TAMPERE
UNIVERSITY OF
TECHNOLOGY
MATHEMATICS
Department of Information Technology.
Mathematics Software Report 9
Keijo
Ruohonen
Designing Sundials by MATLAB and MAPLE
Tampere 1995
1. Introduction
The ancient art of sundial designing and making has produced an astonishing
variety of designs, many of them ingenious and works of art. The history and a
fair sample of various designs can be found in [Ro] or [Pe,Sch,SP]. The principles
of working are in many cases forgotten and understood only by a very tedious reverse-engineering, often made much more difficult by well-meant but incompetent repair work. This is the case e.g. for the famous Schissler bowl sundial,
made in 1578, see [Sa].
Nowadays investigating sundials cannot be considered an important undertaking. Nevertheless, the art is not dead. Some of the more advanced mean solar
time showing sundials are of relatively recent origin, see [Ro,Lo]. The “missing”
latitude-independent sundial was first described in [Fr]. (Any three of the four
variables altitude, declination, azimuth—related to Sun’s position in the sky—
and latitude determine the fourth, so in principle only three of them are needed
to set up a sundial.)
The goal of the present paper is to show that use of mathematics programs can
be a central tool in invigorating the art of sundial designing. The central principle here is to use throughout the usual equal-hour 24 h clock face. This has the
consequence that the shadow must be created by a rather complex surface. Generating pictures of these surfaces requires much numerical computation. The
programs Matlab® and Maple® are used here (actually much of Maple appears
as a toolbox of Matlab). As theoretical tool, vector calculus is used. It is strongly
felt that the more usual method of spherical trigonometry has had its time.
The paper is organized as follows. In Section 2 general principles for designing
and computing shadow caster surfaces are given. In Section 3 a solar time sundial design is obtained and investigated by Maple. The design is further refined
in Section 6. Then in Section 4 the equation of time, giving the difference between mean solar time and solar time, is constructed and consequently used in
Section 5 for designs of various equatorial mean solar time sundials. The surprising difficulties appearing in such designs are thoroughly discussed. The tool here
is Matlab . In Section 7 an attempt to extend the ideas of Section 5 to nonequatorial sundials is taken, without much success.
The codes listed here are stripped of comments (which appear in the text). Not
much effort is taken to optimize the codes, as they are reasonably fast and probably are not run repeatedly. It would be possible to obtain much faster codes by
integrating numerical quadrature and plot file generation, say.
®
Matlab is trademark of The Mathworks Inc.. Maple is trademark Waterloo Maple
Software.
1
2. Some Basic Constructs
The sundial, Earth and Sun are positioned originally in a xyz-coordinate system
where
(i) the origin is in the centre of Earth,
(ii) the xy-plane is the equatorial plane,
(iii) the z-axis is oriented along Earth’s axis, the north pole being on the positive z-axis,
(iv) at noon Sun is in the yz-plane,
the sundial is a circle in the xy-plane with radius 1 and centre in the origin,
(v) the time scale is the usual linear 24 hour scale and is read clockwise
around the perimeter of the circle with the 12 o’clock mark being on the
negative y-axis.
Thus the original situation is as is Figure 1. In
the sequel the time is given in radians, i.e., the
angle w corresponds to
Sun
y
w 
12
+ 1 (mod 24)
π

1
x
w
hours, and to the point (–sin w,–cos w ,0) on the
dial circle.
Figure 1
It will be allowed that the sundial may be tilted, i.e., rotated by an angle δ
around x-axis. After this operation the point corresponding to time w will be
0
0   −sin w  −sin w 
1

p(w) = 0 cos δ sin δ  −cos w =  −cos δ cos w .


 

 0 −sin δ cos δ  0   sin δ cos w 
Thus at noon the situation is as depicted in Figure 2 where σ is Sun’s declination.
z
Sun
sundial
There are many choices for the time. If Earth’s
σ
rotation angle is given by v, then the simplest
y
δ
choice is the solar time given by w = v . Other
choices are the mean solar time, to be treated
Figure 2
later, and, say, a local time, daylight savings
time or sidereal time. In these cases w depends on v and possibly some other parameter, e.g. Sun’s declination σ or (through σ) Earth’s polar angle φ in the eclip2
tic. w is then a function of v and σ or of v and φ.
As concerns measurement and definitions of the various concepts of time in general, [AA], [Se] and [HS] are recommended.
The time is indicated by a shadow cast on the perimeter of the sundial by a surface. This surface has the property that a ray from Sun to the point p(w) touches
the surface. Now, since this ray is given by
p(w) + λd(v,σ)
where
 cos v sin v 0  0   sin v cos σ
d(v ,σ) =  −sin v cos v 0  cos σ =  cos v cos σ ,


 

0
1  sin σ  sin σ 
 0
the surface has the parametric representation
r(v,σ) = p(w(v,σ)) + λ(v,σ)d(v,σ)
for a properly chosen function λ(v,σ) (actually the distance from p(w) to the point
where the ray touches the surface). The function λ(v,σ) can be found by demanding that the normal of the surface is perpendicular to d(v,σ), i.e., setting
0 = rv × rσ • d = (p′wv + λ vd + λdv) × (p′wσ + λ σd + λdσ) • d
= λwvp′ × dσ • d – λwσp′ × dv • d + λ2dv × dσ • d.
Thus
λ=
p′ × (wσ dv − wvdσ ) • d
.
dv × dσ • d
Of course, this surface is also the envelope of the one-parameter family of surfaces given by q(v,σ,λ) = p(w(v,σ)) + λd(v,σ), and λ(v,σ) can be found by setting
det (q′(v,σ,λ)) = 0 and solving for λ.
3
3. A Solar Time Sundial
Choosing w = v above one gets a solar time sundial. In this case
λ=−
p′ × dσ • d
.
dv × dσ • d
Calculating the parametric representation of the shadow caster surface by hand
is somewhat tedious, so it is done below by Maple.
> p:=[-sin(v),-cos(delta)*cos(v),sin(delta)*cos(v)]:
> d:=[sin(v)*cos(sigma),cos(v)*cos(sigma),sin(sigma)]:
> lambda:=evalm(-dotprod(crossprod(diff(p,v),diff(d,sigma)),d))/
dotprod(crossprod(diff(d,v),diff(d,sigma)),d);
lambda := ((cos(delta) sin(v) cos(sigma) - sin(delta) sin(v) cos(v) sin(sigma))
sin(v) cos(sigma)
2
- (sin(delta) sin(v) sin(sigma) + cos(v) cos(sigma)) cos(v) cos(sigma)
2
2
/
- (cos(v) sin(sigma) + cos(delta) sin(v) sin(sigma)) sin(sigma)) /
/
2
3
2
3
(- sin(v) cos(sigma) - cos(v) cos(sigma) +
2
2
(- cos(v) cos(sigma) sin(sigma) - sin(v) cos(sigma) sin(sigma))
sin(sigma))
> lambda:=simplify(");
2
2
- cos(delta) + cos(delta) cos(v) - cos(v)
lambda := - ------------------------------------------cos(sigma)
> r:=evalm(p+lambda*d);
4
r := [ - sin(v) - %1 sin(v), - cos(delta) cos(v) - %1 cos(v),
%1 sin(sigma)
sin(delta) cos(v) - ------------- ]
cos(sigma)
%1 :=
2
2
- cos(delta) + cos(delta) cos(v) - cos(v)
> map(factor,r);
[ - sin(v) (cos(v) - 1) (cos(v) + 1) (cos(delta) - 1),
3
- cos(v) (cos(delta) - 1),
- (- sin(delta) cos(v) cos(sigma) - sin(sigma) cos(delta)
2
2
+ sin(sigma) cos(delta) cos(v) - cos(v) sin(sigma))/cos(sigma) ]
Thus this surface has the parametric representation
 x = −(1 − cos δ)sin3v

