Wyznaczanie optymalnej odległości między budynkami i dopuszczalnej wysokości projektowanego budynku na podstawie wysokości słońca

Małgorzata Bartnicka

doi:10.37190/arc220309

Streszczenie

W artykule poruszono tematykę dostępności światła słonecznego do wnętrz mieszkalnych. Jednym z niezwykle istotnych czynników odpowiedzialnych za przenikanie światła słonecznego do wnętrz jest odpowiednia odległość pomiędzy budynkami. Optymalną odległość można wyliczyć algebraicznie, a do ustalenia danych wyjściowych wystarczy informacja, jakie jest usytuowanie budynku przesłaniającego względem kierunku północy. Celem artykułu jest zaproponowanie alternatywnego sposobu sprawdzania dostępu promieni słonecznych bez konieczności tworzenia modelu 3D oraz symulacji komputerowej. Za pomocą przedstawionej metody można określić optymalną odległość pomiędzy budynkami, przy której zachowaniu nie nastąpi zacienianie budynku istniejącego. Proponowane rozwiązanie wywodzi się z graficznego podejścia do tej tematyki prezentowanego w książce Mieczysława Twarowskiego Słońce w architekturze, czyli do tzw. linijki słońca. Autorka proponuje przemianę graficznego wykreślania cienia na algebraiczne przeliczanie odległości między budynkami. Do przeliczeń wykorzystane zostały znane wzory na wysokość słońca (α) i na azymut słońca (γ). W wyniku analiz układów przestrzennych budynków i możliwości zacieniania wywiedzione zostały autorskie wzory służące do przeliczania odległości między budynkami. Uzyskany wynik odzwierciedla proporcję między odległością a wysokością budynku (W = xH) w konkretnej godzinie dla wybranej szerokości geograficznej. W publikacji ukazano, że istnieje możliwość ustalenia optymalnych odległości pomiędzy budynkami w sposób algebraiczny, bez konieczności wykorzystywania programów CAD. Przedstawione wzory mogą stać się instrumentem sprawdzania poprawności wykresów braku zacieniania przedstawianych przez inwestorów w projektach zabudowy wielorodzinnej. Jednocześnie wzory te można wykorzystywać do wyliczania orientacyjnej wysokości budynku projektowanego w zależności od przewidywanej odległości pomiędzy nim a budynkami istniejącymi.

Podgląd pełnego artykułu jest możliwy wyłącznie na większych ekranach.

2022

3(71)

DOI: 10.37190/arc220309

Małgorzata Bartnicka*

Determining the optimal distance between buildings

based on the solar altitude angle to facilitate appropriate illumination

in residential interiors in accordance with current legislation

Introduction

One can get the impression that in recent years, in

which energy saving and sustainable development have

become one of the main topics, sunlight has been seen

as the predominant alternative source of energy. It seems

that, somehow, many other benets and positive eects of

light have been neglected, all the benets that the hygien-

ists, scientists and architects called for and emphasized

at the turn of the 19

and 20

centuries. The concept of

light, air and sun, which became one of the most import-

ant slogans of modernism in the 1920s and 1930s, was

supported by medical research. At the beginning of the

century, it was discovered that sunlight is a bene-

cial factor in preventing rickets. One of the rst to raise

this issue was Polish doctor Jędrzej Śniadecki. In 1822

in his work on bringing up children, he wrote that rick-

ets […] is not as common in the country as in the city

and that children should be allowed to stay […] in the

sun, which impacts the body in the most eective way to

prevent the disease and facilitate healing [1, p. 184–186].

In subsequent years, it was discovered that sunlight can

destroy bacteria [2]. In 1890, at the 10

International

Congress of Medicine in Berlin, Robert Koch presented

the results of studies that proved that tuberculosis bacteria

die within a few minutes of being exposed to sunlight. In

case of diused light this process is extended to several

hours, however, light still has bactericidal properties [3].

In 1903, Danish doctor Niels Finsen received the Nobel

Prize in medicine for the introduction and use of sunlight

to treat some skin diseases and tuberculosis. In 1876, in

his book on urban development, which in 1880 became

a textbook of urban planning in Germany, urban engineer

Reinhard Baumaister emphasized that […] sunlight and

clean air were necessary for human well-being [4, p. 16].

At the same time, he pointed out the reasons for the poor

health of the society of that time as: […] bad quality of air

in small, overcrowded rooms, the narrowness of buildings

around the courtyards and very small windows [4, p. 16].

