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Palos Verdes model
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To define a particular model, it is important to have at least a concept of the different parts of the house, their locations, and dimensions. If you already have preliminary plans, it's easy to extract the necessary information from them. Otherwise, simply sketch the design, taking care to ensure that the various positions (points) and dimensions (distances between points) do not overlap, thus avoiding situations that do not correspond to real constructions.
ID:(34, 0)
Base design structure
Image
In the case of the Palos Verdes project, the idea is to create a structure with a wide field of view, based on multiple hexagons, each composed of 6 equilateral triangles.
The points of the structure can be defined in this case with three basic parameters:
• $l$: length of the edge.
• $h=\sqrt{3}l/2$: height of the triangle.
• $d=l/2$: half of the base.
The following image shows the parameters and 2D coordinates for one of the hexagons:
In addition to these parameters, the design requires defining the height of the walls ($s_w$) and the loft ($s_l$).
To assist with orientation, cardinal points (North, East, South, West) have been associated with the coordinate system, even though the actual orientation of the constructed house may be different.
ID:(598, 0)
Points at the base of the house
Image
In the following plan, the different corners at ground level (20 cm above the ground) are shown with the edge length $l$, triangle height $h=\sqrt{3}l/2$, and base length $d=l/2$.
| Point | Coordinates |
| $P_{0,1}$ | $(l,0,0)$ |
| $P_{0,2}$ | $(d,h,0)$ |
| $P_{0,3}$ | $(-d,h,0)$ |
| $P_{0,4}$ | $(-l,0,0)$ |
| $P_{0,5}$ | $(-2l,0,0)$ |
| $P_{0,6}$ | $(-2l-d,-h,0)$ |
| $P_{0,7}$ | $(-2l,-2h,0)$ |
| $P_{0,8}$ | $(-l,-2h,0)$ |
| $P_{0,9}$ | $(-d,-h,0)$ |
| $P_{0,10}$ | $(d,-h,0)$ |
| $P_{0,11}$ | $(l,-2h,0)$ |
| $P_{0,12}$ | $(2l,-2h,0)$ |
| $P_{0,13}$ | $(2l+d,-h,0)$ |
| $P_{0,14}$ | $(2l,0,0)$ |
Format
| Code | ID | Name | x | y | z |
| p | 2 | P_0_3 | -2.25 | 3.8971 | 0.2 |
p:2:P_0_3:-2.25,3.8971,0.2
ID:(599, 0)
Points on the roof of the house
Image
In the following plan, the different corners are shown at the roof level ($s_w$ above the floor) with the edge length $l$, the height of the triangle $h=\sqrt{3}l/2$, and the length of the base $d=l/2$.
| Point | Coordinates |
| $P_{1,1}$ | $(l,0,s_w)$ |
| $P_{1,2}$ | $(d,h,s_w)$ |
| $P_{1,3}$ | $(-d,h,s_w)$ |
| $P_{1,4}$ | $(-l,0,s_w)$ |
| $P_{1,5}$ | $(-2l,0,s_w)$ |
| $P_{1,6}$ | $(-2l-d,-h,s_w)$ |
| $P_{1,7}$ | $(-2l,-2h,s_w)$ |
| $P_{1,8}$ | $(-l,-2h,s_w)$ |
| $P_{1,9}$ | $(-d,-h,s_w)$ |
| $P_{1,10}$ | $(d,-h,s_w)$ |
| $P_{1,11}$ | $(l,-2h,s_w)$ |
| $P_{1,12}$ | $(2l,-2h,s_w)$ |
| $P_{1,13}$ | $(2l+d,-h,s_w)$ |
| $P_{1,14}$ | $(2l,0,s_w)$ |
Format
| Code | ID | Name | x | y | z |
| p | 2 | P_0_3 | -2.25 | 3.8971 | 0.2 |
p:2:P_0_3:-2.25,3.8971,0.2
ID:(600, 0)
Points on the house crest
Image
In the following plan, the points for the ridge ($s_w+s_l$ above the floor) are shown with the edge length $l$, the height of the triangle $h=\sqrt{3}l/2$, and the length of the base $d=l/2$.
