Download NASS Scientific Calculator Programs

Sundial Design with a Programmable Scientific Calculator
Designing a proper sundial involves mathematics. There is no way around
this. To lay out the lines to interpret a shadow, we are forced to use mathematical
techniques. Even the “simple” graphical techniques are a form of mathematics.
The more accurate trigonometric calculations look very intimidating and difficult.
The computer programs that seem to do it all for us are filled with such complex
mathematics. The methods described by Waugh and others use the logarithms
of trigonometric functions adding a level of complexity to aid in calculations. How
can an ordinary person with just high school mathematics deal with the
mathematical problems and get on with the art of sundial design?
These days we have excellent tools that can handle the mathematical
calculations for sundials very well. Computers are generally used but a simple
cheap scientific calculator with trigonometric functions is all we really need. If that
calculator has extra memories and can be programmed to remember the steps,
the repetitive calculations can be simplified. The NASS Scientific Calculator with
trigonometric functions, three memories and forty program steps handles most
sundial design calculations very well. Following are example programs designed
specifically for this calculator and adaptable to many similar models.
NASS Scientific Calculator Programs:
Calculator Functions: An introduction for the Chicago Workshop
Trigonometry: Sine, Cosine, Tangent, Inverse-1
Angles: DEG, DMS, Time
Memories: M, K1, K2
Programs: LRN, COMP, HLT, (x)
Sundials Design:
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
Solar Position:
Noon, Sunrise, Sunset
Prime Vertical
Altitude, Azimuth
Declination Lines
Great Circle Distance
Sundial Design:
The hour lines on a horizontal sundial depend on the Latitude of the dial
and the Time. Time is measured from noon meridian as a time angle t, + in the
afternoon and – in the morning. The trig equation is Tan HA = Sin Lat x Tan t.
This same general equation and following calculator program are also applicable
for vertical sundials, including declining sundials, by solving for the horizontal
equivalent dial. The program solves for the Hour Angle HA for various time
angles using the Latitude or equivalent and the Difference in Longitude, when
applicable, stored in the K memories.
Offsets: Lengths are easier to measure accurately than angles so the program
provides an offset option, calculating the length along the borders of a square
with sides equal to a specified size set arbitrarily at 10 in the program.
Horizontal Sundial:
First store the Latitude in the memory K1. Eg. For Chicago 41.88 2ND X->K1.
Clear K2 ON/C 2ND X->K2 as it DL is not used for horizontal dials.
Then start programming in the learn mode for t = 15.
2ND LRN 2ND (X) 15 + K2 = TAN x K1 SIN = X->M x 10 = 2ND HLT
The display shows the X Offset (1.79) based on Y = 10.
For the Hour Angle continue with these steps. MR 2ND tan-1 2ND HLT
If the Hour Angle displayed (10.14º) is greater than 45º you should use the Y
offset. Continue with these program steps MR 2ND 1/X x 10 =
The display now shows the Y Offset (55.90) based on X = 10.
Then enter 2ND LRN to exit the learn mode.
To use the stored program, clear the display, ON/C , press COMP, enter the
time angle, eg 15 and press COMP for the X offset, press COMP again for the
Hour angle and then press COMP again for the Y offset.
This table shows the results for Chicago and a Size of 10 cm.
Time
12:00
1:00
2:00
3:00
4:00
5:00
6:00
7:00
Angle t
0º
15º
30º
45º
60º
75º
90º
105º
X Offset
0
1.79
3.85
6.68
11.56
24.91
error
-24.91
Hour Angle
0º
10.14º
21.08º
33.73º
49.14º
68.13º
90.00º
-68.13
Y Offset
error
55.90
25.95
14.98
8.65
4.01
0
-4.01
Vertical Sundial:
For a south-facing (meridian) sundial, use the same program as above. K2 (DL)
remains 0. The only difference is to store the Co-Latitude in K1. The Co-Latitude
is 90º-Lat so the Co-Latitude for Chicago is 90 – 41.88 = 48.12º 2ND X->K1.
