7/28/2019 sun_diags

1/12

d: \cou rses\b env_ 1\2 00 2_03 \su n_0 2_ 01 .doc Mart in Wi lk inso n

Polar axis Polar axis

hemisphere

Great circle

Meridians Equator

Zenith

Primary Meridian

Polar axis

Equator

Circle of constant Latitude

LONG.

LAT.

Meridian of constant Longitude

Predicting the Suns position

Introduction

There are many different ways of representing the position of the sunin the sky and most of these may be usefully used to help investigate

sunlighting in design. In order to limit explanations, only one or twoof the methods will be considered in any detail, but once the principlesare understood, one should find it easy to apply those other methodsnot described in detail.

The basic astronomical facts will be reviewed but a detailedknowledge of them is not essential for an appreciation of sunlighting.

The Earth

The Earth is effectively a spherical globe that rotates eastwards abouta North-South axis approximately once every 24 hours, as shown in

Figure1.

A globe may be partitioned in various ways as shown in Figure 2.These prove useful in describing parts of the Earth and locatingaccurately different places on the Earth. If a globe is divided into twoequal parts to produce two hemispheres, then the dividing line

between the two parts will be a Great Circle. The axis about whichthe Earth rotates is known as the Polar Axis and this axis intersects theglobe at the North Pole and the South Pole. A great circle passing

through both poles is known as a Meridian. A great circle that isequidistant from the North and South Poles is known as the Equator.

As shown in Figure 3, any place on the Earth may be specified inrelation to;i) a primary meridian,

ii) the equator .

The Longitude describes the position of the appropriate meridian in

relation to the primary meridian - it may be either East or West of theprimary meridian.

The Latitude describes the angle from the equator towards a Polealong a particular meridian it may be either North or South of the

equator.

The Earths orbit around the Sun

The Earth orbits the Sun approximately once every 365 days. Its orbitlies in the same plane as the Sun and is elliptical in shape with the Sunpositioned at one of the ellipses foci as shown in Figure 4. This plane

is known as the Ecliptic Plane because when the moon moves into theplane there is the possibility of an eclipse.

One consequence of the elliptical orbit is that the earth speeds up andslows down as it moves around the sun and this means that the lengthof the day, measured from noon to noon, changes throughout the year.

The changing length of the Solar Day rather complicates time keepingand it is simpler to assume a constant length of day and use the

Figure 1 Earth rotates about Polar Axis

1 R U W K 3 R O H

3 R O D U D [ L V

Figure 2 Divisions of a sphere

Figure 3 Latitude and Longitude

Tilt =23.4

Ecliptic plane

NN

side view

plan view

Figure 4 Earths elliptical orbit around sun

7/28/2019 sun_diags

2/12

Building Environment 1

2

1 R U W K

= H Q L W K

0 H U L G L D Q

( T X D W R U

0 H U L G L D Q 3 O D Q H

7 R W K H 6 X Q

Winter Solstice

N

Summer Solstice

Autumn Equinox

S

N

S

N

S

N

S

Spring Equinox

Equator

tilt

Polar axis Polar axis

Perpendicular to ecliptic plane

Arctic circle

declination

declination

Equator

Ecliptic plane

Tropic of Cancer

Arctic circleTropic of Capricorn

December Solstice June Solstice

1 R U W K

6 X Q V U D \ V

G H F O L Q D W L R Q

G H F O L Q D W L R Q

( T X D W R U L D O S O D Q H

average length of day throughout the year. This leads to the familiartime convention in the UK of Greenwich Mean Time, which is based

upon the mean or average length of day over the whole year.

The difference between solar time and local mean time is called the

equation of time. A correction should be applied for the equation oftime when it is required to know the position of the sun veryaccurately. However, the maximum cumulative difference between

solar time and mean time is in the order of between +15 minutes and -15 minutes and for many architectural purposes it may be ignored.