3
 y = (1 − cos δ)cos v
 z = z,

i.e., it is an astroidal cylinder with maximum radius 1 – cos δ. It is noticed that
λ=
cos δ sin2v + cos2v
cos σ
is nonnegative if and only if cos δ –cot 2 v. Thus this construct is valid for
–π/2 ≤ δ ≤ π/2. Bounds for the values of z are also of interest, at least from the
point of view of practical sundial making. Now
z = sin δ cos v + (cos δ + (1 – cos δ)cos2 v)tan σ,
so that
z ≤ sin δ + tan σ.
Note that when δ = 0 the astroidal cylinder degenerates into a line, the gnomon.
In Finland, nearly all of the most commonly met solar time sundials (armillary
5
spheres, inclined edges, etc.) are of this sort*.
The author has never seen sundials of the above kind with any other choice of δ
than δ = 0. This is probably because the astroidal cylinder cannot be used as
such. Each of the four arcs of the astroid corresponds to a quarter of the face of
the sundial. Thus for the sundial to be usable at all**, these four parts must be
separated so that sunlight reaches the appropriate parts of the surfaces. These
parts can be plotted by the Maple commands
>
>
>
>
>
delta:=Pi/4:
plot3d({evalm(z*p),[r[1],r[2],z]},v=0..Pi/2,z=0..1);
plot3d({evalm(z*p),[r[1],r[2],z]},v=Pi/2..Pi,z=0..1);
plot3d({evalm(z*p),[r[1],r[2],z]},v=Pi..3*Pi/2,z=0..1);
plot3d({evalm(z*p),[r[1],r[2],z]},v=3*Pi/2..2*Pi,z=0..1);
where δ = π/4 (a value chosen for demonstration purposes only). The resulting
plots are in Figures 3–6. A good view is from above (Figures 7–10).
It is also instructive to depict sunrays touching the surface. This can be done by
the Maple commands
> ray:=[p[1]+l*d[1],p[2]+l*d[2],l=0..2]:
ray:=subs(v=k*Pi/24,ray):
rays:=seq(ray,k=0..12):
astroid:=subs(v=l,r[1]),subs(v=l,r[2]):
perimeter:=subs(v=l,p[1]),subs(v=l,p[2]):
>sigma:=0:plot({rays,[astroid,l=0..Pi/2],[perimeter,l=0..Pi/2]});
where σ = 0. The result is Figure 11.
*
A classic is the cast iron garden sundial manufactured by the Finnish metal company Valmet decades ago.
**
But see Section 6.
6
From 12 o’clock to 18 o’clock
1
0.8
0.6
-0.6
-1
0.4
-0.8
-0.4
-0.6
0.2
-0.2
0
00
-0.4
-0.2
0.2
Figure 3
7
From 18 o’clock to 24 o’clock
1
0.8
0.6
-1
0.4
0.2
-0.2
0
0
-0.8
-0.6
-0.4
-0.2
0
-0.20.2
-0.4
-0.6
0.4
0.6
Figure 4
8
From 0 o’clock to 6 o’clock
1
0.8
0.6
0.4
-0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
Figure 5
9
0.2
-0.2
0.4
-0.4
0.6
-0.6
From 6 o’clock to 12 o’clock
1
0.8
0.6
-0.6
-0.4
0.4
-0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
Figure 6
10
0.2
From 12 o’clock to 18 o’clock
Figure 7
From 18 o’clock to 24 o’clock
Figure 8
11
From 0 o’clock to 6 o’clock
Figure 9
From 6 o’clock to 12 o’clock
Figure 10
12
1.2
1
0.8
0.6
0.4
0.2
-1
-0.5
00
0.5
-0.2
-0.4
-0.6
Figure 11
Incidentally, speaking of rays, the rays given by
 −sin v 
 sin v cos σ
p(v) + λd (v , σ) =  −cos δ cos v + λ  cos v cos σ




 sin δ cos v 
 sin σ 
all go through the line segment
13
1
x = 0

 y = (1 − cos δ)cos v
 z = sin δ cos v + tan σ

in the yz-plane, corresponding to λ = 1/cos σ. This line segment can then be used
as a gnomon, changes in σ simply raise or lower it without altering its direction.
This observation leads to certain old classical analemmatic sundial constructions,
see [Ro].
The four quarters of the sundial can be translated to different positions in a variety of ways, some of them leading to quite pleasing designs. An interesting case
is δ = π/2: the sundial is then totally inside the astroidal cylinder. A horizontal
sundial is obtained by taking δ = 90° – local latitude. And a vertical sundial is
obtained by taking δ = – local latitude (this sundial can be fixed on a south facing
window pane, say).
14
4. The Equation of Time
Solar time is not the time used in everyday activity (except in certain parts of the
world). There are the different time zones and the varying velocity of Earth in its
orbit around Sun and the fact that Earth’s axis is not perpendicular to the plane
of ecliptic.
The actual dynamics of Earth in the
solar system is quite complicated. As a
first approximation it might be assumed that Kepler’s laws are accurate.
An xyz-coordinate system may then be
used where the xy-plane is the plane of
ecliptic, the point of perigeum being on
the positive x-axis and Sun is in the
origin, see Figure 1 4. The orbit of
Earth is an ellipse with eccentricity ε,
in polar coordinates
r=
y
Earth
r
φ
x
Sun
Figure 14
a(1 − ε 2 )
1 + ε cos φ
where a is the length of the major semiaxis. The dynamics is then given by the
angular velocity
1
C
C
(1 + ε cos φ)2 =
(1 + ε cos φ)2 ,
φ̇ = 2 = 2
2 2
K
r
a (1 − ε )
where C is a constant (actually C = a(1 − ε 2 )GM where M is Sun’s mass and G
is the gravitational constant), once the initial value φ(0) = 0 is given. The constant K can be obtained from the equation
1
2π
φ˙ (t)
φ˙ (t) dt
dφ
dt = K ∫
1= ∫
=K∫
.
2
2
φ˙ (t)
1
(
ε
cos
φ
)
+
1
+
ε
cos
φ
(
t
)
(
)
0
0
0
1
Denote then r = (r cos φ,r sin φ,0) and let a = (a1 ,a2 ,a3 ) be the direction vector of
Earth’s axis where a3 > 0. Solar time is obtained when a constant velocity (the
value of which is immaterial here) minus the turning velocity of the plane containing Earth’s axis and Sun is integrated. This turning velocity θ̇ is the same as
the angular velocity of the vector s = r × a, i.e.,
15
s × s˙
θ˙ =
.
s2
Now
(r × a) × (r˙ × a) = ((r × a) • a)r˙ − ((r × a) • r˙ )a = (a • r × r˙ )a = C (a • k)a = Ca3a
( r × r˙ = Ck by the conservation of angular momentum or that of areal velocity)
and
r×a
2
= r • a × ( r × a) = r • ( ( a • a) r − ( a • r ) a ) = r 2 − ( a • r ) 2 ,
so
Ca3
θ˙ = 2
=
r − (a • r ) 2
a3φ˙
r