As before, to use the stored program, clear the display, ON/C , press COMP
enter the time angle t, eg 15 and press COMP for the X offset, COMP again for
the Hour angle and COMP again for the Y offset.
Time
12:00
1:00
2:00
3:00
4:00
5:00
6:00
Angle t
0º
15º
30º
45º
60º
75º
90º
X Offset
0
2.00
4.30
7.45
12.90
27.79
error
Hour Angle
0º
11.28º
23.26
36.67º
52.21º
70.21º
90.00º
Y Offset
error
50.13
23.26
13.43
7.75
3.60
0
The drawing below shows on the right how the offsets are used to draw the hour
lines. On the left is a dial design based on these hour lines. The offset square is
just used to set the hour lines. Any size or shape could be used for the dial
design by simply extending the Hour Lines.
Vertical Sundial Chicago
3.60
7
10.00
70°
7.75
52°
8
11°
9
37°
23°
Y Offsets
10 11
2.00
4.30
7.45
10.00
X Offsets
Vertical Declining Sundial:
For a Vertical Declining Sundial, design for the horizontal equivalent sundial
using the program outlined previously. First calculate the parameters for this
horizontal equivalent: the Substyle Distance SD, Substyle Height SH,
Difference in Longitude DL and the Angle to the Vertical AV.
Put Latitude and Wall declination into the K Memories.
Latitude eg 41.88 2ND X->K1, Wall Declination eg 32 2ND X->K2
Calculate and note the parameters (displayed) for the vertical declining dial.
Clear the display and register, ON/C for each calculation
Substyle Distance: Tan SD = Sin Dec / Tan Lat:
ON/C K2 SIN / K1 TAN = 2ND tan-1 (30.58)
`
Substyle Height: Sin SH = Cos Dec x Cos Lat:
ON/C K2 COS x K1 COS = 2ND sin-1 (39.15)
Difference in Longitude: Tan DL = Tan Dec / Sin Lat
ON/C K2 TAN / K1 SIN = 2ND tan-1 (43.11)
To express the Difference in Longitude as time, divide by 15 and convert to
Hours:Minutes:Seconds using DMS: eg 43.11 /15 = 2ND DMS (2:52:26)
Now calculate the horizontal equivalent dial using the same program as for a
horizontal sundial but substitute Substyle Height SH for the Latitude and the Polar
Angle P as the time angle where P = DL + t. Solve for the Hour Angles HA on the
plane of the dial with the modified sundial equation Tan HA = Tan P x Sin SH. Again
calculate the offsets based on a size of 10. These hour angles and offsets are
measyred from the substyle. To lay the angles out on the wall the offset drawing must
be rotated to make the 12 noon meridian vertical as shown in the following sketch.
Start by storing Substyle Height SH in K1 and Difference in Longitude DL in K2.
Enter Substyle Height eg. 39.15 2ND X->K1
Enter Difference in Longitude eg 43.11 2ND X->K2
Now clear the display and enter the same the same program as before.
Clear the display, ON/C , press COMP enter the time angle as (X), eg 15 and
press COMP for the X offset, COMP again for the Hour Angle and then COMP
again for the Y offset. Enter afternoon times as +t and morning times as –t using
the +/- key.
The following table shows the results for a Vertical Sundial in Chicago, Latitude
41.88º, a morning dial on a wall declining +32º east of south.