Perhaps, paradoxically, advances in medicine (antibiot-

ics and the tuberculosis vaccine) initiated the gradual de-

valuation of the antiseptic power of sunlight in the 1940s

[5]. However, for several dozen years, care was taken to

provide natural sunlight to residential interiors. In many

countries the rules for constructing new buildings are

based on regulations aimed at securing the access to light.

The amount of sunlight penetrating the residential in-

teriors depends on several factors. These factors can be

grouped into two basic categories: variables dependent

on us, those which we can inuence, and the independent

variables (Table 1).

* ORCID: 0000-0002-0304-9366. Faculty of Architecture, Bialystok

University of Technology, Poland, e-mail: m.bartnicka@pb.edu.pl

Published in open access. CC BY NC ND license

Factors that determine the penetration

of daylight into residential interiors

independent dependent

Latitude (φ)

sun declination (δ)

height of sun = solar altitude angle (α)

orientation of the building

size of the window opening

distance between buildings

Table 1. The list of factors that determine the amount of

daylight in residential interiors categorized into

dependent and independent variables (elaborated by M. Bartnicka)

Tabela 1. Zestawienie czynników, od których zależy ilość światła

dziennego we wnętrzach mieszkalnych w rozbiciu na czynniki

zależne i niezależne (oprac. M. Bartnicka)

88 Małgorzata Bartnicka

use of solar energy [6]. In the paper quoted, Paul Littlefair

presents guidelines for designing outdoor environments

that facilitate good access to daylight, sunlight, and solar

energy [6]. The author proposes a method of determining

the area of the sky in which no physical obstacle should be

placed. This area is dened horizontally by an angle of 90°

delimited at the level of the center of the window approx-

imately facing south (45° eastwards and westwards from

the south), and vertically through the limit of the maximum

height of buildings, which depends on the latitude. This

method helps determine the optimal height of buildings in

the designated zones. It does not dene the distance be-

tween buildings, nor their position in relation to each other.

Another way to determine the relationship between

buildings is to determine the ratio between the height of the

obscuring building (H) and the distance between the build-

ings in question (W). Attempts to determine the optimal

H/W ratio between buildings were made by numerous re-

searchers who used various methods, such as for example:

– “The daylight access rule” where the basis for deter-

mining the proportion was to maintain accurate Daylight

Factor (DF) in the interior. In this way, the ratio of height

and distance between atrial buildings was proposed [7],

–

Unobstructed Vision Area Method in which the authors

estimated the distance between the tall buildings by provid-

ing the appropriate Vertical Daylight Factor (VDF) [8].

An important and often discussed topic is the H/W re-

lation considered in terms of the so-called street canyons.

Geometry of “urban canyons” is a key factor in energy

consumption in buildings. Positioning and distances be-

tween buildings determine both the amount of light and

energy that comes into the interiors of the buildings and, at

the same time, aect the ability to reect light and increase

the amount of daylight between buildings [9]. As Andrea

Vallati and his team pointed out, the distance between

buildings also means decisions regarding the possibility

of ventilating the area resulting from wind movement, in-

cluding that caused by convection [10]. According to these

authors, the scale of the phenomenon of convection is di-

rectly proportional to the distance between buildings.

If the H/W ratio is high between buildings, then such

urban planning contributes to the formation of Urban Heat

Island (UHI) in the cities. Temperature increases with the

increase of the value, the highest levels are observed in

the range of H/W > 1.5 (W < 0.66H) [11]. The above men-

tioned studies and calculations were performed in temper-

ate areas (Europe and USA). In countries with hot and

dry climates, a strategy of using compact buildings and

deep urban canyons is more recommended (H/W = 2.2;

W = 0.45H), as to alleviate the impact of UHI and achieve

thermal comfort inside and outside [12].

In the discussion about the distance between buildings

depending on solar exposure and sky view factor Gerard

Mills [13] presents an interesting viewpoint, but his con-

clusions seem controversial. The article contains various

charts that show the amount of light on the facades every

day of the year and the sky view factor depending on the

dimensions of buildings and latitude. According to these

tests, at 0° N latitude, the optimal solution is to use high

freestanding buildings spaced relatively to each other so

One of the very important dependent factors listed

above needed to facilitate good lighting is the correct dis-

tance between buildings. According to the author, one can

assess and establish an optimal distance between buildings

facilitating proper light conditions in the interiors without

having to create a 3D model and a computer simulation.

The only physical data to be established in relation to the

building is its position with reference to the north. This pa-

per describes the way to calculate the appropriate distance

between buildings using two variables: the orientation of

the building relatively to the world directions (E/W/N/S)

and the time at which such an optimal distance is checked.