| Point | Coordinates |
| $P_{2,1}$ | $(2l,-2h,s_w+s_l)$ |
| $P_{2,2}$ | $(l,0,s_w+s_l)$ |
| $P_{2,3}$ | $(-l,0,s_w+s_l)$ |
| $P_{2,4}$ | $(-2l,-2h,s_w+s_l)$ |
Format
| Code | ID | Name | x | y | z |
| p | 2 | P_0_3 | -2.25 | 3.8971 | 0.2 |
p:2:P_0_3:-2.25,3.8971,0.2
ID:(601, 0)
Floors
Image
In the case of the floor, which can be either the ground in contact with a concrete slab or a floor mounted on pilings, it consists of three hexagonal elements:
For each wall, we provide a name and the point taken as the origin, which is the first point in the series of points that make up the surface. The name distinguishes the central module from the eastern (right) and western (left) sides. We refer to the front as the anterior side and the back as the posterior side.
| Code | Name | ID | Points (ID) |
| $S_{0,1}$ | Central Module, Floor | 0 | 0,9,8,3,2,1 |
| $S_{0,2}$ | Eastern Module, Floor | 1 | 12,11,10,9,0,13 |
| $S_{0,3}$ | Western Module, Floor | 2 | 8,7,6,5,4,3 |
Format:
| Code | ID | Name | Type | Points (ID) |
| s | 4 | S_1_2: Central Module, North | e | 1,2,16,15 |
Wall Type:
| Code | Meaning |
| f | Floor |
| e | External |
| r | Roof |
ID:(603, 0)
Walls
Image
There are a total of 14 walls defined by 14 ground-level points and 14 points at roof and ridge level. By organizing the points in lists so that the normal vector to the surface points outward according to the right-hand rule, we can define the walls indicated in the image below:
Base
Loft
For each wall, we include a name and the point taken as the origin. The name distinguishes the center module from the east (right side) and west (left side) sides. We refer to the front as the anterior side and the back as the posterior side.
| Code | Name | ID | Points |
| $S_{1,1}$ | Central Module, Northeast | 3 | 0,1,15,29 |
| $S_{1,2}$ | Central Module, North | 4 | 1,2,16,15 |
| $S_{1,3}$ | Central Module, Northwest | 5 | 2,3,30,16 |
| $S_{1,4}$ | West Module, North | 6 | 3,4,18,30 |
| $S_{1,5}$ | West Module, Northwest | 7 | 4,5,19,18 |
| $S_{1,6}$ | West Module, Southwest | 8 | 5,6,31,19 |
| $S_{1,7}$ | West Module, South | 9 | 6,7,21,31 |
| $S_{1,8}$ | West Module, Pantry | 10 | 7,8,22,21 |
| $S_{1,9}$ | Central Module, Kitchen | 11 | 8,9,23,22 |
| $S_{1,10}$ | East Module, Entrance | 12 | 9,10,24,23 |
| $S_{1,11}$ | East Module, South | 13 | 10,11,28,24 |
| $S_{1,12}$ | East Module, Southeast | 14 | 11,12,26,28 |
| $S_{1,13}$ | East Module, Northeast | 15 | 12,13,27,26 |
| $S_{1,14}$ | East Module, North | 16 | 13,0,29,27 |
Format:
| Code | ID | Name | Type | Points (ID) |
| s | 4 | S_1_2: Central Module, North | e | 1,2,16,15 |
Wall Type:
| Code | Meaning |
| f | Floor |
| e | External |
| r | Roof |
ID:(602, 0)
Roofs
Image
In the case of roofs, they are composed of six parallelepiped-type elements:
Loft
Ceiling
For each roof, we provide a name and the point chosen as the origin, which is the first point in the series of points that make up the surface. The name distinguishes the central module from the east (right) and west (left) sides. We refer to the front as the anterior side and the back as the posterior side.