Time
12:00
1:00 pm
2:00 pm
3:00 pm
4:00 pm
11:00 am
10:00 am
-DL
9:00 am
8:00 am
7:00 am
6:00 am
5:00 am
Angle t
0º
15º
30º
45º
60º
-15º
-30º
-43.11º
-45º
-60º
-75º
-90º
-105
X Offset
5.91
10.14
20.79
191.32
-27.11
3.37
1.47
0
-0.21
-1.92
-3.93
-6.74
-11.82
Hour Angle
30.58º
45.42º
64.32º
87.00
-69.75
18.64º
8.36º
0
-1.19º
-10.85º
-21.44º
-34.00º
-49.77º
Y Offset
16.92
9.85
4.81
0.52
-3.69
29.65
68.01
0
-480
-52.16
-25.46
-14.83
-8.46
The offset drawing must be rotated as shown below trough an angle equal to the
substyle distance. The 12 Noon meridian is always a vertical line on a vertical
declining sundial. Extend the hour lines as required to complete the layout of the dial.
Chicago 41.88º N Vertical S 32º E
5
4
-3.
69
-6.
74
6
0.5
2
-3.
93
30.6°
SD
-1.
92
-0.
21
7
3
4.8
1
1.4
7
3.3
7
5.9
1
Vertical
Meridian
8
9
10
11
12
9.8
5
1
2
Analemmatic Sundial:
An analemmatic sundial is the planar (2D) version of an equatorial ring dial.
Projected onto the horizontal plane, the equatorial ring becomes an ellipse and
the polar style becomes the zodiac date line. This calculator program for the
design of an analemmatic sundial is in three parts; the Hour Points, the Zodiac
Date Line and Seasonal Markers.
The Hour Points for an analemmatic sundial are on an ellipse. The minor axis
and all the Y coordinates of the hour points are reduced from a circle by the sine
of the Latitude. For each 15 degree hour angle (t), the (X,Y) coordinates of the
hour points are therefore X= Sin t and Y = Cos t x Sin Lat. These are all scaled
by the size of for the semi-major axis, typically 2.5 meters for a human gnomon.
Store the Latitude (41.88) in K1 X->K1 and the Size (2.5) in K2 X->K2 .
Program 1: Hour Points: Enter time, hours from noon, and calculate the
coordinates (X,Y) for hour points. The program converts time in hours to the time
angle t by multiplying by 15 º/hour.
X = K2 x Sin t 2ND LRN 2ND (x)1 x 15 = X->M SIN x K2 = 2ND HLT
Y = K2 x Cos t x Sin K1 MR COS x K1 SIN x K2 = 2ND LRN
To use the program, clear the register, ON/C press COMP, enter the time in hours
from noon, + pm and – am.
Press COMP for X, and press COMP again for Y.
Time
12:00
1:00 pm
2:00 pm
3:00 pm
4:00 pm
5:00 pm
6:00 pm
7:00 pm
8:00 pm
11:00 am 11:00
Hour Point X
0
0.647
1.25
1.768
2.165
2.415
2.500
2.415
2.165
-0.647
Hour Point Y
1.669
1.612
1.445
1.180
0.834
0.432
0.0
-0.432
-0.834
1.612 etc.
The Zodiac Date Line tells you where on the North/South or Y axis to place the
vertical gnomon for each day of the year. There are two components to this line.
It is a tangent line based on the size of the dial brought to the horizontal plan with
the Cosine of the Latitude, Y = Tan Dec x Size x Cos Lat
Program 2: Zodiac Date Line:
2ND LRN 2ND (x) 1 TAN x K2 x K1 COS = 2ND LRN
Clear ON/C press COMP and enter Solar Declination at (X) to calculate Y.
Continue through the yearly cycle to+/- 23.44º, entering the Declination on various
dates. This declination data is from Sonnenhurh.xls by Sonderegger and Bailey.
Date
Declination
Zodiac Y
-0.790
1 January
-23.01
-0.574
1 February
-17.14
-0.250
1 March
-7.64
0
20 March Equinox
0.18
0.146
1 April
4.49
0.500
1 May
15.03
0.753
1 June
22.03
0.807
21 June Solstice
23.44
0.795
1 July
23.12
0.607
1August
18.05
0.273
1 September
8.33
0.025
21 Sept. Equinox
0.76
-0.102
1 October
-3.13
-0.477
1 November
-14.38
-0.744
1 December
-21.78
-0.807
21 Dec. Solstice
-23.44
.