1− a •

r
2
.
If a is written in the form
a = (cos α sin β,sin α sin β,cos β)
then
−sin σ = a •
r
= a1 cos φ + a2 sin φ = sin β cos(φ − α)
r
(which gives the relation between declination σ and polar angle φ mentioned in
Section 2 and needed later) and
θ˙ =
(cos β)φ˙
1 − sin 2β cos2 (φ − α)
.
The mean solar time is an “artificial” time corresponding to θ̇ = 2π rad/year. It
closely approximates the local time one daily uses (correction to a time zone or a
daylight savings time is not included, of course).
The difference of mean solar time and solar time is the equation of time*
*
Often the equation of time is –E.
16
φ



cos β
2πK
E = ∫
−
d
=
−
2
π
τ

∫  1 − sin2β cos2 (ϕ − α) (1 + ε cos ϕ)2  dϕ
2
2
−
−
1
sin
β
cos
φ
(
τ
)
α
(
)


t
ω
t
(cos β)φ˙ (τ)
0
where ω is chosen to make E equal to zero at a specified time t0 or polar angle. It
is possible to integrate E by hand but the result involves several branches of arctangent and is thus awkward to use.
In order to use the equation of time to map solar time to mean solar time values
of the eccentricity ε and the angles α and β as well as ω are needed. From [AA]
the following values (valid at January 1st 1995) can be read:
ε = 0.01671043
α = –12.852316° = –0.22431523 rad
(mean longitude of winter solstice)
β = 23.439942° = 0.40910416 rad
(mean obliquity of ecliptic)
ν = –3.6567940° = –0.06382310 rad
(mean New Year)
[AA] also gives the following expression for –E accurate to a few seconds of time:
–E = –106.6 sin L + 596.1 sin(2L) + 4.4 sin(3L) – 12.7 sin(4L)
– 428.8 cos L – 2.1 cos(2L) + 19.3 cos(3L)
where
L = 279.196° + 0.985647°d = 4.87289 + 0.0172028d rad
and d is the day of the year. Maple can be used to solve for a day d0 when E = 0:
> L:=4.87289+0.0172028*d:
> fsolve(-106.6*sin(L)+596.1*sin(2*L)+4.4*sin(3*L)-12.7*sin(4*L)
-428.8*cos(L)-2.1*cos(2*L)+19.3*cos(3*L),d);
-5.529862392
Thus d0 = –5.5298624.
To plot and investigate the equation of time Matlab is used because of the extensive numerical calculations needed. First, values of constants are given in Constants.m:
% Constants
17
global alpha beta omega delta epsilon zeta axdir K nu t0
alpha=-0.22431523;
beta=0.40910416;
nu=-0.06382310;
t0=-5.529862392/365.242190;
delta=0;
epsilon=0.01671043;
zeta=0;
K=1/quad('AngVel',0,2*pi);
omega=fzero('OmegaEq',nu);
axdir=[cos(alpha)*sin(beta) sin(alpha)*sin(beta) cos(beta)];
(values of ζ other than 0 will be needed in Section 6). The length of year is
365.242190 days, the tropical year of 1995. The equation giving time t (in years)
in terms of the polar angle φ is
φ (t)
t=K
hence the
∫
dφ
(1 + ε cos φ)2
ν
Matlab-functions
function y=AngVel(f)
global epsilon
y=1 ./(1+epsilon*cos(f)).^2;
and
function y=OmegaEq(a)
global t0 K nu
y=K*quad('AngVel',a,nu)+t0;
The equation of time (E vs. φ) is then given by
18
,
function y=TimEqMin(phi)
global omega zeta
y=(zeta+quad('TimEqD',omega,phi))/2/pi*24*60;
where
function y=TimEqD(phi)
global axdir K epsilon
y=axdir(3)./(1-(axdir(1)*cos(phi)+axdir(2)*sin(phi)).^2)- ...
2*pi*K./(1+epsilon*cos(phi)).^2;
The plot of this function is Figure 15 below.
15
10
5
0
-5
-10
-15
-20
0
1
2
3
4
5
6
Figure 15
The equation of time in [AA] is plotted below, first E vs. d and then E vs. φ.
19
15
10
5
0
-5
-10
-15
-20
0
50
100
150
200
250
300
350
Figure 16
15
10
5
0
-5
-10
-15
-20
0
1
2
3
Figure 17
20
4
5
6
As is seen, Figures 15 and 17 are very close. The difference is plotted in seconds
in Figure 18.
1
0
-1
-2
-3
-4
-5
-6
-7
-8
0
1
2
3
4
5
6
Figure 18
Thus the two equations of time are in agreement within about 7.5 seconds of
time. Interestingly there appears to be a systematic difference of about –3 seconds (plus other low frequency components). This is probably an artifact of forcing the equation of time to be zero near New Year at the same time as that of
[AA].
A more exact modelling of Earth’s dynamics is beyond the scope of this paper, an
interested reader is referred to the excellent presentation of this subject matter
in [Se]. The effects of other planets and Moon should be included as well as certain relativistic effects. It should also be noted that Earth’s axis precesses and
nutates. Indeed, the exact values of α and β slowly change, the formulas given in
[AA] for the year 1995 are
α = –12.852316 + 0.00004707d,
β = 23.439942 – 0.00000036d
21
where d is the day of the year. The very ecliptic slowly changes, e.g. the exact
value of ε in [AA] for 1995 is
ε = 0.01671043 – 0.0000000012d.
In the sequel the integral form of E is used, even though the approximative trigonometric form given in [AA] may be more accurate (and certainly numerically
faster). This E is within some ±10 seconds of time from the correct value, and
likely even more accurate. The accuracy is quite sufficient, it should be noted
that some famous sundial constructs were made using a much more crude equation of time, c.f. e.g. [Lo]. On the other hand, there might be an advantage in
having a formula modelling E, not just an approximate formula.
22
5. An Equatorial Mean Solar Time Sundial
The idea behind designing a mean solar time dundial is to express a ray from
Sun to the point p(w) on the perimeter of the sundial in the form
p(w) + λd(v,φ)
where φ is Earth’s polar angle, cf. Section 2. The connection between Sun’s declination σ and its polar angle φ was obtained in the previous section:
sin σ = − a1 cos φ − a2 sin φ = −sin β cos(φ − α) .
Thus σ is uniquely determined by φ, this dependence is shown in the plot of σ vs.
φ (by Maple) below.
0.4
0.2
-0.20 0
-0.4
1
2
4
3
phi
5
6
Figure 19
So
 sin v
 sin v cos σ

