bi gnomon sundials. - analemma zonnewijzers · let us discuss some nice sundials below: 1- a...

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17 June 2006,

The principle of the bi-gnom sundials is to mount a shadow casting object on a sundial plane whichcasts 2 shadow lines. Each shadow line can be associated with a separated time system and orientationof the sundial plane. This will enable us to build sundials which can be used for 2 different locationson earth or sundials which 2 time systems or sundials which indicate mean time or clock time.

In general the bi-gnom sundial’s manual is: put the sundial in the right position in the sun, look wherethe shadow of the relevant gnomon intersects the date line of today and read the time.

The restrictions for the gnomons are small:1. any location of the sundial for the separate gnomons is possible (also the same location for

both gnomons)2. any orientation of the sundial plane is possible for both gnomons (also the same orientation for

both gnomons)3. any shadow casting object which generates two straight shadow lines is possible (for example

2 straight nails with random positions, or for example a cone if one looks at the left and rightshadow of the cone)

4. any time system can by chosen for the separate gnomons (local, mean, clock time, …)

No formulas are presented in this article. The formulas needed to calculate a sundial with a conegnomon which is parallel to the earth axis are presented in a separate article.

Let us discuss some nice sundials below:

1- a sundial with 2 perpendicular gnomons which can be used for daylight saving time. Thedifferent gnomons indicates simply 1 hour time difference (no Equation of time incorporatedin this design)

2- a sundial which indicates the clock time (or daylight saving time). So including the equationof time but without the analemma’s. Using 2 straight gnomons, not perpendicular to thesundial plane.

3- Sundial with a perpendicular cone gnomon4- sundial with cone gnomon which is parallel to the earth axis

The algorithm are discussed per sundial.

Bi_gnomon sundials. Hendrik Hollander


1. Sundial with 2 perpendicular gnomons which indicate 1 hour difference.

The design algorithm is straightforward- the sundial plane is fixed- time system for gnomon 1 is the local solar time including longitude correction- time system for gnomon 2 is the local solar time including longitude correction+ 1 hour- draw the shadow lines for a fixed date for both the gnomons in their own time system- mark the place where the 2 shadow lines cross for that date. Apparently, the shadow of the

different gnomons (with their own time system) will cover this point on this date.- repeat this procedure for different dates and hours.- connect the points to datelines and hour lines

Below a design for the Netherlands (lat. 52 long. 5):

The 2 gnomons are perpendicular to the sundial plane. They are east-west oriented. The hour lines aremarked with the time. Dependent which gnomon you choose, this is normal time or daylight savingtime (without the equation of time)


Check: draw 2 parallel lines over the gnomons and you will see that they differ 1 hour on a specificdate line.

This design can easily be made for a random vertical wall. When the date lines are drawn for thenightly hours, the ellipse-shaped datelines are completed, see the next drawing.

2. A sundial which indicates the clock time, without analemma’s, with 2 straight gnomons

The algorithm is straightforward- the sundial plane is fixed- time system for gnomon 1 is the clock time during the period winter to summer (Periodw-s )- time system for gnomon 2 is the clock time during the period summer to winter (Periods-w )- choose pares of dates with the same declination of the sun (1 date in the Periodw-s and for 1

date in the Periods-w )- draw the shadow lines for these dates for both the gnomons in there own time system- mark the place where the 2 shadow lines intersect for that declination. Apparently, the shadow

of the different gnomons (with their own time system) will cover this point on these dates.- Repeat this procedure for different dates and hours.- Connect the points to datelines (actually: declination lines) and hour lines

When we place a vertical rod and we draw the horizontal sundial with clock time around it, we willfind the well known horizontal sundial with the analemma’s. I have incorporated the lines for pares of


date as meant above. These date lines are calculated for a multiple of 30 degrees of the longitude ofthe sun. Often these lines are mark with the signs of the zodiac. The lines will mark 1 date in theperiod of winter to summer (Periodw-s ) and 1 date in the summer to winter (Periods-w ). See the nextfigure. The red lines are the datelines for the days that the equation of time is the same during thePeriods-w and Periodw-s . We need these lines later on.

