} | zW_{CD}_{32} |

W_{CD}_{33} | 6838.9 | −429.4 | −708.6 |

W_{AD}_{33} | xW_{BC}_{23} | yW_{BC}_{23} | zW_{BC}_{23} |

W_{BC}_{33} | −529.1 | 5371.0 | 11,378.6 |

**Table A2.**The coordinates of the points S

_{Aij}, S

_{Bij}, S

_{Cij}, S

_{Dij}for (i,j = 1, 2, 3) of the reference surface ω

_{r}.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

S_{A}_{11} | 4500.0 | −4793.6 | 54,790.7 |

S_{B}_{11} | 4500.0 | 4880.7 | 55,786.9 |

S_{C}_{11} | −4500.0 | 4793.6 | 54,790.7 |

S_{D}_{11} | −4500.0 | −4880.7 | 55,786.9 |

S_{A}_{12} | xS_{D}_{11} | yS_{D}_{11} | zS_{D}_{11} |

S_{B}_{12} | xS_{C}_{11} | yS_{C}_{11} | zS_{C}_{11} |

S_{C}_{12} | −13,517.1 | 4793.6 | 52,918.0 |

S_{D}_{12} | −13,517.1 | −4793.6 | 52,918.0 |

S_{A}_{13} | xS_{D}_{12} | yS_{D}_{12} | zS_{D}_{12} |

S_{B}_{13} | xS_{C}_{12} | yS_{C}_{12} | zS_{C}_{12} |

S_{C}_{13} | −21,737.1 | 4793.6 | 49,304.2 |

S_{D}_{13} | −21,737.1 | −4793.6 | 49,304.2 |

S_{A}_{21} | xS_{B}_{11} | yS_{B}_{11} | zS_{B}_{11} |

S_{B}_{21} | 4500.0 | 13,460.5 | 53,340.4 |

S_{C}_{21} | −4500.0 | 13,460.5 | 53,340.4 |

S_{D}_{21} | xS_{C}_{11} | yS_{C}_{11} | zS_{C}_{11} |

S_{A}_{31} | xS_{B}_{21} | yS_{B}_{21} | zS_{B}_{21} |

S_{B}_{31} | 4500.0 | 21,957.5 | 50,497.3 |

S_{C}_{31} | −4500.0 | 21,957.5 | 50,497.3 |

S_{D}_{31} | xS_{C}_{21} | yS_{C}_{21} | zS_{C}_{21} |

S_{A}_{22} | xS_{C}_{11} | yS_{C}_{11} | zS_{C}_{11} |

S_{B}_{22} | xS_{C}_{21} | yS_{C}_{21} | zS_{C}_{21} |

S_{C}_{22} | −13,675.6 | 13,605.3 | 52,270.2 |

S_{D}_{22} | xS_{C}_{12} | yS_{C}_{12} | zS_{C}_{12} |

S_{A}_{23} | xS_{C}_{12} | yS_{C}_{12} | zS_{C}_{12} |

S_{B}_{23} | xS_{C}_{22} | yS_{C}_{22} | zS_{C}_{22} |

S_{C}_{23} | −22,104.0 | 13,660.9 | 49,061.5 |

S_{D}_{23} | xS_{C}_{13} | yS_{C}_{13} | zS_{C}_{13} |

S_{A}_{32} | xS_{C}_{21} | yS_{C}_{21} | zS_{C}_{21} |

S_{B}_{32} | xS_{C}_{31} | yS_{C}_{31} | zS_{C}_{31} |

S_{C}_{32} | −13,692.4 | 22,263.8 | 49,770.7 |

S_{D}_{32} | xS_{C}_{22} | yS_{C}_{22} | zS_{C}_{22} |

S_{A}_{33} | xS_{C}_{22} | yS_{C}_{22} | zS_{C}_{22} |