 

=
d(v ,φ) =  cos v cos σ
cos v

 

 sin σ 

1 − sin 2β cos2 (φ − α) 

1 − sin 2β cos2 (φ − α)  .


−sin β cos(φ − α)

Further, setting the normal of the shadow casting surface perpendicular to
d(v,φ), one gets
23
λ=
p′ × (wφ dv − wvdφ ) • d
dv × dφ • d
where
 cos v 1 − sin 2β cos2 (φ − α) 


2
2

dv = −sin v 1 − sin β cos (φ − α) 




0



sin 2β cos(φ − α)sin(φ − α) 
sin
v


1 − sin 2β cos2 (φ − α) 


sin 2β cos(φ − α)sin(φ − α) 
.
and dφ =  cos v

1 − sin 2β cos2 (φ − α) 


sin β sin(φ − α)






For a mean solar time sundial
w = v + E(φ)
and so the shadow caster surface is given by
r(v,φ) = p(v + E(φ)) + λ(v,φ)d(v,φ)
where
λ=
p′ × ( E ′dv − dφ ) • d
dv × dφ • d
and
E ′(φ) =
cos β
1 − sin 2β cos2 (ϕ − α)
−
2πK
(1 + ε cos ϕ)2
.
It should be noted first that the expanded form of r(v,φ) (even without the formulas for E and E′) is quite formidable, as the reader may verify by expanding it
with Maple. The ruled surface generated by the sunrays for any specific value of
v can however be plotted by Matlab quite speedily. For v = 0 it is the surface
given by
r0(λ,φ) = p(E(φ)) + λd(0,φ).
Figure 20 is plotted by
24
% SunRays
global alpha delta X Y Z
Constants
delta=pi/4;
lambdalow=0;
lambdaup=3;
philow=alpha;
phiup=2*pi+alpha;
nods=30;
X=zeros(nods+1);
Y=zeros(nods+1);
Z=zeros(nods+1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEq(phi);
for m=0:nods
lambda=lambdalow+m*(lambdaup-lambdalow)/nods;
r=Time(0,TimErr)+lambda*DirTan(0,phi);
X(n+1,m+1)=r(1);
Y(n+1,m+1)=r(2);
Z(n+1,m+1)=r(3);
end
end
view([1 3 1])
axis([-1 1 -1 1 -1 1])
mesh(X,Y,Z)
where
function y=Time(v,terror)
global delta
25
y=[-sin(v+terror) -cos(delta)*cos(v+terror) sin(delta)*cos(v+terror)];
and
function y=DirTan(v,phi)
global alpha beta
y=[sin(v)*sqrt(1-sin(beta)^2*cos(alpha-phi)^2) ...
cos(v)*sqrt(1-sin(beta)^2*cos(alpha-phi)^2) -sin(beta)*cos(alpha-phi)];
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
1
0.5
0
-0.5
-1
Figure 20
The cross section of the surface in Figure 20 shows the ubiquitous unsymmetric
form of the “figure of eight” or the analemmic figure. It can be read from Figure
20 that the shadow caster surface must be constructed from two parts, one for
α ≤ φ ≤ π + α (the “spring part”) and the other for π + α ≤ φ ≤ 2π + α (the “autumn
part”), otherwise any point of the surface is almost always covered from Sun by
some other part of the surface and there cannot be a shadow!
26
A serious difficulty presents itself at this point, although it is not apparent in
Figure 20: All components of dφ contain the factor sin(φ – α) and thus dφ = 0 and
dv × d φ • d = 0 when φ = α or φ = π + α. (Actually these are the only values of φ
which make dv × dφ • d equal to zero.) Thus there is a singularity when φ = α or
φ = π + α. Indeed, the shadow caster surface is unbounded and parts of it are not
realistic in that they correspond to negative values of λ. This difficulty must be
addressed and avoided as far as possible.
The purpose of this section is to design an equatorial sundial, i.e., here δ = 0.
Thus
 −sin w  −sin(v + E(φ))
p(w) =  −cos w =  −cos(v + E(φ)) and


 
0
 0  

 −cos w  −cos(v + E(φ))
p′(w) =  sin w  =  sin(v + E(φ))  .


 
0
 0  

It is immediately verified that now application of the rotation matrix
 cos ν sin ν 0
 −sin ν cos ν 0