Although this topic may seem less signicant in the

era of various computer programs, it must be noted that

not all software can track and assess lighting and lighting

conditions accurately. The most used CAD programs are

Autodesk Revit, 3dMAX, SketchUp (Plugin Sun Hours),

SolidWorks, Elite CAD and 2016 and AutoCAD. These

programs require geographical location, date, and time.

Moreover, on top of the high prices of software, there is

no way for the user to verify the results obtained. There

are many reviews in which it has been pointed out that the

software may display discrepancies in the position of the

Sun they apply and the data available in Solar Calculators.

These dierences can reach up to several degrees result-

ing in misleading calculations.

It should be noted that in the case of software light-

ing analysis, the analysis can be carried out only after 3D

spaces are modeled. On the contrary, with the proposed

calculations, the following information can be determined

before the design phase:

– optimal distance between buildings depending on the

planned height of the building and the orientation of the

building overshadowing the building in question,

– approximate height of the designed building/s de-

pending on the expected distance between them.

State of research

The article deals with revision and revitalization of ap-

proaches to daylight that have been changing for several

years now. Year 2015 was declared the International Year

of Light by the UN General Assembly. It was named so

to raise public and international political awareness of

the importance of light for the future of global society. In

the same year the 6

Daylight Symposium “Daylight as

a driver of change” was held in London. The renewed in-

terest in daylight in recent years reects the acknowledge-

ment of its potential not only in terms of energy saving,

but also the extent of impact daylight has on the users of

spaces. The benets of daylight are not only emphasized

and seen in health, but also in the general well-being of

individuals and public.

Studies of urban spaces in terms of buildings obscur-

ing daylight grow in importance in relation to the access

of light to the residential interiors. More and more high-

rise architecture and the increase in the density of build-

ings in urban spaces have a restrictive eect on the amount

of sunlight entering residential interiors. Simultaneously,

such dense urban planning reduces the potential for passive

Determining the optimal distance between buildings based on the solar altitude angle 89

that H/W > 1.0 (W = H); at 30° N latitude rectangular struc-

tures (10 × 5) at spacing 0.5 < H/W < 1.0 (2H > W > H) are

optimal according to Mills; and at 60° latitude, rectangu-

lar structures (10 × 5 or 5 × 10) with spacing of H/W = 2.0

should perform best. This proportion means that buildings

should stand relative to each other at a distance W = 0.5H,

which although it would mean buildings with large widths

would likely minimize heat loss and maximize heat gain

in spring and summer time, they would be unlikely to

achieve such results in the winter months.

Another view related to determining the optimal H/W

ratio is presented in the work on optimization of the ur-

ban building eciency potential using a genetic algorithm

approach [14]. The authors present certain methodology

for optimizing the building energy eciency potential

in urban forms based on the genetic algorithm approach.

The authors investigated optimal shapes and positioning

of buildings at various latitudes to facilitate maximum ab-

sorption of solar radiation in winter and as low as possible

results in the summer. According to the conclusions pre-

sented, architects of the buildings at all latitudes should

strive to reduce the length of the building base. The north-

bound orientated buildings (located on the width of 50° N)

should be in the range from –45 to +45°. The recommend-

ed H/W aspect ratio should be between 0.25 and 0.7, i.e.

4H > W > 1.4H.

All the above-mentioned types of research and studies

were carried out using various types of programs to sim-

ulate the levels of insolation, such as software used for

internal and external thermal analyses e.g., Ecotect and

Dialux Evo. Authors who employ optimization methods

in their research often use Rhinoceros, alongside its plug

ins – Grasshopper (used for parametric design), as well

as a well-known, thanks to You Tube, Ladybug, which is

a solar lighting simulator utilizing a Climate-Based Day-

light Modeling (CBDM) method to predict several mea-

sures of daylight based on sky conditions acquired from

location specic meteorological data. Additionally, evolu-

tionary algorithms are also used, e.g. Galapagos (built into

Grasshopper) [15], [16].

The Polish contribution to the eld of this kind of soft-

ware allowing the analysis of solar lighting of spaces is

called PRC Analysis and was developed by Jacek Marku-

siewicz. It enables the analysis of windows and entire apart-

ments for compliance with local solar regulations [17].

Thanks to the software and additional preliminary analy-

ses, a method was developed to check the interference of

the planned building with the lighting cast onto the win-

dows of the existing building/s [18].

Although all the quoted studies have a common de-

nominator – estimating the distance between buildings

taking into account the eects of sunlight – the research

paths vary. The author of this paper would like to propose

a dierent viewpoint on this issue.