S_{B}_{33} | xS_{C}_{32} | yS_{C}_{32} | zS_{C}_{32} |

S_{C}_{33} | −22,428.4 | 22,611.2 | 47,304.5 |

S_{D}_{33} | xS_{C}_{23} | yS_{C}_{23} | zS_{C}_{23} |

**Table A3.**The coordinates of the vertices A

_{ij}, B

_{ij}, C

_{ij}, D

_{ij}(for i, j = 1, 2, 3) of the eaves edge net B

_{v}

_{1}.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

A_{11} | 4,400.0 | −4,706.4 | 53,794.5 |

B_{11} | 4,600.0 | 4,880.7 | 55,786.9 |

C_{11} | −4,400.0 | 4,706.4 | 53,794.5 |

D_{11} | −4,600.0 | −4,880.7 | 55,786.9 |

A_{12} | −4,390.0 | −4,697.7 | 53,694.9 |

B_{12} | −4,610.0 | 4,889.4 | 55,886.5 |

C_{12} | −13,517.1 | 4,697.9 | 51,869.0 |

D_{12} | −13,852.6 | −4,889.4 | 53,967.0 |

A_{13} | −13,148.2 | −4,688.1 | 51,764.1 |

B_{13} | −13,886.1 | 4,899.0 | 54,071.9 |

C_{13} | −21,136.3 | 4,688.1 | 48,252.2 |

D_{13} | −22,338.0 | −4,899.0 | 50,356.2 |

A_{21} | 4,390.0 | 4,697.7 | 53,694.9 |

B_{21} | 4,610.0 | 13,726.6 | 54,407.7 |

C_{21} | −4,390.0 | 13,194.4 | 52,273.0 |

D_{21} | xB_{12} | yB_{12} | zB_{12} |

A_{31} | 4,379.0 | 13,167.7 | 52,166.3 |

B_{31} | 4,621.0 | 22,430.3 | 51611.1 |

C_{31} | −4,379.0 | 21,484.7 | 49,383.5 |

D_{31} | −4,621.0 | 13,753.2 | 54,514.4 |

A_{22} | xC_{11} | yC_{11} | zC_{11} |

B_{22} | xD_{31} | yD_{31} | zD_{31} |

C_{22} | −13,367.3 | 13,360.7 | 51,326.4 |

D_{22} | xB_{13} | yB_{13} | zB_{13} |

A_{23} | xC_{12} | yC_{12} | zC_{12} |

B_{23} | −13,983.9 | 13,850.0 | 53,213.9 |

C_{23} | −21,549.5 | 13,391.0 | 48,108.1 |

D_{23} | −22,338.0 | 4,899.0 | 50,356.2 |

A_{32} | xC_{21} | yC_{21} | zC_{21} |

B_{32} | −4,621.0 | 22,430.3 | 51,611.1 |

C_{32} | −13,352.3 | 21,827.4 | 48,778.9 |

D_{32} | xB_{23} | yB_{23} | zB_{23} |

A_{33} | xC_{22} | yC_{22} | zC_{22} |

B_{33} | −14,032.4 | 22,700.2 | 50,762.6 |

C_{33} | −21,916.7 | 22,208.4 | 46,465.1 |

D_{33} | −22,658.4 | 13,930.9 | 50,015.0 |

**Table A4.**The coordinates of the vertices W

_{ABijL}, W

_{CDijL}, W

_{ADijL}, W

_{BCijL}(for i, j = 1, 2, 3) of the polyhedral reference network Γ

_{2L}shown in Figure 28.

