Equations for Planar Sundials

Size: px

Start display at page:

Download "Equations for Planar Sundials"

Lester Byrd
5 years ago
Views:

1 Equations for Planar undials Toshimi Taki October 8, 7. Purpose Equations necessar to make planar sundials are derived. You can choose an orientation for the plane of a sundial.. Reference [] Toshimi Taki, atri ethod for Coordinates Transformation, Rev. E, Februar 9, 4. in Takis Homepage( 3. mbols Obliquit of Ecliptic: atitude: Hour Angle of the un: H Declination of the un: Aimuth: A Altitude: h Height of odus from Plane of undial: l Distance between odus G and odus G : d Direction Coe: (,, ) 4. Altitude and Aimuth of the un 4. Position of the un et the declination and hour angle of the un at an given time, and H, respectivel. Declination should be between and + (obliquit of the ecliptic). This coordinates can be epressed with the direction coe in the coordinate sstem O- e - e as follows. e e e ( H ) ( H )

2 Equator West X e enith h ( e, e, e ) eridian orth Celestial Pole e P orth H O outh X h Y h, Y e East Figure. Equatorial Coordinate stem The position of the un is also epressed in aimuth A and altitude h. Aimuth is measured from the south to the west. The direction coe of the uns position in horiontal coordinate sstem O- h - h is epressed in the following equation. h h h h( A) h( A) h eridian enith h West ( h, h, h ) orth h A O outh X h Y h East Figure. Horiontal Coordinate stem

3 The equatorial coordinates of the un can be transformed to the horiontal coordinates ug the following equation. The geographical latitude of the sundial is. h h h e e e Equator eridian X e enith h orth Celestial Pole e P West orth W W un at ummer olstice W A A outh X h Y h, Y e A E E E East un at Winter olstice Figure 3. Position of the un 5. Position of hadows 5. Plane of undial and odus As shown in figure 3, the dial face of the sundial is in O-- plane, where the coordinate sstem of the sundial is O--. ain nodus G is set on -ais normal to the dial face with the distance l. Another nodus G is on the line directed to the north celestial pole from G and distance between G and G is d. The relationship between the horiontal coordinate sstem and the sundial coordinate sstem is as follows. The horiontal coordinate sstem is rotated around h -ais and becomes O--- coordinate sstem. Then, O--- coordinate sstem is rotated around -ais to the sundial coordinate sstem O---. 3

4 enith h, X e P orth Celestial Pole e West G orth eridian G O, outh X h Y h, Y e East Figure 4. undial Coordinate stem The direction coe in O--- coordinate sstem is epressed as follows. Then the direction coe in the horiontal coordinates sstem is transformed to O--- coordinate sstem. h h h Direction coe in O--- coordinate sstem is epressed as follows. Finall, the direction coe in O--- coordinate sstem is transformed to the sundial coordinate sstem, O--- coordinate sstem. 4

5 5 Ug the above equations, the direction coe of the un in the horiontal coordinate sstem can be transformed to the direction coe in the sundial coordinate sstem, ( sh, sh, sh ). 5. Coordinates of odus G in undial Coordinate stem Direction coe of e -ais in the equatorial coordinate sstem is (,, ). This vector is transformed to the sundial coordinate sstem as follows. h h h h h h Then, coordinates of nodus G in the sundial coordinate sstem is epressed as follows. G G G d l 5.3 hadows of odus Figure 5 shows shadows of the nodus. The position of the shadow of nodus G in the sundial coordinate sstem is,

6 G G G l l The position of the shadow of nodus G in the sundial coordinate sstem is, G G G G G It is obvious that G and G are ero. Unit Vector of Direction to the un ength: d hadow of odus G : G Height of odus G : l G G Y hadow of odus G : G, X Figure 5. hadows of odus Finall, the sundial coordinate sstem is transformed to O-X-Y coordinate sstem in figure 5. X Y X Y G G G G G G G G G G G G 6

7 6. ample Calculations I made -Ecel worksheets to calculate the lines of a sundial. You input data of our latitude and orientation of the dial face, then the worksheets calculate the positions of the shadows of the nodus on the dial face. Three eamples are shown below. 6. Vertical undial to be used in agoa, Japan (atitude 35 o orth) A vertical sundial for agoa, Japan (35 o ) which is facing south is calculated b the worksheet. The dial face of the vertical sundial is shown in figure 6. Winter olstice Gnomon (ength = ) h - 7h - Equino 8h -3 6h -4 9h h h h 3h 4h 5h -5-6 ummer olstice Figure 6. Vertical undial for agoa 7

8 6. Horiontal undial in agoa, Japan Figure 7 shows a dial face of a horiontal sundial for agoa calculated b the worksheets. orth 4 Winter olstice 3 5h 9h 6h 4h 3h h h h 8h Equino 7h 7h Gnomon (ength = ) - outh ummer olstice Figure 7. Horiontal undial for agoa 8

6.3 undial of Trinia dei onti in Rome Trinita dei onti (figure 8), which is located above panish teps and Piaa di pagna in Rome, has a sundial on its façade. The latitude of Rome is 4 o 54.

9 6.3 undial of Trinia dei onti in Rome Trinita dei onti (figure 8), which is located above panish teps and Piaa di pagna in Rome, has a sundial on its façade. The latitude of Rome is 4 o 54. The façade is facing 75 o west from the south. I calculated the dial of the vertical sundial with these conditions ug the worksheets. The result is shown in figure 9. The figure also shows the actual sundial for comparison. Figure 8. Trinita dei onti Gnomon (ength = ) Winter olstice Equino h 5h 8h 7h 6h ummer olstice h 3h -3-4 Figure 9. Vertical undial of Trinita dei onti 9

3 N. Follow these instructions carefully to construct a simplified Analemmic Sundial (a vertical gnomon sundial).

3 N. Follow these instructions carefully to construct a simplified Analemmic Sundial (a vertical gnomon sundial). Use your own shadow to tell time. 5 4 3 2 12 11 10 9 8 7 Follow these instructions carefully to construct a simplified Analemmic undial (a vertical gnomon sundial). 1 ach sun dial team should have at least