The solution proposed in this paper comes from a gra-

phic approach to analyzing the amount of sunlight in ur-

ban spaces and residential interiors. In Poland, one of the

best-known publications on this subject is the book by

Mieczysław Twarowski Słońce w architekturze [Sun in

Architecture] from 1960 [19]. The author, referring to the

research of the team including professors from the War-

saw University of Technology and dating back to the oc-

cupation during World War II, presented graphic ways of

determining the access of sunlight to urban and residential

spaces. The charts were made using the sun path diagram

MT (also known as daylight analysis diagram or sundial)

estimated for the selected latitude.

Twarowski’s book included diagrams for the latitudes

of 50°, 52.2° (Warsaw) and 54°. Based on these charts,

one can plot the shadows of the obscuring object project-

ed onto other blocks on a specic day (usually during

the equinox), at specic times. The author shows how to

check the extent of insolation of the interior and proposes

a method for plotting contour lines with reference to inso-

lation, with which one can specify the permissible height

of neighboring buildings that will not cause shading to the

building/s in question. This proposed method has been

used in Poland to determine the shadow cast from the de-

signed building or to demonstrate the lack of shading in

accordance with the regulations.

The diculties in the use of diagrams printed on tracing

paper were eventually overcome by introducing a plug-in

to AutoCAD (version 15 onward). At the same time, as

mentioned previously, it is necessary to use a CAD pro-

gram, and in this case its full version, because the plug-in

does not work with the AutoCAD LT version. The decals

attached to Twarowski’s book are limited and cannot be

used with the above-mentioned latitudes. One can, of

course, calculate the data to plot a local sun ruler, based on

formulas (2) and (3) or (4) – vide infra. The author of this

paper proposes a transformation of the graphic drawing

of the shadow into algebraic calculation of the distance

between buildings and the permissible height of the ob-

scuring object (building).

Method

To be able to calculate the optimal distance between

buildings on a specic day of the year, you need to calcu-

late and establish several values that become xed factors.

The basic constant value is the latitude (local latitude) (φ)

of the place where we make calculations for. The day of

the year (n) in which we check the optimal distance is an-

other constant value. Connected to the day of the year (n)

is the declination of the sun, i.e. the angular position of the

sun at solar noon with respect to the plane of the equator

(north positive); −23.45° ≤ δ ≤ 23.45°. Declination can

be calculated using the formula (1) [20]. In the case of

equinox, which is often referenced in many building reg-

ulations, the sun declination is zero, δ = 0.

δ = 23.45° ∙ sin (1)

where:

n – the following day of the year.

First, two following formulas should be used: for so-

lar altitude angle (α) (2) and solar azimuth angle (γ) (3)

[21] or (4) [22] related to the hourly angle (ω). The solar

altitude angle is the angle between the horizontal and the

δ = 23.45° ∙ sin (360 ∙

n+284

365

n + 284

)

(

365

)

360 ∙

90 Małgorzata Bartnicka

line to the sun. It is the complement of the zenith angle.

The solar azimuth angle is the angular displacement from

south of the projection of beam radiation on the horizontal

plane. Displacements east of south are negative and west

of south are positive.

sin α = cos δ ∙ cos φ ∙ cos ω + sin δ ∙ sin φ

(2)

cos γ

cos δ ∙ cos ω – sin α ∙ cos φ

cos α ∙ sin φ

(3)

sin

(360

– γ)

cos δ ∙ sin ω

cos α

(4)

where:

α – solar altitude angle = solar elevation angle,

δ – declination of the sun,

φ – local latitude,

ω – hour angle in the local solar time,

γ – solar azimuth angle.

The above formulas show that both values (α) and (γ)

depend only on the time in which the test takes place if

latitude (φ) and declination (δ) are constant values in the

chosen site. Hour angle (ω) is an expression describing the

dierence between local solar time and solar noon. Indi-

vidual full hours become a multiple of 15, negative (–) be-

fore solar noon and positive (+) after solar noon. The hour

angle can also be calculated using the formula (5).

ω = 15°(τ – 12.00) (5)

where:

τ – the time being checked.

The main issue investigated in this paper is the possi-

bility of calculating the distance between buildings. These

calculations are made for a specic day and time. The

choice of day depends on the regulations in force in the

country. In the overwhelming number of cases, it is the

day of the equinox; this applies in Poland, Russia, Slove-

nia, Sweden, Great Britain and Germany. Other dates are

also considered, often additional, e.g. December 21 (USA,

South Korea), January 17 (Germany), February 19–

Oc tober 21 (Netherlands), March 1 (Czech Republic),

March 1–October 13 (Slovakia) and April 22 (Estonia).