Vertex | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

W_{AB}_{12L} | −xW_{CD}_{12} | −yW_{CD}_{12} | zW_{CD}_{12} |

W_{CD}_{12L} | xW_{AB}_{11} | yW_{AB}_{11} | zW_{AB}_{11} |

W_{AD}_{12L} | −xW_{BC}_{12} | −yW_{BC}_{12} | zW_{BC}_{12} |

W_{BC}_{12L} | −xW_{AD}_{12} | −yW_{AD}_{12} | zW_{AD}_{12} |

W_{AB}_{13L} | −xW_{CD}_{13} | −yW_{CD}_{13} | zW_{CD}_{13} |

W_{CD}_{13L} | xW_{AB}_{12 L} | yW_{AB}_{12 L} | zW_{AB}_{12 L} |

W_{AD}_{13L} | −xW_{BC}_{13} | −yW_{BC}_{13} | zW_{BC}_{13} |

W_{BC}_{13L} | −xW_{AD}_{13} | −yW_{AD}_{13} | zW_{AD}_{13} |

W_{AB}_{21L} | −xW_{CD}_{21} | −yW_{CD}_{21} | zW_{CD}_{21} |

W_{CD}_{21L} | −xW_{AB}_{21} | −yW_{AB}_{21} | zW_{AB}_{21} |

W_{AD}_{21L} | −xW_{BC}_{21} | −yW_{BC}_{21} | zW_{BC}_{21} |

W_{BC}_{21L} | −xW_{AD}_{21} | −yW_{AD}_{21} | zW_{AD}_{21} |

W_{AB}_{31L} | −xW_{CD}_{31} | −yW_{CD}_{31} | zW_{CD}_{31} |

W_{CD}_{31L} | −xW_{AB}_{31} | −yW_{AB}_{31} | zW_{AB}_{31} |

W_{AD}_{31L} | −xW_{BC}_{31} | −yW_{BC}_{31} | zW_{BC}_{31} |

W_{BC}_{31L} | −xW_{AD}_{31} | −yW_{AD}_{31} | zW_{AD}_{31} |

W_{AB}_{22L} | −xW_{CD}_{22} | −yW_{CD}_{22} | zW_{CD}_{22} |

W_{CD}_{22L} | −xW_{AB}_{22} | −yW_{AB}_{22} | zW_{AB}_{22} |

W_{AD}_{22L} | −xW_{BC}_{22} | −yW_{BC}_{22} | zW_{BC}_{22} |

W_{BC}_{22L} | −xW_{AD}_{22} | −yW_{AD}_{22} | zW_{AD}_{22} |

W_{AB}_{23L} | −xW_{CD}_{23} | −yW_{CD}_{23} | zW_{CD}_{23} |

W_{CD}_{23L} | −xW_{AB}_{23} | −yW_{AB}_{23} | zW_{AB}_{23} |

W_{AD}_{23L} | −xW_{BC}_{23} | −yW_{BC}_{23} | zW_{BC}_{23} |

W_{BC}_{23L} | −xW_{AD}_{23} | −yW_{AD}_{23} | zW_{AD}_{23} |

W_{AB}_{32L} | −xW_{CD}_{32} | −yW_{CD}_{32} | zW_{CD}_{32} |

W_{CD}_{32L} | −xW_{AB}_{32} | −yW_{AB}_{32} | zW_{AB}_{32} |

W_{AD}_{32L} | −xW_{BC}_{32} | −yW_{BC}_{32} | zW_{BC}_{32} |

W_{BC}_{32L} | −xW_{AD}_{32} | −yW_{AD}_{32} | zW_{AD}_{32} |

W_{AB}_{33L} | −xW_{CD}_{33} | −yW_{CD}_{33} | zW_{CD}_{33} |

W_{CD}_{33L} | −xW_{AB}_{3} | −yW_{AB}_{3} | zW_{AB}_{3} |

W_{AD}_{33L} | −xW_{BC}_{33} | −yW_{BC}_{33} | zW_{BC}_{33} |

W_{BC}_{33L} | −xW_{AD}_{33} | −yW_{AD}_{33} | zW_{AD}_{33} |

**Table A5.**The coordinates of the points S

_{AijL}, S

_{BijL}, S

_{CijL}, S

_{DijL}(for i, j = 1, 2, 3) of the reference surface ω

_{r}.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

S_{A}_{12L} | −xS_{C}_{12} | −yS_{C}_{12} | zS_{C}_{12} |

S_{B}_{12L} | −xS_{D}_{12} | −yS_{D}_{12} | zS_{D}_{12} |

S_{C}_{12L} | −xS_{A}_{12} | −yS_{A}_{12} | zS_{A}_{12} |

S_{D}_{12L} | −xS_{B}_{12} | −yS_{B}_{12} | zS_{B}_{12} |

S_{A}_{13L} | −xS_{C}_{12} | −yS_{C}_{13} | zS_{C}_{13} |

S_{B}_{13L} | −xS_{D}_{13} | −yS_{D}_{13} | zS_{D}_{13} |

S_{C}_{13L} | −xS_{A}_{13} | −yS_{A}_{13} | zS_{A}_{13} |

S_{D}_{13L} | −xS_{B}_{13} | −yS_{B}_{13} | zS_{B}_{13} |

S_{A}_{21L} | −xS_{C}_{21} | −yS_{C}_{21} | zS_{C}_{21} |

S_{B}_{21L} | −xS_{D}_{21} | −yS_{D}_{21} | zS_{D}_{21} |

S_{C}_{21L} | −xS_{A}_{21} | −yS_{A}_{21} | zS_{A}_{21} |

S_{D}_{21L} | −xS_{B}_{21} | −yS_{B}_{21} | zS_{B}_{21} |

