1. Sunset Sundial from Scratch
I enjoy sunsets.
Roger Bailey
NASS, Aug 2008
St Louis, Missouri

2. Algarve Coast, Portugal
Last November we vacationed on the Algarve Coast of Portugal. Our prime activity was hiking alone the trails the followed the rugged and scenic coast. The views were spectacular.

3. November Sunsets
We rented an apartment with a view at Roche Brava, near Carvoiero. There we enjoyed watching many sunsets, observing the glowing colours of the sky and high clouds as the sun descended into the sea. This was a great time to share with friends and family as we reminisced and whet our appetite for dinner. Our sunny terrace faced south and west with great views out over the ocean.

4. Sunset Sundial from Scratch
Sunset views south west over Atlantic Ocean
West facing wall on sundeck
Design a sunset sundial, Reverse Italian Hours
Limited tools:
No texts, files computers, programs, internet
Math: geometry, spherical trigonometry
Navigators Equation: Sine Sine Sine Cos Cos Cos
NASS Calculator: scientific & programmable
The blank west facing wall of the deck seemed to be an ideal location for a sundial. What would be more appropriate than a sundial showing the hours to sunset? Designing from scratch a vertical sundial, declining west showing reverse Italian hours became my project.
We travel light so I did not have my usual tools for designing sundials. The only tools available were what I remembered about geometry and spherical trigonometry and a pocket calculator. OK, it was the NASS Programmable Scientific Calculator programmed to design sundials. solve the “sine sine sine cos cos cos” Navigators Equation, etc. This equation calculates the altitude and then the azimuth of the sun for any given latitude, solar declination and time. This is just what we need to calculate the position of a shadow from a perpendicular gnomon on the wall using plane geometry.

5. Wall (Meridian) Plane: West 90º
Project Equatorial Disc EE’ for time from solar noon
Angle to Horizon = Co-Latitude
Mark 15º segments projected to GxTan t on EE’
Draw hour lines perpendicular
In our case for a vertical wall facing directly west, things are simplified. The wall is in the north south meridian plane. The polar axis (P –P’) is true in this plane and makes an angle with the horizontal plane (H – H’) equal to the latitude. The equatorial plane (E – E’) is perpendicular to the polar axis and the meridian plane so it projects as a straight line, the equinoctial, at an angle to the horizontal equal to the co-latitude. The gnomon G is perpendicular to this plane and projects as a point. The hour lines for hours from noon would be lines crossing the equinoctial line at distances (D) equal to the Tangent of the time angle t times G, the gnomon height. D= G x Tan t. This design is a standard sundial design showing the conventional hours from noon. This is a good indicator of what the sunset dial will show.
For the sunset dial we need to solve for the altitude and azimuth and use that information to plot the declination lines and hour line to sunset.

6. Navigation Triangle
Latitude, Altitude & Declination Angles from Horizon on Great Circles
Navigation Triangle contains time and azimuth angles
For the solution, lets look at the Navigation Triangle in Spherical Trigonometry. This is the view from outside the Celestial Sphere. Your horizon is horizontal plane, your Zenith is straight up. The polar axis goes through the horizontal plane at the latitude angle. Declination and and Altitude are also measured as angle of great circles relative to the horizon. Time is the included angle at the pole between your zenith north and the longitude of the sun. The Azimuth is the included angle at your Zenith between the the pole and the sun.

7. Co-Angles in Navigation Triangle
Navigation Triangle sides are complementary angles (90- x)
Solve for Altitude with Cosine Rule, 2 sides and contained angle known
Solve for Azimuth with Sine Rule
We work with the sides of the Navigation Triangle so we use the complementary angles, 90 – the angle. In effect this reverses sines to cosines etc.
Now lets look at the Navigators Equation for Altitude in spherical trigonometry. This is the Cosine Rule but switched around as we are dealing with Co-angles.
Sin Alt = Sin Dec x Sin Lat + Cos Dec x Cos Lat x Cos t
If you know the declination of the sun (Dec) and the latitude (Lat), all trig terms in the equations become just simple numbers. You can easily solve for the altitude (Alt) of the sun at any time angle (t). The Law of Sines for spherical triangles then allows you to solve for the Azimuth (Az) of the sun.
Sin Az = Cos Dec x Sin t / Cos Alt