Solar noon occurs when the sun lies in the Meridian plane as shown inFigure 5. At noon the sun appears to be due South at higher northernlatitudes and due North at higher southern latitudes. Solar time istherefore dependent upon the particular Longitude of a location, and itis clearly rather awkward if clocks need to be changed as one movesfrom one locale to another.

The tilt between the Earths axis and the orbital plane

A most important feature of the Earths circumstance is that the PolarAxis is tilted in relation to the orbital plane as shown in Figure 4 andFigure 6. Within the time spans considered in architecture the

direction of the Polar axis relative to the orbital plane remainsconstant. At the present time the Tilt is at an angle of 23.4.

One consequence of the tilted axis is that seasons of the year areexperienced by those parts of the globe closer to the poles. The closer

a region is to one of the poles, then the more seasonal is the climateexperienced by that region.

Arctic circles and tropics

The tilt also gives rise to the division of the globe into various parts.

The Arctic Circles divide the regions of the Earth into those that willat some time in the year experience a 24 hour day and a 24 hour night,and those regions which always experience a day and a night.

The Tropics divide the regions of the Earth into those where the sun

will be directly overhead at some time in the year and those where thesun will never reach the Zenith.

Declination

The tilt of the earths axis results in a change in the relative position ofthe sun as the earth moves in its orbit. This change in the relative

position of the sun is reflected in the change that occurs in the anglethe suns rays make with the equatorial plane. This angle is known asthe declination.

The declination will vary from a maximum of 234 at the Summer

Solstice to a minimum of -234 at the Winter Solstice. Twice in the

year the declination will be zero and this occurs at the Spring andAutumn Equinox.

Figure 5 Sun in meridian plane at noon

Figure 6 Constant tilt of polar axis as Earth orbits sun

Figure 7 Defining the tropics and arctics

Figure 8 Declination of the sun

7/28/2019 sun_diags

3/12

Sun_02_01.doc

3

North

South

West

East

Zenith

Z

Meridian

Altitude ring

Ground plane

Altitude

Azimuth

There are thus 4 times in the year when the declination takes particularvalues that are especially significant:

Astronomical occurance D Calendar dateWinter Solstice -23.4 23

rd.December

Vernal (Spring) Equinox 0 21st.

MarchSummer Solstice +23.4 23

rd.June

Autumnal Equinox 0 23

rd.

September

The Winter Solstice is that time of year when the declination is a

minimum. Because the direction of Tilt is not exactly in line withmajor axis, it is not the case that this coincides with the shortest day ofthe year.

The Summer Solstice is that time of year when the declination is a

maximum. Similarly to the other solstice, the longest day does notnecessarily coincide with the summer solstice.

The Equinoxes are those times of year when the day and night are of

equal time. Thus the sun will rise at 6am and set at 6pm.

At other times in the year the Declination may be evaluated by theapproximate equation:

( )

+=

365

284360sin4.23

NnDeclinatio degrees

Where N is the day number of the date for which the declination is

being calculated. January 1st.

being day number 1.

A more accurate formulation for declination is given in the margin

and this may be used for computer generated diagrams.

Positioning the sun in the sky

The position of the sun in the sky is given by two angles that areshown in Figure 9,

- the altitude of the sun above the ground or horizon planez the compass direction of the sun on the ground plane.

The Azimuth may be given in two ways; either East or West of South,or clockwise from North. In general, the azimuth is most often given

in terms of the angle from North, but in these notes the angle will begiven as an angle East or West of South.

Figure 10 shows a shadow cast by a vertical pole and how the lengthand position of the shadow are affected by the altitude and azimuth ofthe sun.

( )

radians3sin001480.03cos002697.0

2sin000907.02cos006758.0

sin070257.0cos399912.0006918.0

nDeclinatioSolar

radians365

12angleDay

s

s

dd

dd

dd

d

N

+

+

+=

=

==

/ H Q J W K R I V K D G R Z

+

W D Q

SOUTH

GNOMON

Altitude

Azimuth Vertical Polein ground

+

Figure 10 Shadow cast by a vertical pole

Figure 9 Hemisphere of sky

7/28/2019 sun_diags

4/12

Building Environment 1

4

L

Sy mbol V ari able De fi niti on

D Declination The angle of the sun's rays to

the equatorial plane, positive

in summer.