S_{A}_{31L} | −xS_{C}_{31} | −yS_{C}_{31} | zS_{C}_{31} |

S_{B}_{31L} | −xS_{D}_{31} | −yS_{D}_{31} | zS_{D}_{31} |

S_{C}_{31L} | −xS_{A}_{31} | −yS_{A}_{31} | zS_{A}_{31} |

S_{D}_{31L} | −xS_{B}_{31} | −yS_{B}_{31} | zS_{B}_{31} |

S_{A}_{22L} | −xS_{C}_{22} | −yS_{C}_{22} | zS_{C}_{22} |

S_{B}_{22L} | −xS_{D}_{22} | −yS_{D}_{22} | zS_{D}_{22} |

S_{C}_{22L} | −xS_{A}_{22} | −yS_{A}_{22} | zS_{A}_{22} |

S_{D}_{22L} | −xS_{B}_{22} | −yS_{B}_{22} | zS_{B}_{22} |

S_{A}_{23L} | −xS_{C}_{23} | −yS_{C}_{23} | zS_{C}_{23} |

S_{B}_{23L} | −xS_{D}_{23} | −yS_{D}_{23} | zS_{D}_{23} |

S_{C}_{23L} | −xS_{A}_{23} | −yS_{A}_{23} | zS_{A}_{23} |

S_{D}_{23L} | −xS_{B}_{23} | −yS_{B}_{23} | zS_{B}_{23} |

S_{A}_{32L} | −xS_{C}_{32} | −yS_{C}_{32} | zS_{C}_{32} |

S_{B}_{32L} | −xS_{D}_{32} | −yS_{D}_{32} | zS_{D}_{32} |

S_{C}_{32L} | −xS_{A}_{32} | −yS_{A}_{32} | zS_{A}_{32} |

S_{D}_{32L} | −xS_{B}_{32} | −yS_{B}_{32} | zS_{B}_{32} |

S_{A}_{33L} | −xS_{C}_{33} | −yS_{C}_{33} | zS_{C}_{33} |

S_{B}_{33L} | −xS_{D}_{33} | −yS_{D}_{33} | zS_{D}_{33} |

S_{C}_{33L} | −xS_{A}_{33} | −yS_{A}_{33} | zS_{A}_{33} |

S_{D}_{33L} | −xS_{B}_{33} | −yS_{B}_{33} | zS_{B}_{33} |

**Table A6.**The coordinates of the vertices A

_{ijL}, B

_{ijL}, C

_{ijL}, D

_{ijL}(for i, j = 1, 2, 3) of the eaves edge net B

_{v}

_{2L}.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

A_{12L} | −xC_{12} | −yC_{12} | zC_{12} |

B_{12L} | −xD_{12} | −yD_{12} | zD_{12} |

C_{12L} | −xA_{12} | −yA_{12} | zA_{12} |

D_{12L} | −xB_{12} | −yB_{12} | zB_{12} |

A_{13L} | −xC_{13} | −yC_{13} | zC_{13} |

B_{13L} | −xD_{13} | −yD_{13} | zD_{13} |

C_{13L} | −xA_{13} | −yA_{13} | zA_{13} |

D_{13L} | −xB_{13} | −yB_{13} | zB_{13} |

A_{21L} | −xC_{21} | −yC_{21} | zC_{21} |

B_{21L} | −xD_{21} | −yD_{21} | zD_{21} |

C_{21L} | −xA_{21} | −yA_{21} | zA_{21} |

D_{21L} | −xB_{21} | −yB_{21} | zB_{21} |

A_{31L} | −xC_{31} | −yC_{31} | zC_{31} |

B_{31L} | −xD_{31} | −yD_{31} | zD_{31} |

C_{31L} | −xA_{31} | −yA_{31} | zA_{31} |

D_{31L} | −xB_{31} | −yB_{31} | zB_{31} |

A_{22L} | −xC_{22} | −yC_{22} | zC_{22} |

B_{22L} | −xD_{22} | −yD_{22} | zD_{22} |

C_{22L} | −xA_{22} | −yA_{22} | zA_{22} |

D_{22L} | −xB_{22} | −yB_{22} | zB_{22} |

A_{23L} | −xC_{23} | yC_{23} | zC_{23} |

B_{23L} | −xD_{23} | −yD_{23} | zD_{23} |

C_{23L} | −xA_{23} | −yA_{23} | zA_{23} |

D_{23L} | −xB_{23} | −yB_{23} | zB_{23} |

A_{32L} | −xC_{32} | −yC_{32} | zC_{32} |

B_{32L} | −xD_{32} | −yD_{32} | zD_{32} |

C_{32L} | −xA_{32} | −yA_{32} | zA_{32} |

D_{32L} | −xB_{32} | −yB_{32} | zB_{32} |

A_{33L} | −xC_{33} | −yC_{33} | zC_{33} |

B_{33L} | −xD_{33} | −yD_{33} | zD_{33} |

C_{33L} | −xA_{33} | −yA_{33} | zA_{33} |

D_{33L} | −xB_{33}0 | −yB_{33} | zB_{33} |

**Table A7.**The coordinates of the vertices W

_{ABijp}, W

_{CDijp}, W

_{ADijp}, W

_{BCijp}(for i, j = 1, 2, 3) of the polyhedral reference network Γ

_{3p}shown in Figure 29.

