Equations for Planar Sundials
Toshimi Taki
October 28, 2007
1. Purpose
Equations necessary to make planar sundials are derived. You can choose any orientation for theplane of a sundial.
2. Reference[1] Toshimi Taki, “Matrix Method for Coordinates Transformation,” Rev. E, February 29, 2004. inTaki’s Homepage(http://www.asahi-net.or.jp/~zs3t-tk/)
3. Symbols
Obliquity of Ecliptic:
Latitude:Hour Angle of the Sun: H
Declination of the Sun:
Azimuth: A
Altitude: hHeight of Nodus from Plane of Sundial: lDistance between Nodus G0 and Nodus G1: dDirection Cosine: (L, M, N)
4. Altitude and Azimuth of the Sun
4. Position of the Sun
Let the declination and hour angle of the Sun at any given time,and H, respectively. Declination

should be between – and + (obliquity of the ecliptic). This coordinates can be expressed with thedirection cosine in the coordinate system O-xe’-ye’ as follows.

LSe 'cos cos( H )M Se 'cos sin( H )N Se 'sin

1

ZenithMeridian

Zh

North Celestial Pole

EquatorZe’
Xe’ S (L ’, M ’, N ’)eeePZWestNorthHOSouthXhYh, Ye’

East

Figure 1. Equatorial Coordinate System
The position of the Sun is also expressed in azimuth A and altitude h. Azimuth is measured from thesouth to the west. The direction cosine of the Sun’s position in horizontal coordinate system O-xh-yh isexpressed in the following equation.

LShcos h cos( A)M Sh

cos h sin( A)

N Shsin hZenith
MeridianZhZWest

North

S (Lh, Mh, Nh)hASouthO

XhYh

Figure 2. Horizontal Coordinate System2EastThe equatorial coordinates of the Sun can be transformed to the horizontal coordinates using thefollowing equation. The geographical latitude of the sundial is .

LShM ShN Shcos

200 sin1sin2LSe '200 cos

M Se 'N Se '2Zenith
MeridianZhNorth Celestial Pole

EquatorZe’Xe’PZ

North

West
W0W2

Sun at Summer Solstice

W1

A2
A0A1
E0E1SouthE2

Xh

Yh, Ye’East

Sun at Winter Solstice

Figure 3. Position of the Sun
5. Position of Shadows
5.1 Plane of Sundial and Nodus
As shown in figure 3, the dial face of the sundial is in O-x’’-y’’ plane, where the coordinate system ofthe sundial is O-x’’-y’’. Main nodus G0 is set on z’’-axis normal to the dial face with the distance l.
Another nodus G1 is on the line directed to the north celestial pole from G0 and distance between G0
and G1 is d.
The relationship between the horizontal coordinate system and the sundial coordinate system is asfollows. The horizontal coordinate system is rotatedx

around Zh-axis and becomes O-x’-y-z’

coordinate system. Then, O-x’-y-z’ coordinate system is rotatedcoordinate system O-x’’-y’’-z’’.

3yaround y’-axis to the sundialZenith
Zh , z’North Celestial Pole
Ze’z’’zXe’PZG1WestNorthG0MeridianOy’, y’’yXhSouthYh, Ye’x’

Eastx’’

Figure 4. Sundial Coordinate SystemThe direction cosine in O-x’-y’-z’ coordinate system is expressed as follows.

L'

cos ' cos 'M'

cos ' sin '

N'

sin '

Then the direction cosine in the horizontal coordinates system is transformed to O-x’-y’-z’ coordinatesystem.

L'cos zM'

sinN'0sinzz

cos z00Lh0 Mh1NhDirection cosine in O-x’’-y’’-z’’ coordinate system is expressed as follows.

L' 'cos ' ' cos ' 'M ''cos ' ' sin ' 'N ''sin ' 'Finally, the direction cosine in O-x’-y’-z’ coordinate system is transformed to the sundial coordinatesystem, O-x’’-y’’-z’’ coordinate system.