0
1
 0
to each of the vectors p,p′,d,dv and dφ simply adds ν to the value of v. Since the
values of the triple products p′ × (E′dv – dφ) • d and dv × dφ • d are not changed
by rotations, it follows that the surface given by
r(v,φ) = p(v + E(φ)) + λ(v,φ)d(v,φ)
is a surface of revolution with z-axis as its axis of revolution.
The only way to counter the effect of the numerator of λ being zero when φ = α or
φ = π + α is to make the denominator simultaneously zero (preferably with the
same order although this is not stressed here). Now, if dφ = 0, then
p′ × (E′dv – dφ) • d = 0
if E′ = 0. However, from Figure 15 it is seen that this is not a usable option, the
values of φ making E′ zero are too far from α and π + α. On the other hand, the
vectors p′ and d v are linearly dependent when E = 0. Using this possibility is
tantamount to choosing the value of ω to satisfy E(α ) = 0 or E(π + α) = 0. This is
a viable option since the dates of winter and summer solstices are not too far
27
from zeros of the equation of time. In fact, ω can be made to slowly change with φ
to satisfy both of these equations, a construct used later.
To see how severe an effect the “blow-up” near the values φ = α and φ = π + α will
have, the surface or its profile can be plotted. Matlab is used because of the
amount of numerical computation. The profile of the shadow caster surface in the
interval α + 0.001 ≤ φ ≤ π + α – 0.001 (spring) is obtained by
% ArmillaryProfile
global alpha delta zeta
Constants
delta=0;
v=0;
zeta=0;
philow=alpha+0.001;
phiup=pi+alpha-0.001;
nods=100;
R=zeros(nods+1,1);
Z=zeros(nods+1,1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEq(phi);
TimErrD=TimEqD(phi);
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
R(n+1,1)=sqrt(r(1)^2+r(2)^2);
Z(n+1,1)=r(3);
end
plot(R,Z)
axis('equal')
hold
28
plot(-R,Z)
hold
where p′ is given by
function y=TimeD(v,terror)
global delta
y=[-cos(v+terror) cos(delta)*sin(v+terror) -sin(delta)*sin(v+terror)];
and dφ and dv by
function y=DirTanDphi(v,phi)
global alpha beta
y=[sin(v)*sin(beta)^2*cos(phi-alpha)*sin(phi-alpha)/ ...
sqrt(1-sin(beta)^2*cos(phi-alpha)^2) ...
cos(v)*sin(beta)^2*cos(phi-alpha)*sin(phi-alpha)/ ...
sqrt(1-sin(beta)^2*cos(phi-alpha)^2) sin(beta)*sin(phi-alpha)];
and
function y=DirTanDv(v,phi)
global alpha beta
y=[cos(v)*sqrt(1-sin(beta)^2*cos(phi-alpha)^2) ...
-sin(v)*sqrt(1-sin(beta)^2*cos(phi-alpha)^2) 0];
The resulting plot is in Figure 21. The “explosion” is clearly seen but it is not
very serious because the angle 0.001 radians is about 1.4 h of time. It is well possible to choose a much larger angle and still obtain a reasonably small period of
29
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Figure 21
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.4
-0.2
0
Figure 22
30
0.2
0.4
unavailability. For instance letting φ be in the interval α + 0.03 ≤ φ ≤ π + α – 0.03
gives Figure 22. There the explosion is much less pronounced and can be seen
clearly only in an enlargement (Figures 23 and 24). The angle 0.03 corresponds
to about 1.7 days, so the total time of unavailability is about a week. In Finland
this is of no consequence in winter and might be considered as acceptable.
The shape of the shadow caster surface is different in autumn. Plotting for the
values π + α + 0.001 ≤ φ ≤ 2π + α – 0.001 gives Figure 25. Again choosing the interval π + α + 0.03 ≤ φ ≤ 2π + α – 0.03 nearly disposes of the “explosive” parts
(Figure 26) (but not quite, as could be seen in enlarged pictures).
There are thus two problems: First, getting rid of the “explosive” parts of the surfaces, and second, the fact that two surfaces are needed, one for spring and the
other for autumn. Making the surfaces removable may lead to some rather cumbersome arrangements, note that both surfaces taper to a point. One possibility
is to cut the profiles in thin sheets and position them perpendicular to each other
with z-axis as the line of intersection. The proper profile is then turned to face
the sun (the other profile can be used to do this as its shadow will disappear in
the correct position).
0.42
0.4
0.38
0.36
0.34
0.32
0.3
0.28
0.26
0.24
0.22
-0.1
-0.05
0
Figure 23
31
0.05
0.1
-0.24
-0.26
-0.28
-0.3
-0.32
-0.34
-0.36
-0.38
-0.4
-0.42
-0.1
-0.05
0
0.05
0.1
0.15
Figure 24
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.6
-0.4
-0.2
0
Figure 25
32
0.2
0.4
0.6
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.4
-0.2
0
0.2
0.4
Figure 26
Plotting the two surfaces reveals the lines of constant v and φ. Since these surfaces are surfaces of revolution, the lines of constant φ are circles. It should be
noted that as φ changes, only one point in each of these circles may serve as the
touching point of a sunray giving time on the dial during a day. The lines of constant v are spirals of small torsion.
The “spring” surface (α + 0.03 ≤ φ ≤ π + α – 0.03) is plotted by
% Armillary
global alpha delta zeta X Y Z
Constants
delta=0;
zeta=0;
vlow=0;
vup=2*pi;
philow=alpha+0.03;
33
phiup=pi+alpha-0.03;
nods=30;
X=zeros(nods+1);
Y=zeros(nods+1);
Z=zeros(nods+1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEq(phi);
TimErrD=TimEqD(phi);
for m=0:nods
v=vlow+m*(vup-vlow)/nods;
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
X(n+1,m+1)=r(1);
Y(n+1,m+1)=r(2);
Z(n+1,m+1)=r(3);
end
end
mesh(X,Y,Z)
view([1 1 0.5])
axis([-0.1 0.1 -0.1 0.1 -0.5 0.5])
The result is in Figure 27. The “autumn” surface is obtained by letting φ range in
the interval π + α + 0.03 ≤ φ ≤ 2π + α – 0.03 (Figure 28). The locus of the touching
points of sunrays on the shadow casting surface is a tightly woven spiral. This
spiral can be plotted by
% ArmillarySpiral
global alpha delta zeta
Constants
34
0.5
0
-0.5
-0.1
-0.1
-0.05
-0.05
0
0
0.05
0.05
0.1 0.1
Figure 27
35
0.5
0
-0.5
-0.1
-0.1
-0.05
-0.05
0
0
0.05
0.05
0.1 0.1
Figure 28
36
delta=0;
zeta=0;
philow=alpha+0.03;
phiup=pi+alpha-0.03;
nods=1000;
X=zeros(nods+1,1);
Y=zeros(nods+1,1);
Z=zeros(nods+1,1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEq(phi);
TimErrD=TimEqD(phi);
v=K*quad('AngVel',nu,phi)*365.242192*2*pi;
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
X(n+1)=r(1);
Y(n+1)=r(2);
Z(n+1)=r(3);
end
plot3(X,Y,Z)
view([1 1 0.5])
axis([-0.1 0.1 -0.1 0.1 -0.5 0.5])
The results are in Figure 29 (spring)and Figure 30 (autumn).
As indicated above, a way to remove the effect of the “blow-up” near φ = α and
φ = π + α is to make the equation of time zero for these values of φ. This of course
sacrifices some accuracy. Now, one way to define such an equation of time is to
define
φ


2πK
τ
cos β
ESpring (φ) = (φ − α) + ∫ 
−
 dϕ
2
2
2
π
 1 − sin β cos (ϕ − α) (1 + ε cos ϕ) 
α
37
0.5
0
-0.5
-0.1
-0.1
-0.05
-0.05
0
0
0.05
0.05
0.1 0.1
Figure 29
38
0.5
0
-0.5
-0.1
-0.1
-0.05
-0.05
0
0
0.05
0.05
0.1 0.1
Figure 30
and
39
EAutumn (φ) =
φ


τ
cos β
2πK
(2π + α − φ) + ∫ 
−
 dϕ
π
 1 − sin 2β cos2 (ϕ − α) (1 + ε cos ϕ)2 
α
where
τ=−
α+π


2πK
cos β
−
dϕ = –0.01486978.