Vertex | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

W_{AB}_{22p} | xW_{AB}_{22} | −yW_{AB}_{22} | zW_{AB}_{22} |

W_{CD}_{22p} | xW_{CD}_{22} | −yW_{CD}_{22} | zW_{CD}_{22} |

W_{AD}_{22p} | xW_{BC}_{22} | −yW_{BC}_{22} | zW_{BC}_{22} |

W_{BC}_{22p} | xW_{AD}_{22} | −yW_{AD}_{22} | zW_{AD}_{22} |

W_{AB}_{23p} | xW_{AB}_{23} | −yW_{AB}_{23} | zW_{AB}_{23} |

W_{CD}_{23p} | xW_{CD}_{23} | −yW_{CD}_{23} | zW_{CD}_{23} |

W_{AD}_{23p} | xW_{BC}_{23} | −yW_{BC}_{23} | zW_{BC}_{23} |

W_{BC}_{23p} | xW_{AD}_{23} | −yW_{AD}_{23} | zW_{AD}_{23} |

W_{AB}_{32p} | xW_{AB}_{32} | −yW_{AB}_{32} | zW_{AB}_{32} |

W_{CD}_{32p} | xW_{CD}_{32} | −yW_{CD}_{32} | zW_{CD}_{32} |

W_{AD}_{32p} | xW_{BC}_{32} | −yW_{BC}_{32} | zW_{BC}_{32} |

W_{BC}_{32p} | xW_{AD}_{32} | −yW_{AD}_{32} | zW_{AD}_{32} |

W_{AB}_{33p} | xW_{AB}_{33} | −yW_{AB}_{33} | zW_{AB}_{33} |

W_{CD}_{33p} | xW_{CD}_{33} | −yW_{CD}_{33} | zW_{CD}_{33} |

W_{AD}_{33p} | xW_{BC}_{33} | −yW_{BC}_{33} | zW_{BC}_{33} |

W_{BC}_{33p} | xW_{AD}_{33} | −yW_{AD}_{33} | zW_{AD}_{33} |

**Table A8.**The coordinates of the points S

_{Aijp}, S

_{Bijp}, S

_{Cijp}, S

_{Dijp}for (i, j = 1, 2, 3) of the reference surface ω

_{r}.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

S_{A}_{22p} | xS_{B}_{22} | −yS_{B}_{22} | zS_{B}_{22} |

S_{B}_{22p} | xS_{A}_{22} | −yS_{A}_{22} | zS_{A}_{22} |

S_{C}_{22p} | xS_{D}_{22} | −yS_{D}_{22} | zS_{D}_{22} |

S_{D}_{22p} | xS_{C}_{22} | −yS_{C}_{22} | zS_{C}_{22} |

S_{A}_{23p} | xS_{B}_{23} | −yS_{B}_{23} | zS_{B}_{23} |

S_{B}_{23p} | xS_{A}_{23} | −yS_{A}_{23} | zS_{A}_{23} |

S_{C}_{23p} | xS_{D}_{23} | −yS_{D}_{23} | zS_{D}_{23} |

S_{D}_{23p} | xS_{C}_{23} | −yS_{C}_{23} | zS_{C}_{23} |

S_{A}_{32p} | xS_{B}_{32} | −yS_{B}_{32} | zS_{B}_{32} |

S_{B}_{32p} | xS_{A}_{32} | −yS_{A}_{32} | zS_{A}_{32} |

S_{C}_{32p} | xS_{D}_{32} | −yS_{D}_{32} | zS_{D}_{32} |

S_{D}_{32p} | xS_{C}_{32} | −yS_{C}_{32} | zS_{C}_{32} |

S_{A}_{33p} | xS_{B}_{33} | −yS_{B}_{33} | zS_{B}_{33} |

S_{B}_{33p} | xS_{A}_{33} | −yS_{A}_{33} | zS_{A}_{33} |

S_{C}_{33p} | xS_{D}_{33} | −yS_{D}_{33} | zS_{D}_{33} |

S_{D}_{33p} | xS_{C}_{33} | −yS_{C}_{33} | zS_{C}_{33} |

**Table A9.**The coordinates of the vertices A

_{ijp}, B

_{ijp}, C

_{ijp}, D

_{ijp}, (for i, j = 1, 2, 3) of the eaves edge net B

_{v}

_{3p}.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

A_{22p} | xA_{32} | −yA_{32} | zA_{32} |

B_{22p} | xB_{12} | −yB_{12} | zB_{12} |

C_{22p} | xC_{12} | −yC_{12} | zC_{12} |

D_{22p} | xD_{32} | −yD_{32} | zD_{32} |

A_{23p} | xA_{33} | −yA_{33} | zA_{33} |

B_{23p} | xB_{13} | −yB_{13} | zB_{13} |

C_{23p} | xC_{13} | −yC_{13} | zC_{13} |

D_{23p} | xD_{33} | −yD_{33} | zD_{33} |

A_{32p} | xC_{31} | −yC_{31} | zC_{31} |

B_{32p} | xB_{22} | −yB_{22} | zB_{22} |

C_{32p} | xC_{22} | −yC_{22} | zC_{22} |

D_{32p} | xB_{33} | −yB_{33} | zB_{33} |

A_{33p} | xC_{32} | −yC_{32} | zC_{32} |

B_{33p} | xB_{23} | −yB_{23} | zB_{23} |

C_{33p} | xC_{23} | −yC_{23} | zC_{23} |

D_{33p} | −22,940.0 | 23,014.0 | 48,143.90 |

**Table A10.**The coordinates of the vertices W

_{ABijr}, W

_{CDijr}, W

_{ADijr}, W

_{BCijr}(for i, j = 1, 2, 3) of the polyhedral reference network Γ

_{4r}shown in Figure 21.