Latitude The angle from the equator to

a position on Earth's surface.

H Hour angle The angle the Earth needs to

rotate to bring the meridian to

noon. Each hour of time is

equivalent to 15 deg.

N Day Number The day number, January 1st

is 1.

$ O W L W X G H

+

/

: L Q G R Z

6 X Q 3 D W F K

6 R O D U

: D O O

$ ] L P X W K

The suns position may be determined from the following equations:

LDHLD sinsincoscoscossin += ,

cos

cossinsincoscoscos

LDLHDz

= ,

cos

cossinsin

DHz

= ,

DLHL

Hz

tancoscossin

sintan

= .

Application of the above formulae are not necessarily the best way to

appreciate the various effects of the suns position and a more

graphical approach may usefully be adopted. However the aboveformulae may be usefully used where more precise information is

needed.

Sundials

The Gnomon is a point in space through which the rays of the sun pass

to later shine upon some surface. The shadow on the ground cast by aflagpole will depend upon the suns altitude and azimuth, and if thetopmost tip of the flagpole is considered, then it will sweep out a path

on the ground as the sun moves across the sky. The topmost tip of theflagpole may be considered as a gnomon.

It may occur to you that a simple perspective is also constructed as agnomic projection.

Plotting the paths of the tip of the shadow for different timesthroughout the year will produce a sun dial as shown in Figure 11.

Using the horizontal sundial

If the height of the gnomon is known then the sundial can be used toconstruct the shadows created by buildings at different times of year.This is done simply by measuring the length of the shadow on the

sundial and increasing the length in proportion to how much greater isthe height of the building to the height of the gnomon. The directionof the shadow will be the same as that on the sundial. This is showndiagrammatically in Figure 12.

Sun patches created by sunlight shining through windows can beconstructed in a similar manner to show the effect of sunlight through

different types of window at different times of year as in Figure 13.

The sundial may also be used in what is sometimes called the aviators

method. If the sundial is placed on a model and is viewed so that thegnomon is lined up with a particular time of year, then the eye ispositioned in the direction of the sun at that time of year as in Figure

14. All that the eye will see on the model will therefore be exposed tosunlight at that time of year. Clearly, as the eye is very much nearer

SOUTH

Winter

Equinox

Summer

Noon

10am

8am

2pm 4pm

Winter Noon

3 O D Q R I % X L O G L Q J

/

+

/

$

+

$

S

S

B

B

B

B

S

S

LH

HL

L

H

L

H

=

=

Figure 11 Sun dial

Figure 12 Sundial used to draw shadows

Figure 13 Drawing s un patches

7/28/2019 sun_diags

5/12

Sun_02_01.doc

5

EastWest

North

South

=

r

R =0000

r

Horizon Line , altitude = 0 = 0 = 0 = 0

Zenith = 90 = 90 = 90 = 90

Constant altitude

$ ] L P X W K =

Projection Plane

Sky Vault

altitude ring onprojection plane

EW

N

S

r

#

to the model than the sun, there will be increasing discrepancies asones gaze moves away from the centre of the sundial. For this reason

it is best to view the model from some distance away so that onesdirection of view stays fairly constant as one views the whole model.

Sunpath diagrams

There are many different ways of graphically displaying the relativeposition of the sun at different times of the day and year. These range

from,

i) Sun dials, these are gnomic projections,

ii) Rectangular projections of the sky,iii) Circular sunpath diagrams.

There is no one method that is preferable to all others, and each has itsown advantages and disadvantages. For hand sketching diagrams, Ibelieve that the Stereographic diagram has distinct advantages. The

diagram allows the whole sky to be considered and this means there is

no need to construct a new diagram for each different orientation ofthe faade. Also, all the sunpaths and time lines are arcs of circles and

these are reasonably easy to sketch for even the non artistic.