winter line

summer line

red linesindicate thedates for whichthe equation oftime is equalduring theperiods-w andperiodw-s

periods-w is the period from Summer to Winterperiodw-s is the period from Winter to Summer

To read the time, one has to know which side of the analemma to use. We can build a bi-gnom sundialwith 1 gnomon for the Periods-w and one gnomon for the Periodw-s. Doing so, the analemma’sdisappear. I have chosen for 2 gnomons which are not perpendicular to the sundial plane. See the nextdrawing. The 2 gnomons intersect above the sundial plane, above the small circle. The 2 gnomons aremounted to the sundial plane at the small + signs. If the distance between the + signs is 2, the distanceof the intersection point of the gnomons is 0.5 above the plane. The line through the + signs is east towest.


1 gnomon for the periodes-w

1 gnomon for the periodew-s

gnomon for theperiodew-s

gnomon for theperiodes-w

The datelines are marked with the zodiac signs. Note that the declination of the sun is the same duringPeriods-w and Periodw-s for each dateline.

A photo of a paper model of this sundial is printed below:


The shape of the double gnomon is visible on the photograph. Note that this gnomon construction canindeed mark 2 different times at 1 dateline. When the shadow of the intersection point of the gnomonsis above a declination line, the shadow of each gnomon will mark a different time. This time can beassociated with the 2 dates of the declination line.

13.0014.00

13.00 14.00


When the shadow of the intersection point of the gnomon is exact on a declination line, both gnomonsindicate the same time. This has to be the case when the equation of time is equal for that sundeclination during the Periods-w and Periodw-s. In the next figure I have marked these datelines red.

red lines indicatethe declinationlines for which theequation of time isequal during theperiodw-s and theperiods-w


We can easily check whether these lines are the same for the bi-gnom sundial and the standard sundialwith the analemma’s. Both sundials are printed together in the above figure. Although a lot off linesare visible, it is clear that the red lines are the same for both sundials.


CONE’SThe sundial with the straight line gnomons as described above has some shortcomings.1. the datelines and hour lines appear mostly at 1 side of the gnomons2. therefore it is not always possible to show all hours in the sundial

To overcome this issue the 2 straight line gnomons can be replaced by a cone. The shadow of the left sideof the cone (with the sun behind you) will indicate the first time system. The shadow of the right side ofthe cone (with the sun behind you) will indicate the second time system

3. Sundial with a perpendicular cone gnomon

To get a feel for the date lines and hour lines, let’s use a cone which is perpendicular to thesundial plane. During the periodw-S the shadow of the right side of the cone is used. During theperiods-w the left side. In the figure below we see how an hour line is build up.

The algorithm:- choose a declination of the sun- find both dates on which the declination is valid- draw the shadow lines, using the valid side of the cone- mark the intersection point of the 2 shadow lines- repeat the steps above several times- connect the points to the date lines and hour lines


creation of an hour line. The shadow lines of thedifferent sides of the cone are combined

the cone with the apex abovethe red cross

A full sundial design is shown below. The intersection of the cone and the sundial plane is drawnas a blue circle. The apex of the cone is above the red + sign. When the radius of the blue circleis 1, the apex of the cone is 3 above the sundial plane. So: look at the shadow of the proper sideof the cone (the Periods-w or Periodw-s.), extrapolate the shadow line to the date line if neededand read the clock time.


the radius of the cone is 1the height is 3

Summer linedaylight saving time

winter lineclock time

Indeed all hours can be shown on the sundial. To undertand how the sundial marks two differenttimes on a specific date line, the next figure is drawn.


22 oct19 feb line

The shadow of the different sides of the cone do indicate different times. For example, the dateline of 22 october and 19 february is shown. The shadow of the right side of the cone indicates10:00 “wintertijd” on february 19th . The shadow of the left side of the cone indicates 10:15“wintertijd” on october 22th.

4. sundial with a cone gnomon which is parallel to the earth axisAs will be noticed, the hour lines are curved. To overcome this, the cone can be tilted in a waythat the central axis of the cone is parallel to the earth axis. All hour lines will become straight(well, a very small deviation as we will see later on)


12:00

14:00

Cone gnonom with axis parallel to the earth axis indicatesmean time with straight hour lines

apex of the cone is above the red + sign, intersection of thecone with the sundial plane is the blue ellipse.