Vertex | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

W_{AB}_{22r} | −xW_{CD}_{22} | yW_{CD}_{22} | zW_{CD}_{22} |

W_{CD}_{22r} | −xW_{AB}_{22} | yW_{AB}_{22} | zW_{AB}_{22} |

W_{AD}_{22r} | −xW_{AD}_{22} | yW_{AD}_{22} | zW_{AD}_{22} |

W_{BC}_{22r} | −xW_{BC}_{22} | yW_{BC}_{22} | zW_{BC}_{22} |

W_{AB}_{23r} | −xW_{CD}_{23} | yW_{CD}_{23} | zW_{CD}_{23} |

W_{CD}_{23r} | −xW_{AB}_{23} | yW_{AB}_{23} | zW_{AB}_{23} |

W_{AD}_{23r} | −xW_{AD}_{23} | yW_{AD}_{23} | zW_{AD}_{23} |

W_{BC}_{23r} | −xW_{BC}_{23} | yW_{BC}_{23} | zW_{BC}_{23} |

W_{AB}_{32r} | −xW_{CD}_{32} | yW_{CD}_{32} | zW_{CD}_{32} |

W_{CD}_{32r} | −xW_{AB}_{32} | yW_{AB}_{32} | zW_{AB}_{32} |

W_{AD}_{32r} | −xW_{AD}_{32} | yW_{AD}_{32} | zW_{AD}_{32} |

W_{BC}_{32r} | −xW_{BC}_{32} | yW_{BC}_{32} | zW_{BC}_{32} |

W_{AB}_{33r} | −xW_{CD}_{33} | yW_{CD}_{33} | zW_{CD}_{33} |

W_{CD}_{33r} | −xW_{AB}_{33} | yW_{AB}_{33} | zW_{AB}_{33} |

W_{AD}_{33r} | −xW_{AD}_{33} | yW_{AD}_{33} | zW_{AD}_{33} |

W_{BC}_{33r} | −xW_{BC}_{33} | yW_{BC}_{33} | zW_{BC}_{33} |

**Table A11.**The coordinates of the points S

_{Aijr}, S

_{Bijr}, S

_{Cijr}, S

_{Dijr}for (i, j = 1, 2, 3) of the reference surface ω

_{r}.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

S_{A}_{22r} | −xS_{D}_{22} | yS_{D}_{22} | zS_{D}_{22} |

S_{B}_{22r} | −xS_{C}_{22} | yS_{C}_{22} | zS_{C}_{22} |

S_{C}_{22r} | −xS_{B}_{22} | yS_{B}_{22} | zS_{B}_{22} |

S_{D}_{22r} | −xS_{A}_{22} | yS_{A}_{22} | zS_{A}_{22} |

S_{A}_{23r} | −xS_{D}_{23} | yS_{D}_{23} | zS_{D}_{23} |

S_{B}_{23r} | −xS_{C}_{23} | yS_{C}_{23} | zS_{C}_{23} |

S_{C}_{23r} | −xS_{B}_{23} | yS_{B}_{23} | zS_{B}_{23} |

S_{D}_{23r} | −xS_{A}_{23} | yS_{A}_{23} | zS_{A}_{23} |

S_{A}_{32r} | −xS_{D}_{32} | yS_{D}_{32} | zS_{D}_{32} |

S_{B}_{32r} | −xS_{C}_{32} | yS_{C}_{32} | zS_{C}_{32} |

S_{C}_{32r} | −xS_{B}_{32} | yS_{B}_{32} | zS_{B}_{32} |

S_{D}_{32r} | −xS_{A}_{32} | yS_{A}_{32} | zS_{A}_{32} |

S_{A}_{33r} | −xS_{D}_{33} | yS_{D}_{33} | zS_{D}_{33} |

S_{B}_{33r} | −xS_{C}_{33} | yS_{C}_{33} | zS_{C}_{33} |

S_{C}_{33r} | −xS_{B}_{33} | yS_{B}_{33} | zS_{B}_{33} |

S_{D}_{33r} | −xS_{A}_{33} | yS_{A}_{33} | zS_{A}_{33} |

**Table A12.**The coordinates of the vertices A

_{ijr}, B

_{ijr}, C

_{ijr}, D

_{ijr}(for i, j = 1, 2, 3) of the eaves edge net B

_{v}

_{4r}.

Point | x-Coordinate [mm] | y-Coordinate [mm] | z-Coordinate [mm] |
---|---|---|---|