Regardless of the day on which we consider the opti-

mal distance between buildings, the following reasoning

presented below could be adopted as seen in Figure 1.

Figure 1 shows two buildings. The rst is the building

being obscured and the second, located below, is the “ob-

scuring” building. They are divided by the W

distance

dened as the optimal distance between the longitudi-

nal sides of the buildings (the W

distance is the optimal

distance between the leading edges of the building). The

height of the obscuring building (H) used for the calcu-

lation casts a shadow on a plane of length (c), such as

not to obscure the other building. Between the directions

, c and the longitudinal edge of the obscuring building,

a triangle is formed, in which at the corner H we have an-

gle x. This angle follows the following relationship:

sin x =

⇒

= c

∙ sin x

cos x =

⇒

= c

∙ cos x

Due to the fact that the value of angle x depends on

the azimuth of the sun, it can be read from Figure 1 that

x is the sum of two angles: the deviation of the obscuring

building from the north (β) and the azimuth of the sun (γ):

x = γ + β

Knowing that the relationship between the length of

the shadow of building c depends on its height (H) and the

height of the sun (α), it can be represented as:

c = H ∙ cot α

Assuming all of the above inter-dependencies, we can

see that:

Fig. 1. A diagram of relationships between building orientation,

solar azimuth angle, and distance between buildings. Where:

β – building’s angle in relation to north, γ – solar azimuth angle,

H – building height, c – shadow length from height H,

– optimal distance between longer sides of buildings,

– optimal distance between shorter sides of buildings

(elaborated by M. Bartnicka)

Il. 1. Wykres zależności pomiędzy orientacją budynku,

wysokością słońca oraz odległościami między budynkami:

β – kąt odchylenia budynku od kierunku północy, γ – azymut słońca,

H – wysokość budynku, c – długość cienia od wysokości H,

– optymalna odległość między wzdłużnymi bokami budynków,

– optymalna odległość między krótkimi bokami budynków

(oprac. M. Bartnicka)

Determining the optimal distance between buildings based on the solar altitude angle 91

The obtained result will reect the proportion between

the distance and the height of the building (W = xH) at

a specic time for the selected latitude.

Practical application of the proposed formulas

If a new building is to be built near an existing one, one

can specify at what distance it should be built. To limit the

number of calculations, a set of specic data for a given

latitude should be prepared. Table 2, for example, con-

tains data for latitude 53°07′59″.

With this proposed table, the calculation of the inter-de-

pendencies between buildings becomes much simpler. If

you want to check at what distance buildings located at

an angle of 15° (β = 15°) relative to the north should be

placed, assuming that we check this at various hours during

the equinox period, use the formula (9), starting from 7.00.

and, respectively:

= 3.13, R

= 1.9, R

= 1.27, R

= 0.77, R

= 0.34

These calculations show that the optimal distance on the

day of the equinox at certain hours is:

= 6.35H; W

= 3.13H;

= 1.9H; W

= 1.27H; W

= 0.77H; W

= 0.34H.

Discussion and conclusions

Virtually all current research related to lighting seems

to be based on CAD software, which, to varying degrees

of accuracy, allows one to create realistic lighting simu-

lations. The results are usually depicted in a visual way,

showing the extent of shading or the intensity of sunlight.

= c ∙ sin

(

γ + β

)

= H ∙ cot α ∙ sin

(

γ + β

)

(6)

= H ∙ cot α ∙ cos

(

γ+β

)

(7)

Finally, assuming that we are looking for a relationship

R (cf. [23]) between the height of the building H and the

optimal distance W

(or W

), at a specic time, the follow-

ing general formula should be used (8). With this assump-

tion, the result from the calculations shows the distance

as a specic multiple of height H.

= ω

∙

cot α

∙ sin

(

γ + β

)

(8)

The above formula is true for the spatial arrangement

proposed in Figure 1, when the azimuth γ reects the an-

gle corresponding to (visible at) approximately 11.00 a.m.