However its advantages are less pronounced with the advent of the

computer, as now calculations and redrawing can be undertakenwithout effort. Never the less, it still is a most useful way ofconsidering the suns position. In computer applications, it does lenditself to serious design because it allows the consideration of a numberof variables at one time, and this is not always possible with some of

the other techniques of displaying sunpaths.

The Stereographic projection

The basis of the circular projections is that a hemisphere of sky isprojected down onto a horizontal plane.

This results in a diagram of the form shown in Figure 15, where thepoints of the compass are defined by the direction out from the centreof the diagram, and the altitude is defined by the distance out from the

centre.

The construction of the Stereographic projection is shown in Figure 16

and results in the relation between radius and altitude,

=

2

90tan0

Rr

SOUTH

Winter

Equinox

Summer

Noon

10am

8am

2pm 4pm

* Q R P R Q

Figure 14 The Aviators method

Figure 15 Basis of circular Sunpath diagram

Figure 16 The Stereographic projection

7/28/2019 sun_diags

6/12

Building Environment 1

6

Ecliptic plane

Direction of sun

Zenith

Polar axis

LAT.

Declination

Equatorial plane

+ R U L ] R Q S O D Q H

Suns position in the sky

The sunpath loci are arcs of circles on the stereographic projection and

the arc of a circle can be sketched quite easily if three positions on thearc are identified.

Three positions in the sky are used to sketch a Sunpath for a given

day:i) the azimuth of the sun rise,ii) the altitude of the sun at solar noon,iii) the azimuth of the sun set.

The days course of the sun across the sky is caused by the Earthsrotation about its own axis, and as the declination changes only but a

little during the course of a day, the sun rise and the sun set may beassumed to be symmetrically located on either side of south.

When sketching a sunpath for a particular location, it is really onlynecessary to consider four times in the year,

i) the Winter Solstice,ii) the Summer Solstice,iii) the Vernal and Autumnal Equinoxes.

The two Equinoxes have the same sun path loci and therefore it is

rarely the case that more than thr ee sun path loci need be plotted onthe sun path diagram. The suns altitude at noon can be derived fromFigure 17 and is given by,

DLNoon

+= 90 .

During the summer months, above the arctic circle the sun will not set,

and therefore the sun path locus will be a circle on a stereographicprojection. All that is needed to draw a circle is its diameter. One end

of the diameter will be given by the position of the sun a noon, and theother by the suns position at midnight, as shown in Figure 18. If thealtitude is measured from the southern direction it will be given by theformula,

DLMidnight = 270

and if the altitude is measured from the northern direction it will begiven by,

+==

90

)270(180

DL

DLMidnight

The azimuth of the sun at sun rise and sun set may be found using therelationship given below.

L

Dz

cos

sincos 0

== .

It is worthwhile noting that, where the sun does not cross the horizon,the above relation does not hold. Therefore, there will be no solution

when the latitude is >66 for either the summer, or the winter solstice.

Equatorial plane

Declination

Horizon plane

Direction of sun

LAT.

Ecliptic plane

= H Q L W K

1 R U W K

6 R X W K

V R X W K

Figure 17 Altitude of Sun at Noon

Figure 18 Altitude of sun at Midnight

7/28/2019 sun_diags

7/12

Sun_02_01.doc

7

6

: H V W ( D V W

6 X Q D W O R F D O 1 R R Q

6 X Q D W V X Q V H W

6 X Q D W V X Q U L V H

6

: H V W ( D V W

6 X Q D W O R F D O 1 R R Q

6 X Q D W V X Q V H W

6 X Q D W V X Q U L V H

6

: H V W

( D V

6

: H V W

( D V W

Sketching a Sunpath

Sketching sunpaths for the City of Bath which is at a Latitude of 513

North.

Noting that at solar noon the sun is due South and at the maximum

altitude given by the relation:

DLNoon

+= 90 .

