Storyboard

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.

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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.

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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 $d=l/2$.

hcc
Point &&& Coordinates
$P_{0,1}$ &&& $(l,0,0)$
$P_{0,2}$ &&& $(d,h,0)$
$P_{0,3}$ &&& $(-d,h,0)$
$P_{0,4}$ &&& $(-l,0,0)$
$P_{0,5}$ &&& $(-2l,0,0)$
$P_{0,6}$ &&& $(-2l-d,-h,0)$
$P_{0,7}$ &&& $(-2l,-2h,0)$
$P_{0,8}$ &&& $(-l,-2h,0)$
$P_{0,9}$ &&& $(-d,-h,0)$
$P_{0,10}$ &&& $(d,-h,0)$
$P_{0,11}$ &&& $(l,-2h,0)$
$P_{0,12}$ &&& $(2l,-2h,0)$
$P_{0,13}$ &&& $(2l+d,-h,0)$
$P_{0,14}$ &&& $(2l,0,0)$

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p:2:P_0_3:-2.25,3.8971,0.2

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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 $d=l/2$.

hcc
Point &&& Coordinates
$P_{1,1}$ &&& $(l,0,s_w)$
$P_{1,2}$ &&& $(d,h,s_w)$
$P_{1,3}$ &&& $(-d,h,s_w)$
$P_{1,4}$ &&& $(-l,0,s_w)$
$P_{1,5}$ &&& $(-2l,0,s_w)$
$P_{1,6}$ &&& $(-2l-d,-h,s_w)$
$P_{1,7}$ &&& $(-2l,-2h,s_w)$
$P_{1,8}$ &&& $(-l,-2h,s_w)$
$P_{1,9}$ &&& $(-d,-h,s_w)$
$P_{1,10}$ &&& $(d,-h,s_w)$
$P_{1,11}$ &&& $(l,-2h,s_w)$
$P_{1,12}$ &&& $(2l,-2h,s_w)$
$P_{1,13}$ &&& $(2l+d,-h,s_w)$
$P_{1,14}$ &&& $(2l,0,s_w)$

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p:2:P_0_3:-2.25,3.8971,0.2

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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 $d=l/2$.

hcc
Point &&& Coordinates
$P_{2,1}$ &&& $(2l,-2h,s_w+s_l)$
$P_{2,2}$ &&& $(l,0,s_w+s_l)$
$P_{2,3}$ &&& $(-l,0,s_w+s_l)$
$P_{2,4}$ &&& $(-2l,-2h,s_w+s_l)$

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p:2:P_0_3:-2.25,3.8971,0.2

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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.

hclcl
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

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ID:(603, 0)

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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.

hclcl
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

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ID:(602, 0)

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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.

hclcl
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

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ID:(604, 0)

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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 $h$:

hclccl
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

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ID:(605, 0)

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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 $h$:

hclccl
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

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Wall Type:

ID:(606, 0)

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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 $h$:

hclccl
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

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ID:(607, 0)

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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 $h$:

hclccl
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

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ID:(608, 0)