8. Design Technique for Sunset Sundial
Calculate polar sundial using equatorial disc
Solve for sunset times at solstices
Cos t = Tan Lat x Tan Dec
Tabulate t = sunset time – N hours (15º/hour)
Calculate Altitude for t knowing Lat & Dec
Sin Alt = Sin Lat x Sin Dec + Cos Lat x Cos Dec x Cos t
Calculate Azimuth for t knowing Altitude
Sin Az = Cos Dec x Sin t / Cos Alt
Project point onto wall knowing declination of wall, altitude and azimuth
Plot the results and join the points for hours to sunset and solstice declination lines
To determine the hours lines for hours to sunset we need to solve the time of sunset for the Summer and Winter Solstices. This is our starting time base for Reverse Italian Hour Lines, the Hours until Sunset. For sunset the Altitude is zero. The Altitude equation then simplifies at sunset to:
Cos t = -Tan Dec x Tan Lat
For our Latitude in Portugal (37.09º) and the Solstice declinations of =/ º, the extreme sunset times will be Cos t = +/- Tan x Tan = Taking the ArcCos gives a time angle t of º and º. Dividing by 15º per hour and expressing as time gives us 4:43:28 PM and 7:16:32 PM for the Winter and Summer Solstices. The scientific calculator easily calculates these trig functions and does the conversion to D:M:S

9. Horizontal Plane: Azimuth True
Lets use plane geometry of determine the projected shadow on the wall. Looking down on the horizontal plane gives us the view where the azimuth angle is true. This gives us the x value for the point where the shadow of the tip of the gnomon hits the wall and the horizontal distance H from the gnomon tip to the wall shadow.
X = G/Tan Az and H = G/Sin Az

10. Solar Plane: Altitude True
Now lets look at the vertical plane through the sun, the Solar Plane: Here the Altitude angle and the horizontal distance H are true.
We now have what we need, the (x,y) coordinates of the point on the wall of the gnomon tip in terms of the gnomon Height G and altitude and azimuth of the sun.
x = G / Tan Az and y = G Tan Alt / Sin Az

11. NASS Programmable Scientific Calculator Programs:
Workshop at NASS Conference in Chicago 2005
Calculator Functions:
Trigonometry: Sine, Cosine, Tangent, Inverse-1
Angles: DEG, DMS, Time
Memories: M, K1, K2
Programs: LRN, COMP, HLT, (x)
Solar Position:
Noon, Sunrise, Sunset
Prime Vertical
Altitude, Azimuth
Declination Lines
Great Circle Distance and Direction (Qiblah)
To assist with the calculations, I used the NASS Calculator provided to those who attended the NASS conference in Chicago in I gave a workshop on the use of the calculator and provided a number of useful sundial design programs. Now I am demonstrating that the calculator and these programs are actually useful.
This inexpensive calculator has trig functions, angle format conversions, three memories and stores up to 40 program steps. These programs pushed the limits of its capability. The programs can be modified for similar calculators from Sharp, Radio Shack etc.
We will be using the solar position programs that calculate sunset time and altitude and azimuth at any time knowing the latitude and declination.
The great circle distance calculations are quite useful for those collecting airline points. I have used the Great Circle direction calculation to determine the Qiblah, the direction to Mecca

12. NASS Programmable Scientific Calculator Programs:
Sundials Design Programs:
Horizontal: Time & Hour Angles Offsets
Vertical: Time & Hour Angles, Offsets
Vertical Declining:
Substyle Distance, Substyle Height, Difference in Longitude
Time, Polar & Hour Angles, Offsets
Analemmatic: Hour Points, Zodiac Date Line, Seasonal Markers
Longitude Correction
As you can see, these calculator programs are useful for most common types of sundials. Now to show how theey can simplify the design of a sunset sundial.