In particular, note and store X->M Y for the Declination of 18º for August 1 for the
Seasonal Marker calculation.
Seasonal Markers are points on the X axis that indicate where and when the
sun rises and sets through the seasons. Stand on the Zodiac date line and sight
over the seasonal markers to the Hour Points to see the time when the sun rises
or sets. Stand on the seasonal marker and sight over the Zodiac Dates to see the
azimuth, where the sun rises or sets.
Where the sun rises is given is: Sunrise Azimuth: Cos Az = Sin Dec / Cos Lat.
The position on the X axis for the Seasonal Marker is Distance X = Y x Tan Az
where Y is the position for that date on the Zodiac Date Line. There is some minor
variation in the Seasonal Marker position through the year but the position for the
Solar Declination = 18º is a good average. This you should have calculated using
the previous program and stored in M. This Y value also includes the Size
parameter. Now calculate the Seasonal Marker Distance with these steps.
Enter 18 SIN / K1 COS = cos –1 TAN x MR = for the Seasonal Marker Distances
on the X (East / West) axis, - for sunrise and + for sunset. For this 5 meter in
Chicago the Seasonal Markers are at X = +/- 1.33 meters. The sketch below
showing a typical analemmatic sundial layout is for a different latitude and size.
Sunrise
Direction
North
10
11
12
1
2
9
3
June
8
May
7
West
Sunrise
Time
April
4
July
5
August
Sept
6
6
Mar
Sunrise
5
4
Feb
Oct
Sunset
Jan Nov
Dec
East
7
8
5.00
South
Chicago
Analemmatic
Longitude Correction for Clock Time:
All these calculations are based on true local time determined by the position of
the sun. At noon, the sun is due south at its maximum altitude. Clocks show an
arbitrary average legal civil time based on time zones, daylight savings and an
instrument correction, the equation of time. Unfortunately you will need to be able
to convert clock time to solar time to show that the sundial is correct. The
calculator makes this conversion as it has a conversion to Degrees: Minutes:
Seconds 2ND -> DMS.
Enter your longitude, say Chicago 87.63º W. Divide by 15 to change the degrees
of longitude to hours. (5.842). Press 2ND -> DMS to convert this to time. (5:50:31)
Chicago: 87.63º = 5.842 hours = 5:50:31 hh:mm:ss.
Chicago is in the Central Time Zone, centered on 90º degrees or 6 hours west.
The sun is earlier in Chicago than 90º longitude by 6:00:00 - 5:50:31 = 00:09:29.
For daylight savings time (spring forward) add one hour to the clock. Solar noon
in Chicago in the summer is therefore typically at 12:50:31 CDST.
All the calculations in these programs use the time angle t, the time in degrees
from solar noon. You can calculate the lines on any sundial to include the
longitude correction. Divide the longitude by 15 and subtract the time zone
longitude: eg Chicago 87.63º / 15 – 6 = -0.158º Add this to all your time angle t
to give the clock hour lines on your dial. Eg. t + (-0.158). For each hour add 15º
to the previous corrected time angle.
Solar Position:
Sundials tell time by showing the position of the sun. If you know your
location and the solar declination of the day, you can calculate a number of
interesting things related to the position of the sun. Simple trig functions using
your latitude and solar declination can be easily solved with a scientific
calculator.
In these examples calculator buttons are in boxes like 2ND COS-1 ,numbers are
just entered, and answers displayed are shown in brackets like (60.9)
Start by putting the Latitude and Declination into the K Memories.
For example: Chicago on 18 August:
Latitude 41.88 2ND X->K1, Solar Declination 12.78 2ND X ->K2
The following special cases with simplified geometry can be calculated using the
values in the K Memories and trig functions.