2
2
2
1
1
−
sin
β
cos
(
ϕ
−
α
)
(
+
ε
cos
ϕ
)


α
∫
Then
ESpring(α) = ESpring(π + α) = 0,
EAutumn(α + π) = EAutumn(2π + α) = 0.
The former is plotted in the interval α + 10–10 ≤ φ ≤ π + α – 10–10 by
% ArmillarySpringPr
global alpha delta tau
Constants
delta=0;
v=0;
tau=-quad('TimEqD',alpha,pi+alpha);
philow=alpha+1.e-10;
phiup=pi+alpha-1.e-10;
nods=100;
R=zeros(nods+1,1);
Z=zeros(nods+1,1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEqSpring(phi);
TimErrD=tau/pi+TimEqD(phi);
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
40
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
R(n+1,1)=sqrt(r(1)^2+r(2)^2);
Z(n+1,1)=r(3);
end
plot(R,Z)
axis('equal')
hold
plot(-R,Z)
hold
where ESpring is given by
function y=TimEqSpring(phi)
global alpha tau
y=tau/pi*(phi-alpha)+quad('TimEqD',alpha,phi);
The latter is plotted in the interval π + α + 10–10 ≤ φ ≤ 2π + α – 10–5 by
% ArmillaryAutumnPr
global alpha delta tau
Constants
delta=0;
v=0;
tau=-quad('TimEqD',alpha,pi+alpha);
philow=pi+alpha+1.e-10;
phiup=2*pi+alpha-1.e-5;
nods=100;
R=zeros(nods+1,1);
Z=zeros(nods+1,1);
for n=0:nods
41
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEqAutumn(phi);
TimErrD=-tau/pi+TimEqD(phi);
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(dial,phi)),DirTan(dial,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
R(n+1,1)=sqrt(r(1)^2+r(2)^2);
Z(n+1,1)=r(3);
end
plot(R,Z)
axis('equal')
hold
plot(-R,Z)
hold
where EAutumn is given by
function y=TimEqAutumn(phi)
global alpha tau
y=tau/pi*(2*pi+alpha-phi)+quad('TimEqD',alpha,phi);
The resulting plots are Figures 31 and 32, respectively.
42
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.4
-0.2
0
0.2
0.4
0.2
0.4
Figure 31
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.4
-0.2
0
Figure 32
43
As can be seen, these profiles are rather similar, which suggests that
EAutumn(2π + 2α – φ) ≅ –ESpring(φ) for α ≤ φ ≤ π + α.
Indeed, plotting EAutumn(2π + 2α – φ) + ESpring(φ) in seconds of time gives Figure
33 and shows that the difference is at most some 45 seconds of time.
50
40
30
20
10
0
-10
-20
-30
-40
-50
0
0.5
1
1.5
2
2.5
Figure 33
Thus, both the effect of the “blow-up” and the problem of having two shadow
caster surfaces are removed if the equation of time given by
ECommon (φ) =
1
2
( ESpring (φ) − EAutumn (2π + 2α − φ))
is used. The plot of this equation of time in minutes of time is in Figure 34. The
difference E(φ) – ECommon(φ) is plotted in Figure 35. It can be seen that there is
considerable loss of accuracy, an accuracy of some ±115 seconds of time is, however, achieved. Since
44
20
15
10
5
0
-5
-10
-15
-20
0
1
2
3
4
5
6
4
5
6
Figure 34
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
1
2
3
Figure 35
45
ESpring(φ) + ESpring(2π + 2α – φ) = EAutumn(φ) + EAutumn(2π + 2α – φ)
it follows that
ECommon(2π + 2α – φ) = –ECommon(φ),
i.e., ECommon is antisymmetric about φ = π + α, and only one shadow caster surface is needed, as told. A calculation shows that
φ

cos β
πK
πK
ECommon (φ) = ∫ 
−
−
dϕ
2
2
2
2
1
1
−
sin
β
cos
(
ϕ
−
α
)
(
+
ε
cos
ϕ
)
+
−
1
2
ε
cos
(
ϕ
α
)
(
)


α
so the numerical values of ECommon are obtained by
function y=TimEqCommon(phi)
global alpha
y=quad('TimEqCommonD',alpha,phi);
is given by
where ECommon
′
function y=TimEqCommonD(phi)
global alpha axdir K epsilon
y=axdir(3)./(1-(axdir(1)*cos(phi)+axdir(2)*sin(phi)).^2)- ...
pi*K./(1+epsilon*cos(phi)).^2-pi*K./(1+epsilon*cos(phi-2*alpha)).^2;
The profile of the shadow caster surface corresponding to ECommon can be plotted
by
% ArmillaryCommonPr
global alpha delta
Constants
delta=0;
46
v=0;
philow=alpha+1.e-5;
phiup=pi+alpha-1.e-5;
nods=100;
R=zeros(nods+1,1);
Z=zeros(nods+1,1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEqCommon(phi);
TimErrD=TimEqCommonD(phi);
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
R(n+1,1)=sqrt(r(1)^2+r(2)^2);
Z(n+1,1)=r(3);
end
plot(R,Z)
axis('equal')
hold
plot(-R,Z)
hold
The plot is in Figure 36. The surface itself can be plotted by
% ArmillaryCommon
global alpha delta
Constants
delta=0;
vlow=0;
vup=2*pi;
47
philow=alpha+1.e-10;
phiup=pi+alpha-1.e-10;
nods=30;
X=zeros(nods+1);
Y=zeros(nods+1);
Z=zeros(nods+1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEqCommon(phi);
TimErrD=TimEqCommonD(phi);
for m=0:nods
v=vlow+m*(vup-vlow)/nods;
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.4
-0.2
0
Figure 36
48
0.2
0.4
0.5
0
-0.5
-0.1
-0.1
-0.05
-0.05
0
0
0.05
0.05
0.1 0.1
Figure 37
49
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
X(n+1,m+1)=r(1);
Y(n+1,m+1)=r(2);
Z(n+1,m+1)=r(3);
end
end
mesh(X,Y,Z)
view([1 1 0.5])
axis([-0.1 0.1 -0.1 0.1 -0.5 0.5])
The resulting plot is in Figure 37.
50
6. Timeshifts
It may be necessary to show a shifted time to compensate for the difference between local time or, say, daylight savings time. Suppose a timeshift of ζ is
required. There are three methods of achieving this:
(1) Rotate the face of the sundial anticlockwise by ζ. (This works because of
the equal-hour scale used throughout.)
(2) Rotate the whole assembly (the face of the sundial + the shadow caster
surface) around the z-axis by the angle ζ.
(3) Construct a new shadow caster surface corresponding to w = v + ζ.
Methods 1 and 2 are very easily implemented, so there would not appear to be
any need for both of them or for Method 3. However, one can apply one of the
methods and then compensate the effect using the other methods. In this way a
lot of new sundial designs are obtained.
For example, using Method 2 to rotate the sundial to the desired position, the resulting error in time can be corrected by using Method 1. The unit normal of the
face of the sundial is then
0
0   0  −sin ζ sin δ
 cos ζ −sin ζ 0  1
 sin ζ cos ζ 0  0 cos δ sin δ   0 =  cos ζ sin δ  .