| Code | Name | ID | Points |
| $S_{2,1}$ | Central Module, Roof north | 17 | 29,15,16,30 |
| $S_{2,2}$ | West Module, Roof north | 18 | 30,18,19,31 |
| $S_{2,3}$ | West Module, Roof south | 19 | 31,21,22,30 |
| $S_{2,4}$ | Central Module, Roof south | 20 | 22,23,29,30 |
| $S_{2,5}$ | East Module, Roof south | 21 | 23,24,28,29 |
| $S_{2,6}$ | East Module, Roof north | 22 | 28,26,27,29 |
Format:
| Code | ID | Name | Type | Points (ID) |
| s | 4 | S_1_2: Central Module, North | e | 1,2,16,15 |
Roof Type:
| Code | Meaning |
| f | Floor |
| e | External |
| r | Roof |
ID:(604, 0)
Windows and Doors, north side
Image
The front view displays the walls $S_{1,13}, S_{1,14}, S_{1,1}, S_{1,2}, S_{1,3}, S_{1,4}, S_{1,5}$ with origins defined by points $P_{1,13}, P_{1,14}, P_{1,1}, P_{1,2}, P_{1,3}, P_{1,4}, P_{1,5}$:
From the plans, the following windows (w) and doors (d) are obtained, indicating their coordinates $(x,y)$ relative to the origin and their width $w$ and height $h$:
| Code | Name | Surface (ID) | ID | x,y,w,h |
| $W_{2,1}$ | North 1 | 1 | 0 | 0.06,0.4,2.10,1.6 |
| $W_{2,2}$ | North 2 | 1 | 2 | 2.28,0.4,2.10,1.6 |
| $W_{1,1}$ | East 1 | 0 | 4 | 0.06,0.4,2.10,1.6 |
| $W_{1,2}$ | East 2 | 0 | 6 | 2.28,0.4,2.10,1.6 |
| $W_{3,1}$ | West 1 | 2 | 8 | 0.06,0.4,2.10,1.6 |
| $W_{3,2}$ | West 2 | 2 | 10 | 2.28,0.4,2.10,1.6 |
| $W_{13}$ | Bedroom 1b | 12 | 14 | 2.3,0.7,2.0,1.3 |
| $W_{14}$ | Bedroom 1a | 13 | 15 | 0.2,0.7,2.0,1.3 |
| $W_{4}$ | Study 1a | 3 | 16 | 2.3,0.7,2.0,1.3 |
| $W_{5}$ | Study 1b | 4 | 17 | 0.2,0.7,2.0,1.3 |
| $D_{14}$ | Bedroom | 13 | 2 | 0.2,0.0,1.2,2.2 |
| $D_{4}$ | Study | 3 | 3 | 3.1,0.0,1.2,2.2 |
Format:
| Code | ID | Name | Surface | Type | x,y,w,h |
| w | 0 | North 1 | 0 | e | 0.06,0.4,1.05,1.6 |
Wall Type:
| Code | Meaning |
| f | Floor |
| e | External |
| r | Roof |
ID:(605, 0)
Windows and Doors, south side
Image
The front view displays the walls $S_{1,6}, S_{1,7}, S_{1,8}, S_{1,9}, S_{1,10}, S_{1,11}, S_{1,12}$ with origins defined by points $P_{1,6}, P_{1,7}, P_{1,8}, P_{1,9}, P_{1,10}, P_{1,11}, P_{1,12}$:
From the plans, the following windows (w) and doors (d) are obtained, indicating their coordinates $(x,y)$ relative to the origin and their width $w$ and height $h$:
| Code | Name | Surface (ID) | ID | x,y,w,h |