Noon: At noon when sun is due South, the solar Altitude = 90º-Lat + Dec
ON/C 90 – K1 + K2 = Altitude at Noon (60.90º) for Chicago 18 August.
Sunrise, Sunset:
Time: When does the sun set? Cos t = - Tan (Lat) x Tan (Dec)
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 + 12 =
(18.7823)
Convert to Hours:Minutes:Seconds with 2ND ->DMS
Sunset Time is 18:46:56
*For Sunrise subtract from noon – 12 = +/Azimuth: Where does the sun set? Cos (Az) = Sin(Dec) / Cos (Lat)
ON/C K2 SIN ÷ K1 COS = 2ND COS-1
(72.71º)
Path: At what angle does the sun set? Cos (Phi) = Sin (Lat) / Cos(Dec)
ON/C K1 SIN ÷ K2 COS = 2ND COS-1
(46.80º)
Prime Vertical: Time when Sun is due East or West
Cos t = Tan (Dec) / Tan (Lat)
ON/C K2 TAN ÷ K1 TAN = 2ND COS-1 (75.346º)
This gives you t, the time as an angle from noon when the
sun is due east. Divide by 15 for hours. ÷15 = (5.023)
For west. add 12 for noon.
(17.023) or (17:01:23)
Converted to Hours:Minutes:Seconds with 2ND ->DMS
Altitude and Azimuth:
The position of the sun can be calculated for any time any day of the year
if you know the local solar time and solar declination using the Navigators’
Equation from spherical trigonometry.
Sin Alt = Sin Lat x Sin Dec + Cos Lat x Cos Dec x Cos t
This is easily remembered as the sine,sine, sine, cos, cos, cos equation.
The calculation can be programmed into the calculator for repetitive calculations.
As before, Start as before by putting the Latitude and Declination into the K
Memories and the time angle t, 15º per hour from noon, in memory X->M.
Now clear the display and start programming in the learn mode.
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 (58.1064º). 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
The display shows the Azimuth of the sun at 1 pm (28.5375º).
The program is complete. To use the stored program for other times, , enter the
time angle into the memory, clear the display, press COMP to start the program
calculating the Altitude and COMP again for the Azimuth. The calculator is slow,
taking over 10 seconds for each complex calculation.
Time
Noon
1:00pm
2:00pm
3:00pm
4:00pm
5:00pm
6:00pm
7:00pm
11:00am
tº
0
15 X->M: ON/C
30 X->M: ON/C
45 X->M: ON/C
60 X->M: ON/C
75 X->M: ON/C
90 X->M: ON/C
105 X->M: ON/C
-15 X->M: ON/C
COMP
COMP
COMP
COMP
COMP
COMP
COMP
COMP
Altitude
Azimuth
60.90º
0º
(58.106º) COMP (28.537º) SW
(50.941º) COMP (50.700º) SW
(41.384º) COMP (66.793º) SW
(30.712º) COMP (79.191º) SW
(19.609º) COMP (89.769º) SW
( 8.492º) COMP (80.414º)*NW
(-2.31º)** COMP (70.520º)*NW
(58.106º) COMP (-28.537º) SE
For morning the numbers are the same for each hour from noon but the sign
changes for t and the Azimuth is SE
*Trig Ambiguity: 180-80.414 = 99.586º for SW: ** After sunset.
Declination Lines:
Knowing the altitude and azimuth of the sun, you can now calculate and
plot Declination Lines, the path of the shadow of the tip of a gnomon.
N
SL Sin Az
Az
SL
W
SL Cos Az
E
Using polar coordinates (r,θ), the
shadow length SL, the horizontal
distance from the base of the vertical
gnomon is (r) the gnomon height times
the Tangent of the Altitude.
SL = G / Tan Alt.
The direction (θ) is the Azimuth.
For Cartesian (X,Y) coordinates, the
East/West (X) coordinate (r Sin ø) is
X = SL x Sin Az. The North/South (Y)
coordinate (r Cos ø) is Y = SL x Cos Az.