Buildings can be situated relatively to each other in dif-

ferent directions. The analysis of individual spatial sys-

tems showed that the nal formula depends on the time at

which this distance is checked. Regardless of the spatial

arrangement of the object, i.e. the value of β, which is the

angle of building deviation from the north, the following

two sets of formulas will be valid:

– distance between buildings calculated in the morning

(before solar noon):

= cot α ∙ sin

(

γ + β

)

= cot α ∙

cos

(

γ + β

(9)

= cot α ∙ sin

(

γ + β

)

= cot α ∙

cos

(

γ + β

(10)

– distances between buildings calculated in the after-

noon (after solar noon):

= cot α ∙

sin

(

β – γ

= cot α ∙

cos

(

β – γ

(11)

= cot α ∙

sin

(

β – γ

= cot α ∙

cos

(

β – γ

(12)

Time

Data 7.00 8.00 9.00 10.00 11.00 12.00

ω –75 –60 –45 –30 –15 0

cos ω 0.2588 0.5 0.7071 0.8660 0.9659 1

sin ω –0.9659 –0.8660 –0.7071 –0.5 –0.2588 0

cos φ 0.6

sin φ 0.8

sin α (formula (2)) 0.1553 0.3 0.4243 0.5196 0.5795 0.6

cos α 0.9877 0.9539 0.9052 0.8544 0.8149 0.8

cos γ (formula (3)) 0.2096 0.4193 0.6249 0.8109 0.9483 1

sin γ 0.9778 0.9078 0.7807 0.5852 0.3174 0

γ 77.90° 65.20° 51.32° 35.82° 18.51° 0°

Table 2. Set of data necessary to calculate the optimal distance between buildings.

The data correspond to the day of the equinox (δ = 0), for latitude 53°07′59″ (53.1331)

(elaborated by M. Bartnicka)

Tabela 2. Zestaw danych niezbędnych do wyliczania optymalnej odległości między budynkami.

Dane odpowiadają dniu równonocy (δ = 0), dla szerokości geograficznej 53°07′59″

(oprac. M. Bartnicka)

= c ∙ sin

(

γ + β

)

= H ∙ cot α ∙ sin

(

γ + β

)

= ω ∙ cot α ∙ sin

(

γ + β

)

= ω ∙ cot α ∙ sin

(

γ + β

)

= ω ∙ cot α ∙ sin

(

γ + β

)

cos α

sin α

∙

sin (γ + β) =

∙

sin (77.9 + 15) =

= 6.3599 ∙ 0.9987 = 6.35

0.9877

0.1553

92 Małgorzata Bartnicka

The proposed method gives the possibility of establishing

the optimal distances between buildings that provide and

facilitate better interior lighting. This relationship is ex-

pressed by the R

or R

values specifying the distance

between the buildings as a multiple of the height of the

obscuring building H.

The scope of interior lighting depends on several pa-

rameters, including the location of the building relatively

to the north, the size of the opening, or wall thickness.

The subject of the optimal positioning of the buildings has

been researched for centuries. Socrates was looking for

the optimal orientation of a house that would be warm

in winter and cool in summer. Vitruvius paid attention to

the overheating of the southern side of the building and

suggested positioning streets (and therefore buildings) in

accordance with the diagonal of the octagon (i.e. at an an-

gle of 22.5° from the N–S direction). At the beginning of

the 20

century, the slogans of hygienists and modernist

architects demanded access to light, air and sun. Research

at the time led to the conclusion that the best lighting

conditions are provided in buildings located in the north-

south direction, which enabled lighting of ats from the

east and west. It is worth investigating what lighting and

spatial conditions are being created by individual building

positioning.

According to the data from Table 3, if lighting is as-

sessed in the morning (and respectively in the evening),

due to the low solar altitude angle largest possible distanc-

es between buildings are required. Interesting results are

noted in case of the data obtained when the buildings are

aligned with the longitudinal axis in the east-west direc-

tion, i.e. the preferred arrangement due to the energy-sav-

ing qualities – in this case, one of the façades faces south

and the opposite façade faces north. Importantly, the re-

quired distance between buildings is constant in this case;

in this example it is W = 1.33H, but in any other example

it would be possible to quickly determine it on the basis of

local latitude (φ), because this value, on the day of equi-

nox, is cot (90° – φ), i.e. tg φ.

In line with the title of the article, the author presented

ways of determining the optimal distance between build-

ings using the formulas she had investigated and proposed.

Moreover, the table can also be used to estimate the rec-

ommended building height at a known distance. Usually,

the minimum distance between buildings is determined by

building codes. In Poland, this value is a minimum of 4 m

from the border of the plot and maintaining the distance

relationship between buildings W ≥ H. If one of the build-

ings exceeds the height of 8 m (2 × 4 m from the border

of the plot), the distance should be increased accordingly

to maintain the dependence W ≥ H. According to the reg-

ulations, this applies to buildings that are up to 35 m high,

with buildings higher than 35 m the distance between

them becomes the maximum value. Tables obtained from

calculations show the relationship of distance and height

as the equation W = xH. Based on this, building height can

be estimated with the simple formula H = (1/x)W.