  
0
1  0 −sin δ cos δ  1 
cos δ 
 0
Thus an east-west sundial can be obtained by choosing δ = π/2 and ζ = π/2 (a six
hour timeshift).
In what follows, Method 3 is used (and the correction is made by Method 1, say).
First, the solar time sundial of Section 3 is considered. Maple can be used to
compute the parametric representation of the shadow caster surface, exactly as
in Section 3:
> with(linalg):
Warning: new definition for norm
Warning: new definition for trace
_
> p:=[-sin(v+zeta),-cos(delta)*cos(v+zeta),sin(delta)*cos(v+zeta)]:
_
> d:=[sin(v)*cos(sigma),cos(v)*cos(sigma),sin(sigma)]:
_
> lambda:=evalm(-dotprod(crossprod(diff(p,v),diff(d,sigma)),d))/
dotprod(crossprod(diff(d,v),diff(d,sigma)),d);
51
lambda := (- (cos(delta) sin(v + zeta) cos(sigma)
- sin(delta) sin(v + zeta) cos(v) sin(sigma)) sin(v) cos(sigma) (sin(delta) sin(v + zeta) sin(v) sin(sigma) + cos(v + zeta) cos(sigma))
cos(v) cos(sigma) - (cos(v + zeta) cos(v) sin(sigma)
/
+ cos(delta) sin(v + zeta) sin(v) sin(sigma)) sin(sigma)) / (
/
2
3
2
3
- sin(v) cos(sigma) - cos(v) cos(sigma) +
2
2
(- cos(v) cos(sigma) sin(sigma) - sin(v) cos(sigma) sin(sigma))
sin(sigma))
_
> lambda:=simplify(");
sin(v) cos(delta) sin(v + zeta) + cos(v) cos(v + zeta)
lambda := -----------------------------------------------------cos(sigma)
_
> r:=evalm(p+lambda*d);
r := [ - sin(v + zeta) + %1 sin(v),
- cos(delta) cos(v + zeta) + %1 cos(v),
%1 :=
%1 sin(sigma)
sin(delta) cos(v + zeta) + ------------- ]
cos(sigma)
sin(v) cos(delta) sin(v + zeta) + cos(v) cos(v + zeta)
The shadow caster surface has then the parametric representation
 x = −sin(v + ζ) + cos δ sin 2v sin(v + ζ) + sin v cos v cos(v + ζ)

2
 y = −cos δ cos(v + ζ) + cos δ sin v cos v sin(v + ζ) + cos v cos(v + ζ)
 z = z.

52
This is a cylindrical surface with z-axis as its axis. For instance, for δ = 0 the
cylinder is circular with radius |sin ζ|. An animation showing how the curve
(x,y) changes when ζ ranges in [0,π/2] (and δ = π/4) is obtained by
> delta:=Pi/4:with(plots):
_
> animate([r[1],r[2],v=0..2*Pi],zeta=0..Pi/2,numpoints=50,frames=20);
It shows that for a large enough |ζ| this curve is smooth and encloses a convex
area. Indeed, for ζ = π/2 this curve is as shown in Figure 38.
zeta = 6 h
0.6
0.4
0.2
-1
-0.5
00
-0.2
0.5
1
-0.4
-0.6
Figure 38
To obtain the break point value of ζ the tangent of the curve is calculated:
> diff([r[1],r[2]],v);
[- cos(v + zeta) + (cos(v) cos(delta) sin(v + zeta)
+ sin(v) cos(delta) cos(v + zeta) - sin(v) cos(v + zeta)
53
- cos(v) sin(v + zeta)) sin(v)
+ (sin(v) cos(delta) sin(v + zeta) + cos(v) cos(v + zeta)) cos(v),
cos(delta) sin(v + zeta) + (cos(v) cos(delta) sin(v + zeta)
+ sin(v) cos(delta) cos(v + zeta) - sin(v) cos(v + zeta)
- cos(v) sin(v + zeta)) cos(v)
- (sin(v) cos(delta) sin(v + zeta) + cos(v) cos(v + zeta)) sin(v)
]
_
> map(simplify,");
[- 2 cos(v + zeta) + 2 sin(v) cos(v) cos(delta) sin(v + zeta)
2
+ cos(delta) cos(v + zeta) - cos(delta) cos(v + zeta) cos(v)
2
+ 2 cos(v) cos(v + zeta) - sin(v) cos(v) sin(v + zeta),
2
2 cos(v) cos(delta) sin(v + zeta)
+ cos(v) sin(v) cos(delta) cos(v + zeta)
2
- 2 cos(v) sin(v) cos(v + zeta) - cos(v) sin(v + zeta)]
Thus
 x′(v) = sin v((cos δ − 2)sin v cos(v + ζ) + (2 cos δ − 1)cos v sin(v + ζ))