For the equinoxes, March 21st.

and September 23rd.

, when thedeclination is 0.

=

=

7.38

3.5190Noon

Noting also that at the equinoxes, the sun rises due East and sets dueWest, there are three known positions on the Sunpath locus for the

equinox, and these are shown on Figure 19. These may be used tosketch the first of the Sunpath loci, as is shown in Figure 20.

For the summer solstice, June 21st.

, when the declination is 234,

=

+=

1.62

4.237.38Noon

Using the relation for the azimuth when the sun rises and sets,

L

Dz cos

sincos 0

==

The declination is 23.4 and therefore,

==

==

=

=

=

4.129)635.0(cos

635.0625.0

397.0

3.51cos

4.23sincos

1

0

0

z

z

Therefore the azimuth of sunrise and sunset are respectively1294

East and West of South, as is shown in Figure 21.

There are then three positions of the sun that can be connected

together by an arc of a circle that denotes its Sunpath, as shown inFigure 4.

For the winter solstice, December 23rd.

, when declination is 234,

=

=

3.15

4.237.38Noon

The declination is 234 and therefore using the relation for azimuth

at sunrise and sunset,

Figure 19 Times needed to sketch sun paths

Figure 20 Sun path at Equinoxes

Figure 21 Times at summer solstice

Figure 22 Sun path at Summer Solstice

7/28/2019 sun_diags

8/12

Building Environment 1

8

6

: H V W

( D V W

6

: H V W ( D V W

6

: H V W ( D V W

D P

D P

D P

S P

S P

S P

1 R R Q

==

==+

=

=

=

=

6.50)635.0(cos

635.0625.0

397.0

3.51cos

4.23sin

3.51cos

)4.23sin(cos

1

0

0

z

z

It is worthwhile noting at this point that,

zz

zzz

cos)180cos(

sin180sincos180cos)180cos(

=

+=

and this is confirmed by the observation that,

1294+506=180.

Therefore, there is no need to go through the calculation of theazimuth twice. It is simpler to use the fact that the sunrise and sunsetfor the two Solstices are symmetrically positioned about the East-

West axis. This is shown diagrammatically in Figure 23 where theknown sun positions for the Winter Solstice are plotted.

Figure 24 shows the three points connected by the arc of a circle togive the Sunpath locus for the Winter Solstice.

The three Sunpath loci can then be collected together on the samediagram to give the range of sun positions throughout the year, as is

shown in Figure 25.

Sketching solar time lines

The hour lines on the stereographic projection are also arcs of circles.These hour lines always cross the sunpaths at 90 and this helps inconstructing them.

The easiest hour line is that of noon, which is a straight line towardsthe South. At the Equinoxes the 6am and 6pm hour lines pass throughthe horizon line due East and due West respectively as shown in

Figure 26.

Although not exactly correct, for the purposes of sketching, theintermediate hour lines may be positioned on the basis of spacingthem equally between the noon and 6 clock hour lines, as in

Figure 27.

6

: H V W

( D V W

6

: H V W ( D V W

Figure 23 Times at Winter Solstice

Figure 24 Sunpath at Winter Solstice

Figure 25 Sun path diagram

Figure 26 Construction of Hour Lines Figure 27 Complete Sunpath diagram

7/28/2019 sun_diags

9/12

Sun_02_01.doc

9

For Latitudes beyond the Arctic circle

At latitudes greater than 666N, the sun will not rise above thehorizon at the Winter Solstice. Therefore only two Sunpath loci are

required to show the extreme ranges of the suns position in the sky,that at the Equinoxes and that of the Summer Solstice.

At the Summer Solstice the sun will be above the horizon for thewhole day and therefore its Sunpath is sketched using the position ofthe sun at noon and midnight. Considering the Figure 18 used to

obtain the relation for the altitude of the sun at midnight, it should benoted that the sun will appear to be due North.

Considering the Sunpath for a Latitude of 70 North:

At the Equinoxes the Declination is zero and the maximum altitude atnoon will be,

== 207090Noon

.