A_{22r} | −xA_{23} | yA_{23} | zA_{23} |

B_{22r} | −xB_{23} | yB_{23} | zB_{23} |

C_{22r} | −xC_{21} | yC_{21} | zC_{21} |

D_{22r} | −xD_{21} | yD_{21} | zD_{21} |

A_{23r} | −xC_{13} | yC_{13} | zC_{13} |

B_{23r} | −xD_{33} | yD_{33} | zD_{33} |

C_{23r} | −xC_{22} | yC_{22} | zC_{22} |

D_{23r} | −xD_{22} | yD_{22} | zD_{22} |

A_{32r} | −xA_{33} | yA_{33} | zA_{33} |

B_{32r} | −xB_{33} | yB_{33} | zB_{33} |

C_{32r} | −xC_{31} | yC_{31} | zC_{31} |

D_{32r} | −xD_{31} | yD_{31} | zD_{31} |

A_{33r} | −xC_{23} | yC_{23} | zC_{23} |

B_{33r} | −xD_{33p} | −yD_{33p} | zD_{33p} |

C_{33r} | −xC_{32} | yC_{32} | zC_{32} |

D_{33r} | −xD_{32} | yD_{32} | zD_{32} |

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**Figure 3.**(

**a**) An axis-symmetric experimental thin-walled corrugated steel shell supported by two curved skew directrices; (

**b**) elements of a smooth model used for shaping the transformed shells.

**Figure 4.**An accurate mechanical thin-walled folded model of a nominally plane folded sheet transformed elastically into a shell shape and the graphical expression of the “effective” stresses in MPa on its top surface.

**Figure 5.**Two unconventional symmetric shell-free forms: (

**a**) an erected roof shell by Reichhart [10]; (

**b**) a computer model of a bar structural system intended for the transformed shell roof sheeting.

**Figure 6.**A symmetric shell structure roofing the experimental hall at Rzeszow University of Technology: (

**a**) an internal view; (

**b**) an external view.

**Figure 7.**Symmetrically arranged hyperbolic paraboloid units by Petcu and Gioncu [5]: (

**a**) an erected corrugated umbrella shed; (

**b**) various configurations of umbrella shell structures.

**Figure 9.**The external view of one elevation of the experimental hall at Rzeszow University of Technology roofed with the shell structure.

**Figure 10.**The first Field House structure proposed by Biswas and Iffland [26]: (

**a**) concept and elevation; (

**b**) plan.

**Figure 11.**The second structure proposed by Biswas and Iffland [26]: (

**a**) elevation; (

**b**) framing plan of a quarter of the structure.

**Figure 12.**Two complex building free forms covered with multi-segment roof shell structures and plane-walled folded glass elevations characterized by (

**a**) curved roof directrices; (

**b**) straight roof directrices.

**Figure 13.**Creation of a coherent complex free form arranged over a double-curved reference ellipsoid: (

**a**) a quarter of the final structure; (

**b**) normals to the reference ellipsoid.

**Figure 14.**The architectural study of a free form optimized on the basis of the investigated reference ellipsoid.

**Figure 15.**Two steel structural systems supporting transformed folded sheets by Żwirek [38]: (

**a**) the erection of the final shell roof; (

**b**) experimental tests.

**Figure 16.**Creation of a complex free form by means of a reference network Γ composed of tetrahedral meshes Γ

_{ij}(i = j = 1): (

**a**) Γ

_{ij}before setting (

**b**) Γ

_{ij}after setting in Γ.

**Figure 17.**A complex free form building structure Σ roofed with a continuous shell structure: (

**a**) edge model; (

**b**) architectural stadium.

**Figure 18.**Creation of Γ

_{11}: (

**a**) vertices: W

_{CD}

_{11}, W

_{AB}

_{11}, W

_{AD}

_{11}and W

_{BC}

_{11}; (

**b**) side edges: a

_{11}= (W

_{AD}

_{11}, W

_{AB}

_{11}), b

_{11}= (W

_{BC}

_{11}, W

_{AB}

_{11}), c

_{11}= (W

_{CD}

_{11}, W

_{BC}

_{11}) and d

_{11}= (W

_{AD}

_{11}, W

_{CD}

_{11}).

**Figure 19.**Two subsequent steps of the creation of Σ

_{11}: (

**a**) vertices A

_{11}, B

_{11}, C

_{11}and D

_{11}; (

**b**) eaves B

_{v}

**.**

_{11}**Figure 20.**Two subsequent steps of the determination of (

**a**) the vertices W

_{AB}

_{12}, W

_{BC}

_{12}, W

_{AD}

_{12}, W

_{CD}

_{12}of Γ

_{12}; (

**b**) the vertices A

_{12}, B

_{12}, C

_{12}, D

_{12}of B

_{v}

_{12}.

**Figure 21.**Two subsequent steps of the method’s algorithm related to the construction of; (

**a**) the vertices of Γ

_{21}; (

**b**) the eaves side edges of Σ

_{21}.

**Figure 22.**Two subsequent steps of the method’s algorithm related to the construction of (

**a**) all vertices of Γ

_{22}; (

**b**) all side edges of Σ

_{22}.

**Figure 23.**Subsequent steps of the method’s algorithm concerning: (

**a**) definition of the Γ’s axes; (

**b**) creation of the vertices of Σ

_{21}, Σ

_{22}.

**Figure 24.**Subsequent steps of the method’s algorithm related to the creation: (

**a**) a quarter of Γ; (

**b**) the entire network Γ and the reference surface ω

_{r}symmetrical about the z-axis.