Many more countries dened their regulations based

on the date of the equinox, which somehow divides the

conditions of sunlight into two six-month periods, the rst

one in which light access is much easier for half a year

and the second one in which there is much less light for

the other half. As can be seen from Table 3, this seem-

ingly most “reasonable” date also places restrictions as to

the distance between buildings at high latitudes. That is

why many mitigating provisions have been added to the

regulations (e.g. in Poland). The main regulation (“Tech-

nical conditions to be met by buildings and their location”

§60.1 [24]) requires three hours of sunshine in the time

slot from 7

a.m. to 17

p.m. At the same time, it is not

specied whether the sunshine duration should be a con-

tinuous process or the sum of “light episodes”.

Another factor reducing the amount of light in the

apartment is the limitation of the above-mentioned pro-

vision to one room in the apartment, without specify-

ing which room the regulation applies to. Additionally,

a provision was granted that allows for the lack of any

natural light regulations for one-bedroom apartments (the

so-called studio apartments consisting of one room and

a kitchen or a kitchenette). However, according to the

same regulations, in the space dened as “downtown” it

was allowed to reduce by half the sun exposure so that

the W/H dependency could be W ≥ 0.5H, which allowed

for the density of urban development to increase to the

levels prevailing at the end of the 19

century [25]. These

types of allowances can be seen not only in the exam-

ples from Poland. This much more global problem results

from long-term housing shortages. Housing estates from

Spatial orientation of

the obscuring building

Optimal distance between buildings expressed as a multiple of the height of the obstructing building (xH)

depending on the time of day (sun time)

7.00 8.00 9.00 10.00 11.00 12.00

β = 0°

orientation N–S

6.22H 2.88H 1.66H 0.96H 0.44H 0H

β = 22.5° 6.25H 3.17H 2.05H 1.40H 0.91H 0.51H

β = 45° 5.33H 2.98H 2.12H 1.62H 1.25H 0.94H

β = 90°

orientation E–W

1.33H 1.33H 1.33H 1.33H 1.33H 1.33H

Table 3. Optimal distance between buildings (on the day of equinox) facilitating lighting for the façade of the obstructed building

depending on the spatial orientation of the obstructing building (β) (elaborated by M. Bartnicka)

Tabela 3. Optymalna odległość między budynkami (w dniu równonocy) zapewniająca doświetlenie elewacji budynku

przesłanianego w zależności od orientacji przestrzennej budynku przesłaniającego (β) (oprac. M. Bartnicka)

Determining the optimal distance between buildings based on the solar altitude angle 93

the times of the Polish People’s Republic (and during the

subsequent years) were characterized by an urban concept

on the scale of the entire housing estate, quarter, district,

in accordance with the regulations, including maintaining

appropriate distances and providing recreational and green

spaces. Unfortunately, current housing developments are

focused mainly on prot and not on creating suitable and

comfortable, also in terms of light, residential spaces. This

leads to further densication of urban spaces and the loss

of the intellectual achievements of modernism. Gradual

abandonment of the requirements for sunlight access to

residential interiors will have its consequences.

Phil Leather’s team in his studies on stress Windows in

the Workplace: Sunlight, View, and Occupational Stress

[26] points out that the most signicant impact on job sat-

isfaction and comfort of using a particular space is the

presence of direct solar light penetrating the interior. Al-

lowing the gradual reduction of sunlight into the interior

will lead to an excessive density of residential areas. This

phenomenon is one of the so-called stressors of space. The

psychological reaction of users of spaces lacking in sun-

light is stress. This stress is caused by an internal sense

of disruption of balance and a lack of inuence on the

following emerging factors [27]: urban over density, in-

cluding the scale of buildings, the short distances between

the windows of opposite apartments, or simply the loss of

views from the windows. Lack of distance between private

spaces causes sensory overload and, as a result, dicul-

ties in the use of spaces and the risk of creating oppressive

spaces [28].

Undoubtedly, the issues of daylight, its access to interior

spaces and its regulation are in the interest of designers,

albeit to a much lesser extent, of developers and people in-

volved in real estate. Surely, access to daylight and decent

views should become an asset of high-quality buildings

and good architecture, after all, slogans about daylight de-

sign have been increasingly appearing. Determining opti-

mal distances and spatial orientation of buildings is, with-

out doubt, an essential part of good design. Authorities

checking compliance with regulations, including access

to daylight, are also those who issue building permits.