The intersection of the cone with the sundial plane is an ellipse and is depicted blue. The top ofthe cone is above the red + sign. Design for lat. 52, long. 0,including equation of time. The top of the cone is 0,79 above theplane (relative to the scale in the edge of the drawing). Below apicture of a similair sundial with correction for the equation of timeand longitude correction, so it indicates clock time.

5. The equatorial sundial with a cone gnomon

The sundial with the cone gnomon can be designed as an equatorial sundial.

First we take a look at the northern part (summer part for northern lat.). See the picture below.


Northern side of the equatorial sundialwith cone gnomon,height of the cone is 1radius of the cone is 1indication: clock time

10 febr. 2006

As can be expected, the hourlines are the same as for a sundial which indicates the local solartime. However, this sundial indicates the clock time. The inner red circle is the zodiac line forthe start of the summer. The midle red circle is the next zodiac line. Unfortunately, the equinoxline is not available, it is for out this picture (although, not at infinity). So this side of the sundialwill indicate clock time from approx. 21 april to 21 august. The southern part of the sundial willtell the time from approx. 21 october to 21 february, for the same reason: the equinox date line isat “infinity”. The design for the northern part is generated for the radius 1, the top is 1 above thered + sign.


The southern part of the sundial is depicted below.

Southern side of the equatorial sundial with coneheight of the apex is 3.6radius is 1indication: clock time10 febr. 2006

Here we have some unexpected features. Since the equation of time changes rapidly, the datelines move rapidly towards the centre or outside the centre. Carefully designing the size of thecone will put the datelines (zodiac lines) together. So, moving from infinity towards the centrewe have:

- infinity: 21 september *)


- outer circle: 21 october- inner circle: 21 november- outer circle again: 21 december- inner circle again: 21 january- outer circle again: 21 february- infinity: 21 march *)

(actually the datelines are the zodiac lines, so the 21th is for reference only, actually the redcircles are the lines for multiple 30 degrees of the logitude of the sun, which is what I haveimplemented).

*)in detail: the line for spring and autumn (approx. at 21 march and 21 september) are not atinfinty. Since the equation of time is not equal on these dates, the shadow of the cone has tointersect this dateline.

When the radius is 1, the to of the cone is 3.6 above the plane.

To design the sundials with an oblique cone gnomon as user friendly as possible it is suggestedby Fred Sawyer to associate the left and right shadow of the cone in such a way with the dates ofthe year that the shadow will always intersects the current date line.

This definitely improves the concept and is incorporated in for example this paper cut out.


Some last remarks about the “straightness” of the hour lines.

The hour lines (indicating clocktime with the cone gnomon) are straight because the equation oftime is almost symmetrical with respect to the sun declination. The very small deviations areshown in the picture below.

intersection about august30th and april 12th

approx. june 20th toaug. 30th

approx. april 12th todec 20th

approx.augustus30th todec 20th

approx.april 12thtojune 20th

left shadowside of thecone

right shadowside of thecone


the well known analemma’s

the local solar time

the clock time with conegnomon

The actual shape of the equation of time is determined by the elliptic orbit of the earth and theobliquity of the ecliptic. Currently, the perihelium (the point where the earth is the closest to thesun) is at January 3rd. When the perihelium coincide with the start of a season, the equation oftime is symmetric. During the year 1246 the perihelium coincide with the start of the winter atdecember 20st and therefore the equation of time was 100% symmetrical. Building the samepicture as above for the year 1246 returns the picture below.


clock time with cone gnomonlocal solar time with standard gnomon

the analemma’s withstandard gnomon

as can be noticed, the hour lines are 100% straight.

I want to thank Fred Sawyer (president of the North American Sundial Society) for his supportand good suggestions on the algorithm (see above) and also Fer de Vries (secretary of the DutchSundial Society “de Zonnewijzerkring”) for his support and checks on the algorithms

bi gnomon sundials. - analemma zonnewijzers · let us discuss some nice sundials below: 1- a...

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