The sunrise and sunset are respectively due East and West.

At the Summer Solstice the Declination is +23.4 and therefore themaximum altitude at noon will be,

=+=+= 4.434.23204.237090Noon

.

And the minimum altitude of the sun at midnight will be,

=== 6.1764.2370270270 DLMidnight .

measured from South, and

= 4.36.176180

measured from North.

These are then used to plot the sunpaths as is shown in Figure 28 .

For Latitudes within the Tropics

For Latitudes within the tropics the sun will pass overhead through thezenith at some time of year. Therefore, it is important to realise that atthe summer solstice the sun may be to the North in Northern latitudes

and to the south in southern latitudes. This should be apparent fromthe cross section through the earth at noon shown in Figure 29.

As an example, sketching the sunpaths for the Latitude of 10 N:

At the Equinoxes, the Declination = 0, and the max altitude is,

=+=+= 800109090 DLNoon

At the Summer solstice, the declination is 23.4, and the altitude is,

Norththeto6.76103.4-180

souththeto4.1034.23109090

=

=+=+= DLNoon

The azimuth at sunrise and sunset is,

6

: H V W ( D V W

Direction of su

Polar axis

LAT.

Declination

Equatorial plane+ R U L ] R Q S O D Q H

Zenith

Ecliptic plane

6

: H V W ( D V W

Figure 28 Sunpaths for 70North

Figure 29 Section of Earth at noon

Figure 30 Sunpaths for 10North

7/28/2019 sun_diags

10/12

Building Environment 1

10

/ D W L W X G H

$ ]

$ W

6 X Q U L V H

$ O W L W X G H

$ W

P L G Q L J K W

$ O W L W X G H

$ W

1 R R Q

$ ]

$ W

6 X Q U L V H

$ O W L W X G H

$ W

P L G Q L J K W

$ O W L W X G H

$ W

1 R R Q

$ ]

$ W

6 X Q U L V H

$ O W L W X G H

$ W

1 R R Q

1 $ 6

2 U

1

1 $

6

2 U

1

( D V W 6

2 U

1

1 $ 6

2 U

1

6

2 U

1

1 $1 $

6

2 U

1

1 R V X Q

( D V W 6

2 U

1

1 $

6

2 U

1

1 $

6

2 U

1

( D V W

6

2 U

1

' H F H P E H U

7 /

- X Q H

8 9

( T X L Q R [

==

=

=

=

=

= 8.113)403.0(cos

403.0985.0

397.0

10cos

4.23sin

cos

sincos

1

0z

L

Dz

At The Winter Solstice, the declination is 23.4, and the altitude is,

souththeto6.564.23109090 ==+= DLNoon

and the azimuth of sunrise and sunset will be given by,

Z=0 = 180 - 113.8 = 66.2.

The whole Sunpath diagram for a latitude of 10 N is sketched in

Figure 30.

For Southern Latitudes

In southern latitudes the sun will primarily be due north at solar noon.

The sun will still rise in the East and set in the West. If there is doubtin your mind about where the sun is, then consider again a diagram ofthe cross section through the ecliptic plane as in Figure 31.

If a convention is adopted that the altitude of the sun at noon andmidnight is always measured from the south, then by convention

Northern latitudes are positive and Southern Latitudes are negative.

From Figure 31, it will be seen that the altitude of the sun at noon willbe given by,

)(9090 DLNoon

++=+=

but as the Latitude is to the South, by convention L is negative, andso,

( )

DL

DL

Noon

Noon

+=

++=

90

)(90

The three latitudes previously considered for northern latitudes are

reconsidered here as being southern latitudes, and the altitudes andazimuths needed for sketching the diagrams are listed in the table.The diagrams for southern latitudes are shown in the margin in

Figures 32-34.