13. NASS Calculator: Sunset Time
Latitude and Declination into the K Memories.
Latitude: ND X->K1, Declination: ND X ->K2
Sunset Time: Cos t = - Tan (Lat) x Tan (Dec)
Program: ON/C K1 TAN x K2 TAN = +/- 2ND COS-1
This gives you t, the time as an angle from noon. (101.73)
Divide by 15 for hours. ÷15 = (6.782)
For Sunset* add 12 for noon = (18.782)
Convert to Hours:Minutes:Seconds with 2ND ->DMS
Sunset Time is 18:46:56
First lets look at the little program to calculate sunset time from the usual simplification of the Navigators equation for Altitude = zero.
Cos t = - Tan Lat x Tan Dec. Put the latitude in memory K1 and the declination in K2. This example is from the Chicago notes. From the sunset time and time angle, we determine the time and time angle for hours before sunset, the variable t for the alt/az calculations,

14. NASS Calculator: Altitude & Azimuth
Sin Alt = Sin Lat x Sin Dec + Cos Lat x Cos Dec x Cos t
Program: 2ND LRN MR COS x K1 COS x K2 COS + K1 SIN x K2 SIN =
2ND sin-1 2ND HLT
The display shows the Altitude of the sun at 1 pm ( º).
Continue for Azimuth using the equation:
Sin Az = Cos Dec x Sin t / Cos Alt
COS 2ND 1/X x MR SIN x K2 COS = 2ND sin-1 2ND LRN
Here are the equations and program steps for altitude and azimuth. Latitude and declination are still stored in K1 and K2. Put in a declination, loop through and solve for various times. Note down on a table the altitude and azimuth each time and then solve for x and y to plot the point on the plane. Then do it again for another declination. I did three, for the solstices (+/ º) and the equinox, zero

15. NASS Calculator: Sample Output Altitude & Azimuth
Time tº Altitude Azimuth
Noon º 0º
1:00pm 15 X->M: ON/C COMP (58.106º) COMP (28.537º) SW
2:00pm 30 X->M: ON/C COMP (50.941º) COMP (50.700º) SW
3:00pm 45 X->M: ON/C COMP (41.384º) COMP (66.793º) SW
4:00pm 60 X->M: ON/C COMP (30.712º) COMP (79.191º) SW
5:00pm 75 X->M: ON/C COMP (19.609º) COMP (89.769º) SW
6:00pm 90 X->M: ON/C COMP ( º) COMP (80.414º)*NW
7:00pm 105 X->M: ON/C COMP (-2.31º)** COMP (70.520º)*NW
11:00am -15 X->M: ON/C COMP (58.106º) COMP ( º) SE
Chicago Latitude (41.88º N), Aug 18 Declination (12.78º)
Here is a sample from the Chicago notes for a typical loop through of for the declination on the day of the workshop, August 18

16. Times from Noon and Sunset Equinox Line
Here is a table collecting the data for the latitude in Portugal, ( º) and the equinox, declination = zero. I noted the altitude and azimuth and then calculated the x,y coordinates on the plane of the wall. This in effect is the design for a normal polar sundial but the x y coordinates are that of the vertical plane, not the distance along the equinox line, d = G x Tan t. On the equinox sunset is at 6 PM so the hours to sunset are just subtracted from 6.

17. Solstice Hours from Noon Hour and Declination Lines
This is the loop through for the solstice declinations using the time angle based on noon. This gives us the hour lines and declination lines for a normal west facing polar sundial

18. Sunset Sundial: Winter Solstice
Now we do the calculations for hours before sunset, using the time angle as hours from noon. This table is for the winter solstice.

19. Sunset Sundial: Summer Solstice
And one final loop through, this time for the summer solstice declination and sunset time.

20. Final Sunset Sundial Design
Here is the final result from plotting all the (x.y) coordinates determined with the calculator and shown on the tables. The technique works.

21. This shows the prototype I plotted and left for the owner of the apartment a our “gift for the house.” I have also sent him proper Delta Cad drawings that he can print full size, say one meter square. One of the local tile shops would be happy to use the drawing to create the custom dial on classical Portuguese tiles (Azulejos). I have encouraged him to add this custom sundial to the west wall of his deck but I don’t think he has the same passion for sundials as I do.