S
Set up a table for of Time, time angle t, Altitude and Azimuth with the data
already calculated, and do repeated calculations with the following program
entering the Altitude and Azimuth to find the polar and Cartesian coordinates of
the declination line. The vertical Gnomon Height G is programmed in as 5, an
arbitrary size. The length of the gnomon base is Y = 5 / Tan Lat. Calculate this Y
first and store it in memory to be added to the Y values of the shadow to bring
them to the origin of the graph, where the sloping style intersects the horizontal
plane. For Chicago this is: 5 / 41.88 TAN = X->M
Program: Start in the learn mode and enter the altitude as 2 at (X).
2ND LRN 2ND (X) 2 TAN 2ND 1/x X 5 = 2ND X->K1 2ND HLT
The display shows the Shadow Length SL stored in K1.
Enter the Azimuth and calculate (X,Y):
2ND (X) 3 2ND X->K2 SIN x K1 = 2ND HLT The displayed value is X.
Then enter1K2 COS x K1 + MR = 2ND LRN for Y and to leave the LRN mode..
To use the stored program, clear the display, ON/C press COMP , enter Altitude,
and press COMP for Shadow Length. Press COMP and enter Azimuth, press
COMP for X and then COMP for Y.
This table shows the results for Chicago, August 21, gnomon height 5 cm.
Bold Altitude and Azimuth are entered as (X1) and (X2).
Displayed Shadow Length, and (X,Y) are italicized in the table.
Time
12:00
1:00
2:00
3:00
4:00
5:00
6:00
Angle t
0º
15º
30º
45º
60º
75º
90º
Altitude
60.90º
58.10º
50.94º
41.38º
30.71º
19.61º
8.49º
Shadow
2.78
3.11
4.058
5.675
8.418
14.034
33.496
Azimuth
0.0º
28.53º
50.70º
66.79º
79.19º
89.77º
99.59º
X
0
1.49
3.14
5.22
8.27
14.03
33.03
Y
8.36
8.31
8.15
7.81
7.16
5.63
-0.0
Horizontal Sundial Chicago 41.88º N
9
10 11 12 1
2
4
8
August 18
Declination
Line
7
6
3
Gnomon
Shadow
5
East
West
6
South
5
7
Great Circle Distance:
It is often interesting to know the great circle distance between two locations. For
example if you are flying somewhere this distance is useful to estimate how long
the flight will take and how many air mile points you will collect. The spherical
trigonometry formula to calculate the Great Circle Distance is degrees is:
Cos D = Sin Lat1 x Sin Lat2 + Cos Lat1 x Cos Lat2 x Cos (Long1 – Long2)
This can be calculated directly but you can program the steps for repetitive
calculations like different legs of the trip. The answer is in degrees. To convert to
statute miles multiply by 60 x 1.15078 = 69.05 miles per degree.
First store the Latitudes in the K Memories. Enter Lat1 2ND X-> K1 and
Lat2 2ND X-> K1 Store 69.05 X->M.
Program: 2ND LRN 2ND (X) 1 - 2ND X 2 = COS x K1 COS x K2 COS + K1
SIN x K2 SIN = 2ND cos-1 x MR = 2ND LRN
To use the program, clear the display, ON/C COMP and enter Long1 at (X1)
Press COMP and enter Long2 at (X2) Press COMP to calculate the Great Circle
Distance in statute miles. The calculation takes about 12 seconds.
For example:
Sidney BC (N 48.6, W 123.4) to Chicago (N 41.88, W 87.63) is 1782 miles.
Toronto ON (N 43.67, W 79.61) to Chicago is 425 miles.
New York NY (N 40.71, W74.00) to Chicago is 711 miles.
London UK (N 51.50. W 0.46) to Chicago is 3932 miles.
Frankfurt DE (N 50.13, E 8.669) to Chicago is 4323 miles.