Their control is often based on checking that the design

documentation includes drawings indicating the absence of

shading; however, they do not perform any adequate veri-

cation. The algebraic method presented in the article could

easily become a useful control tool not only for the design-

ers and architects, but also for the building control ocials.

Translated by

Ewa Youngman

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94 Małgorzata Bartnicka

Abstract

Determining the optimal distance between buildings based on the solar altitude angle to facilitate appropriate illumination

in residential interiors in accordance with current legislation

The article touches upon the subject of the availability of sunlight in residential interiors. The distance between buildings is one of the crucial

factors responsible for the penetration of sunlight into interiors. The optimal distance can be calculated algebraically and to determine the output data

it is sucient to know the location of the obscuring building in relation to the north.

The aim of the article is to propose an alternative way to check the access of sunlight without the need to create a 3D model and computer simula-

tion. Using the presented method, it is possible to determine the optimal distance between buildings, which will guarantee no shading to the existing

building.

The proposed solution derives from the graphic approach to this subject presented in Mieczysław Twarowski’s book Słońce w architekturze [The

Sun in Architecture], i.e. to the so-called sun ruler. The author proposes to change the graphic plotting of the shadow into an algebraic conversion of

the distance between buildings. For the calculations, known formulas for the height of the sun (α) and for the azimuth of the sun (γ) were used. As a re-

sult of the analysis of spatial layouts of buildings and the possibility of shading, original formulas for calculating the distance between buildings were

derived. The obtained result reects the proportion between the distance and height of the building (W = xH) at a specic time for the selected latitude.

The publication proves that it is possible to determine the optimal distances between buildings in an algebraic way without the need for CAD

programs. The presented formulas may be used as an instrument for checking the correctness of the non-shading charts presented by investors in

multi-family housing projects. At the same time, these formulas can be used to calculate the approximate height of the proposed building depending

on the expected distance between that building and existing buildings.

Key words: distance between buildings, N–S orientation of the building, solar altitude angle, H/W ratio, access of sunlight

Streszczenie

Wyznaczanie optymalnej odległości między budynkami i dopuszczalnej wysokości projektowanego budynku na podstawie wysokości słońca

W artykule poruszono tematykę dostępności światła słonecznego do wnętrz mieszkalnych. Jednym z niezwykle istotnych czynników odpowiada-

jących za przenikanie światła słonecznego do wnętrz jest odpowiednia odległość pomiędzy budynkami. Optymalną odległość można wyliczyć alge-

braicznie, a do ustalenia danych wyjściowych wystarczy informacja, jakie jest usytuowanie budynku przesłaniającego względem kierunku północy.

Celem artykułu jest zaproponowanie alternatywnego sposobu sprawdzania dostępu promieni słonecznych bez konieczności tworzenia modelu 3D

oraz symulacji komputerowej. Za pomocą przedstawionej metody można określić optymalną odległość pomiędzy budynkami, przy której zachowa-

niu nie nastąpi zacienianie budynku istniejącego.

Proponowane rozwiązanie wywodzi się z gracznego podejścia do tej tematyki prezentowanego w książce Mieczysława Twarowskiego Słońce

w architekturze, czyli do tzw. linijki słońca. Autorka proponuje przemianę gracznego wykreślania cienia na algebraiczne przeliczanie odległości

między budynkami. Do przeliczeń wykorzystane zostały znane wzory na wysokość słońca (α) i na azymut słońca (γ). W wyniku analiz układów

przestrzennych budynków i możliwości zacieniania wywiedzione zostały autorskie wzory służące do przeliczania odległości między budynkami.

Uzyskany wynik odzwierciedla proporcję między odległością a wysokością budynku (W = xH) w konkretnej godzinie dla wybranej szerokości

geogracznej.

W publikacji ukazano, że istnieje możliwość ustalenia optymalnych odległości pomiędzy budynkami w sposób algebraiczny, bez konieczności

wykorzystywania programów CAD. Przedstawione wzory mogą stać się instrumentem sprawdzania poprawności wykresów braku zacieniania przed-

stawianych przez inwestorów w projektach zabudowy wielorodzinnej. Jednocześnie wzory te można wykorzystywać do wyliczania orientacyjnej

wysokości budynku projektowanego w zależności od przewidywanej odległości pomiędzy nim a budynkami istniejącymi.

Słowa kluczowe: odległość między budynkami, orientacja N–S budynku, wysokość słońca, proporcja H/W, dostęp światła słonecznego

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