4

L' 'cos y0

sin

M ''
N ''

0
sin y

100
cos y

L'y

M'
N'

Using the above equations, the direction cosine of the Sun in the horizontal coordinate system can betransformed to the direction cosine in the sundial coordinate system, (Lsh’’, Msh’’, Nsh’’).
5.2 Coordinates of Nodus G1 in Sundial Coordinate System
Direction cosine of Ze’-axis in the equatorial coordinate system is (0, 0, 1). This vector istransformed to the sundial coordinate system as follows.

LNhcos0 sin

2M Nh
N Nh0
sin

1

LN 'cos z

MN 'sin

NN '0

LN ' 'cos y0sin

M N ''
N N ''

0sin y

100
cos y

sinz

00 cos2z

cos z0020012LNh0 M Nh1N Nhy

LN 'MN 'N N'

Then, coordinates of nodus G1 in the sundial coordinate system is expressed as follows.

xG1 ' '

0

LN ' 'y G1 ' '0d M N ''z G1 ' 'l

N N ''

5.3 Shadows of Nodus
Figure 5 shows shadows of the nodus.
The position of the shadow of nodus G0 in the sundial coordinate system is,

5

xGS 0 ' '0y GS 0 ' '0z GS 0 ' 'l

LS ' 'lM S ''NS ''N S ''The position of the shadow of nodus G1 in the sundial coordinate system is,

xGS1 ' '0y GS1 ' '0z GS1 ' 'N G1 ' 'LS ' 'N G1 ' 'M S ''NS ''N S ''

It is obvious that zGS0’’ and zGS1’’ are zero.
z’’
Unit Vector of Direction to the Sun
Length: dHeight of Nodus G0: lG0G1
YShadow of Nodus G0 : GS0

Shadow of Nodus G1: GS1y’’, Xx’’

Figure 5. Shadows of Nodus
Finally, the sundial coordinate system is transformed to O-X-Y coordinate system in figure 5.

X GS 0y GS 0 ' 'YGS 0xGS 0 ' 'Z GS 0z GS 0 ' 'X GS1y GS1 ' 'YGS1xGS1 ' 'Z GS1z GS 1 ' '6

6. Sample CalculationsI made MS-Excel worksheets to calculate the lines of a sundial. You input data of your latitude andorientation of the dial face, then the worksheets calculate the positions of the shadows of the nodus on
the dial face. Three examples are shown below.
6.1 Vertical Sundial to be used in Nagoya, Japan (Latitude 35o10’ North)
A vertical sundial for Nagoya, Japan (35o10’) which is facing south is calculated by the worksheet.
The dial face of the vertical sundial is shown in figure 6.

Winter SolsticeGnomon (Length = 1)
10-6-5-4-3-2-17h0123417h-2

Equinox16h-49h10h

11h

6-1-38h

512h

13h

14h-5-6

Summer Solstice

Figure 6. Vertical Sundial for Nagoya

715h6.2 Horizontal Sundial in Nagoya, Japan
Figure 7 shows a dial face of a horizontal sundial for Nagoya calculated by the worksheets.

North4Winter Solstice
315h

9h
14h

16h

13h

212h

11h

10h

8h
Equinox117h7h0-5-4-3-2-1

01-1

Gnomon (Length = 1)

Summer Solstice-2

SouthFigure 7. Horizontal Sundial for Nagoya

823456.3 Sundial of Trinia dei Monti in Rome
Trinita dei Monti (figure 8), which is located above
Spanish Steps and Piazza di Spagna in Rome, has asundial on its façade. The latitude of Rome is 41o54’. Thefaçade is facing 75o west from the south. I calculated thedial of the vertical sundial with these conditions using theworksheets. The result is shown in figure 9. The figure
also shows the actual sundial for comparison.

Figure 8. Trinita dei Monti
Gnomon (Length = 1)
EquinoxWinter Solstice10-4-3-2-1

01

18h
17h
16h-115h
14h-2

12h-3

13h-4

Figure 9. Vertical Sundial of Trinita dei Monti

923

Summer Solstice4