: H V W ( D V W

1 R U W K

6 R X W K

- X Q H

' H F H P E H U

: H V W ( D V W

1 R U W K

6 R X W K

' H F H P E H U

( T X L Q R [ H V

: H V W ( D V W

1 R U W K

6 R X W K

' H F H P E H U

- X Q H

North

Direction of sun

Zenith

LAT.

Declination

Equatorial plane

+ R U L ] R Q S O D Q H

Ecliptic plane

Figure 31 Section of Earth at noon

Figure 32 Sunpaths for 51.3South

Figure 33 Sunpaths for 70South

Figure 34 Sunpaths for 10South

7/28/2019 sun_diags

11/12

Sun_02_01.doc

11

The Stereographic projection

The stereographic projection displays the whole sky on a flat plane.

In order to fully utilise the diagram, it is necessary to relate it to thethree dimensional reality, and to only consider the diagram as a twodimensional drawing will lead to misunderstandings.

Particularly useful in conjunction with the projection itself is ashading protractor such as is shown in Figure xx. This aid divides thehemisphere of sky into a series of inclined plane and vertical planes.

A single inclined plane is shown in Figure xx, and it can be seen that itdivides the hemisphere of sky into two parts; that above the inclinedplane and that below the inclined plane.

This inclined plane is positioned within the hemisphere of sky by twoquantities:

i) The Vertical Shading Angle VSAii) The orientation of the Ground Line GL

Figure xx shows the single inclined plane of Figure xx plotted on astereographic projection. Some aspects should be noted about thelocus of the line positioning of the inclined plane on the diagram:

i) the altitude of the inclined plane ( VSA), and the altitude circle ofVSA coincide on a line normal to the Ground Line,

ii) the altitude of the inclined plane is zero where the GroundLine meets the horizon line,

iii) the locus is an arc of a circle on the stereographic

projection.

A special case of the inclined plane is when the VSA is 90. Such avertical plane is shown in Figure xx, and Figure xx shows this plane

plotted on a stereographic projection. Being simply a vertical planepassing through the centre of the projection, it has a constant azimuthand will just be a radial line emanating from the centre of theprojection.

A protractor will normally have a series of radial lines representingvertical planes on the opposite side of the ground line, as is shown inthe example of a protractor in Figure xx.

Viewing a room

The stereographic projection may be used to display a room as seenfrom some point. Consider the 10m square room shown in Figure xx.Assume that the projection is centred on the centre of the room at aheight of 1m.

Baseline

Inclined Shadingplanes

20

0

80

60

40

9 H U W L F D O V K D G L Q J

S O D Q H V

VSA

GL Ground Line

Sky Vault

= H Q L W K

Orientation ofGround line

Inclined plane

' $

= H Q L W K

* U R X Q G / L Q H

2 U L H Q W D W L R Q R I S O D Q H

Ground

Line

9 H U W L F D O S O D Q H

2 U L H Q W D W L R Q R I S O D Q H

7/28/2019 sun_diags

12/12

Building Environment 1

12

]

2

3 O D Q

6 H F W L R Q

]

]

1 2

]

Each of the walls intersection with the ceiling will lie in an inclined

plane emanating from the rooms centre. The inclined planes, each of

a vertical shading angle 1

, will therefore represent the junction of theceiling and walls.

In this particular example;

== 315

3tan 11

These are plotted on a stereographic projection with the shadingprotractor and using the angle of VSA=31.

The walls intersect in a vertical plane with a constant azimuth andtherefore are shown by radial lines. These may be constructed either

by using the angle z2 , or drawn as a radial lines emanating from thecorners of the ceiling.

The vertical sides of the window subtend an azimuth of z 1 and the

head of the window lies in an inclined plane of angle 2 where;

== 8.215

2tan 12

and

==

7.385

4tan

1

1z

Thus Figure xx shows a stereographic projection of the room seenfrom its centre. This projection may then be superimposed over a

Sunpath diagram as is shown in Figure xx, and the sunpaths seenthrough the aperture of the window will be seen by the point at thecentre of the room.

Clearly the room projection should be correctly orientated withrespect to the Sunpath diagram.