**Figure 26.**A multi-segment reference network Γ composed of four symmetric subnets Γ

_{1}, Γ

_{2L}, Γ

_{3p}, and Γ

_{4r}.

**Figure 27.**Creation of two initial reference tetrahedrons: (

**a**) Γ

_{11}; (

**b**) Γ

_{12}; and (

**c**) a detail of Γ

_{12}.

**Figure 30.**The step related to the creation of the reference tetrahedron Γ

_{31}: (

**a**) a general shape; (

**b**) vertices.

**Figure 31.**Determination of Γ

_{22}: (

**a**) a general shape; (

**b**) vertices of B

_{v}

_{22}; (

**c**) vertices of Γ

_{22}.

**Figure 33.**Two steps of the determination of the second quarter Γ

_{2}

_{L}of the designed z-axis-symmetric reference network Γ.

**Figure 35.**A sum of many spatial quadrangular meshes arranged compatible with a regular surface of the positive Gaussian curvature by means of a polyhedral reference network determined on the basis of the pair of auxiliary orthogonal networks composed of plane meshes.

**Figure 36.**A sum of many spatial quadrangular meshes arranged compatible with a regular surface of the negative Gaussian curvature by means of a polyhedral reference network determined on the basis of the pair of auxiliary orthogonal networks composed of plane meshes.

Variable or Division Coefficient | Value | Unit |
---|---|---|

α_{11} | 10.0 | |

W_{AB}_{11}W_{CD}_{11} | 2000.0 | mm |

OW_{BC}_{11} | 100,00.0 | mm |

d_{OW}_{11} | 1.0 | - |

d_{S}_{11} | 2.5 | - |

dd_{11} | 0.1 | - |

Division Coefficient | Value |
---|---|

dα_{CD}_{12} | 1.2 |

d_{O}_{12WCD12} | 1.0 |

d_{WBC}_{12}, d_{WAD}_{12} | 1.1 |

d_{SC}_{12}, d_{SD}_{12} | 2.5 |

dd_{12} | 0.1 |

Division Coefficient | Value |
---|---|

dα_{CD}_{13} | 1.2 |

d_{O}_{13WCD13} | 1.0 |

d_{WBC}_{13}, d_{WAD}_{13} | 1.1 |

d_{SC}_{13}, d_{SD}_{13} | 2.5 |

dd_{13} | 0.1 |

Division Coefficient | Value |
---|---|

dα_{CD}_{21} | 1.2 |

d_{O}_{21WBC21} | 1.0 |

d_{WCD}_{21}, d_{WAB}_{21} | 1.1 |

d_{SB}_{21}, d_{SC}_{21} | 2.5 |

dd_{21} | 0.1 |

Division Coefficient | Value |
---|---|

dα_{CD}_{31} | 1.0 |

d_{O}_{31WBC31} | 1.0 |

d_{WCD}_{31}, d_{WAB}_{31} | 1.1 |

d_{SB}_{31}, d_{SC}_{31} | 2.5 |

dd_{31} | 0.1 |

Division Coefficient | Value |
---|---|

d_{WCD}_{22} | 1.0 |

d_{WBC}_{22} | 1.0 |

d_{SC}_{22} | 2.5 |

dd_{21} | −0.1 |

**Table 7.**The initial data defining the meshes Γ

_{23}, Γ

_{32}, Γ

_{33}and B

_{v}

_{23}, B

_{v}

_{32}, B

_{v}

_{33}.

Division Coefficient | Value |
---|---|

d_{WCD}_{23} | 1.0 |

d_{WBC}_{23} | 1.0 |

d_{SC}_{23} | 2.5 |

dd_{C}_{23} | 0.1 |

d_{WCD}_{32} | 1.0 |

d_{WBC}_{32} | 1.0 |

d_{SC}_{32} | 2.5 |

dd_{C}_{32} | 0.1 |

d_{WCD}_{33} | 1.0 |

d_{WBC}_{233} | 1.0 |

d_{SC}_{233} | 2.5 |

dd_{C}_{33} | −0.1 |

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**MDPI and ACS Style**

Abramczyk, J.
Symmetric Free Form Building Structures Arranged Regularly on Smooth Surfaces with Polyhedral Nets. *Symmetry* **2020**, *12*, 763.
https://doi.org/10.3390/sym12050763

**AMA Style**

Abramczyk J.
Symmetric Free Form Building Structures Arranged Regularly on Smooth Surfaces with Polyhedral Nets. *Symmetry*. 2020; 12(5):763.
https://doi.org/10.3390/sym12050763

**Chicago/Turabian Style**

Abramczyk, Jacek.
2020. "Symmetric Free Form Building Structures Arranged Regularly on Smooth Surfaces with Polyhedral Nets" *Symmetry* 12, no. 5: 763.
https://doi.org/10.3390/sym12050763