| $W_{9}$ | Cocina | 8 | 12 | 11.2,1.3,2.0,0.7 |
| $W_{10}$ | Baño 1 | 9 | 13 | 0.3,0.8,0.6,1.2 |
| $W_{6}$ | Baño 2 | 5 | 18 | 1.95,1.4,0.6,0.6 |
| $W_{7,1}$ | Lavandería | 6 | 19 | 1.0,1.2,0.6,0.8 |
| $W_{7,2}$ | Taller | 6 | 20 | 2.9,1.2,0.6,0.8 |
| $D_{13}$ | Entrada | 12 | 0 | 0.2,0.2,1.0,2.0 |
| $D_{10}$ | Servicio | 9 | 1 | 1.8,0.2,0.9,2.0 |
Format:
| Code | ID | Name | Surface | Type | x,y,w,h |
| w | 0 | North 1 | 0 | e | 0.06,0.4,1.05,1.6 |
Wall Type:
| Code | Meaning |
| f | Floor |
| e | External |
| r | Roof |
ID:(606, 0)
View from the east
Image
The house viewed from the east shows the walls $S_{1,12}$, $S_{1,13}$, and $S_{1,1}$, with the wall $S_{1,14}$ not being visible.
From the plans, the following windows (w) and doors (d) are obtained, indicating their coordinates $(x, y)$ relative to the origin and their width $w$ and height $h$:
| Code | Name | Surface (ID) | ID | x, y, w, h |
| $W_{1,1}$ | Right 1 | 0 | 4 | 0.06, 0.4, 1.05, 1.6 |
| $W_{1,2}$ | Right 2 | 0 | 5 | 1.17, 0.4, 1.05, 1.6 |
| $W_{1,3}$ | Right 3 | 0 | 6 | 2.28, 0.4, 1.05, 1.6 |
| $W_{1,4}$ | Right 4 | 0 | 7 | 3.39, 0.4, 1.05, 1.6 |
| $W_{13}$ | Bedroom 1b | 12 | 14 | 2.3,0.7,2.0,1.3 |
| $D_{12}$ | Bedroom 2 | 11 | 2 | 0.2, 0.0, 1.2, 2.2 |
Format:
| Code | ID | Name | Surface | Type | x, y, w, h |
| w | 0 | Front 1 | 0 | e | 0.06, 0.4, 1.05, 1.6 |
Wall Type:
| Code | Meaning |
| f | Floor |
| e | External |
| r | Roof |
ID:(607, 0)
View from the west
Image
The house seen from the west shows the walls $S_{1,3}$, $S_{1,5}$, and $S_{1,6}$, with the wall $S_{1,4}$ not visible.
From the plans, the following windows (w) and doors (d) are obtained, indicating their coordinates $(x, y)$ relative to the origin and their width $w$ and height $h$:
| Code | Name | Surface (ID) | ID | x, y, w, h |
| $W_{3,1}$ | Left 1 | 3 | 8 | 0.06, 0.4, 1.05, 1.6 |
| $W_{3,2}$ | Left 2 | 3 | 9 | 1.17, 0.4, 1.05, 1.6 |
| $W_{3,3}$ | Left 3 | 3 | 10 | 2.28, 0.4, 1.05, 1.6 |
| $W_{3,4}$ | Left 4 | 3 | 11 | 3.39, 0.4, 1.05, 1.6 |
| $W_{5}$ | Study 1b | 5 | 17 | 0.2, 0.7, 2.0, 1.3 |
| $W_{6}$ | Bathroom 2 | 6 | 18 | 1.95, 1.4, 0.6, 0.6 |
Format:
| Code | ID | Name | Surface | Type | x, y, w, h |
| w | 0 | Front 1 | 0 | e | 0.06, 0.4, 1.05, 1.6 |
Wall Type:
| Code | Meaning |
| f | Floor |
| e | External |
| r | Roof |
ID:(608, 0)