 y′(v) = cos v((cos δ − 2)sin v cos(v + ζ) + (2 cos δ − 1)cos v sin(v + ζ)).
The breakpoint is thus determined by the solvability of the equation
(cos δ – 2)sin v cos(v + ζ) + (2 cos δ –1)cos v sin(v + ζ) = 0,
that is,
54
(cos δ – 2)tan v + (2 cos δ –1)tan(v + ζ) = 0
or
(2 – cos δ)tan ζ tan2v –3(1 – cos δ)tan v + (2 cos δ – 1)tan ζ = 0.
Note that cos v = 0 or cos(v + ζ) = 0 cannot give a solution unless sin ζ = 0 (which
possibility is here omitted) or cos δ = 1/2. The breakpoint is then seen because
this equation is unsolvable if and only if
tan 2ζ >
9(1 − cos δ)2
4(2 cos δ − 1)(2 − cos δ)
zeta = 2.0652432 h
0.3
0.2
0.1
-0.4 -0.2 0 0
-0.1
0.2
0.4
-0.2
-0.3
Figure 39
when cos δ > 1/2. When cos δ ≤ 1/2 the curve is never smooth. Thus for δ = π/4 the
breakpoint is ζ = ±0.54067943 rad = ±2.0652432 h (see Figure 39).
Furthermore it is seen that
λ=
cos δ sin v sin(v + ζ) + cos v cos(v + ζ)
.
cos σ
55
When there is no timeshift any value of δ in the interval [–π/2,π/2] can be chosen
without making λ negative. For δ = 0 any value of ζ in the interval –π/2 ≤ ζ ≤ π/2
can be chosen as then λ = cos ζ cos σ ≥ 0. For δ 0 the equation
cos δ = –cot v cot(v + ζ)
defines the smallest value of δ which makes λ zero. The solutions are “somewhere near” the values v = ±π/2,±3π/2 and correspond to (local) extrema of the
function –cot v cot(v + ζ), plotted by Maple in Figure 40 for ζ = π/4.
zeta = Pi/4 rad
10
5
-3
-2
-1
00
1
v
2
3
-5
-10
Figure 40
The extrema are easily found. The minimum value is cot 2
ζ
2
which is not less
than 1 in the interval –π/2 ≤ ζ ≤ π/2. The maximum value is tan 2
proper range for δ is given by
ζ
2
cos δ ≥ tan 2 .
56
ζ
2
and so a
The conclusion is that it is possible to get a shadow caster surface which need not
be divided into separate parts, provided that the timeshift ζ can be chosen from
the interval
arctan
3(1 − cos δ)
≤ ζ ≤ 2 arctan cos δ .
2 (2 cos δ − 1)(2 − cos δ)
The trick is to use Method 3 and then correct using Method 1. The endpoints of
the interval vs. δ are depicted (by Maple) in Figure 41. Thus this figure gives the
region of proper pairs (δ,ζ).
1.6
1.4
1.2
1
zeta
0.8
0.6
0.4
0.2
-1
00
-0.5
0.5
delta
1
Figure 41
The points (±δ0 ,ζ0 ) of intersection can be found by Maple
> delta0:=fsolve(2*arctan(sqrt(cos(delta)))=arctan(3*(1-cos(delta))/2/
sqrt((2*cos(delta)-1)*(2-cos(delta)))),delta=0..2);
delta0 := 1.024537832
_
57
> zeta0:=evalf(subs(delta=delta0,2*arctan(sqrt(cos(delta)))));
zeta0 := 1.249045772
Note that δ0 is only slightly less than π/3 = 1.047197551. For –δ0 ≤ δ ≤ δ0 the
value ζ = ζ0 is always allowed.
For the equatorial sundial design in Section 5 adopting timeshifts has the effect
of aggravating the problems: The “explosive behaviour” near the values φ = α and
φ = π + α becomes more pronounced, as does the difference between the “spring”
surface and the “autumn” surface. These designs appear to have been used, however, see e.g. [Ro], and so they are treated here. Take for instance ζ = π/12
(= 1 h). This is set for Matlab in Constants.m (see Section 4). The resulting
spring surface profile (with α + 0.1 ≤ φ ≤ π + α – 0.05) can be plotted by
% GnomonProfile
global alpha delta
Constants
delta=0;
v=0;
philow=alpha+0.1;
phiup=pi+alpha-0.05;
nods=100;
R=zeros(nods+1,1);
Z=zeros(nods+1,1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEq(phi);
TimErrD=TimEqD(phi);
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
58
r=Time(v,TimErr)+distance*DirTan(v,phi);
R(n+1,1)=sqrt(r(1)^2+r(2)^2);
Z(n+1,1)=r(3);
end
plot(R,Z)
axis('equal')
hold
plot(-R,Z)
hold
The result is in Figure 42 and the autumn surface profile (with π + α + 0.05 ≤ φ ≤
2π + α – 0.1) is in Figure 43.
0.4
0.2
0
-0.2
-0.4
-0.2
0
Figure 42
59
0.2
0.4
0.2
0
-0.2
-0.4
-0.4
-0.2
0
Figure 43
The surfaces (Figures 44 and 45) can be plotted by
% Gnomon
global alpha delta
Constants
delta=0;
vlow=0;
vup=2*pi;
philow=alpha+0.1;
phiup=pi+alpha-0.05;
nods=30;
X=zeros(nods+1);
Y=zeros(nods+1);
60
0.2
0.4
Z=zeros(nods+1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEq(phi);
TimErrD=TimEqD(phi);
for m=0:nods
v=vlow+m*(vup-vlow)/nods;
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
X(n+1,m+1)=r(1);
Y(n+1,m+1)=r(2);
Z(n+1,m+1)=r(3);
end
end
mesh(X,Y,Z)
view([1 1 0.5])
axis([-0.5 0.5 -0.5 0.5 -0.5 0.5])
The time of unavailability is nearly six days in summer and nearly twelve days
in winter. This must be considered as unacceptable. One way to deal with this is
to use a solar time sundial with δ = 0 and ζ = π/12 for the periods α ± 0.1 and π +
α ± 0.05, that is, a circular cylinder of radius |sin ζ| = 0.25881905. In an otherwise quite accurate design (error within ±10 seconds of time) this however creates a large extra error
max(E(α + 0.1),E(2π + α – 0.1)) ≅ 271 s
in winter and
max(E(π + α + 0.05),E(π + α – 0.05)) ≅ 140 s
in summer. It would appear to be advisable to create a timeshift here by applying Method 1 or Method 2, but not Method 3.
61
0.5
0
-0.5
-0.5
-0.5
0
0
0.5
0.5
Figure 44
0.5
0
-0.5
-0.5
-0.5
0
0
0.5
0.5
Figure 45
62
7. A Nonequatorial Mean Solar Time Sundial
For a nonequatorial sundial δ
0. A construct such as in Section 5 may be at-
tempted but the resulting shadow caster surface is quite complicated. Altogether
eigth surfaces are needed (assuming that there is no timeshift) corresponding to
the four quarters of the sundial face and the two halves of the year. The surface
corresponding to spring and time from 12 o’clock to 18 o’clock (i.e., α + 0.1 ≤ φ ≤
π + α – 0.1 and 0 ≤ v ≤ π/2) and δ = π/4 is plotted by Matlab by
% SunDial
global alpha zeta
Constants
zeta=0;
vlow=0;
vup=pi/2;
philow=alpha+0.1;
phiup=pi+alpha-0.1;
nods=30;
X=zeros(nods+1);
Y=zeros(nods+1);
Z=zeros(nods+1);
for n=0:nods
phi=philow+n*(phiup-philow)/nods;
TimErr=TimEq(phi);
TimErrD=TimEqD(phi);
for m=0:nods
v=vlow+m*(vup-vlow)/nods;
distance=dot(cross(TimeD(v,TimErr),TimErrD*DirTanDv(v,phi)- ...
DirTanDphi(v,phi)),DirTan(v,phi))/ ...
dot(cross(DirTanDv(v,phi), ...
DirTanDphi(v,phi)),DirTan(v,phi));
r=Time(v,TimErr)+distance*DirTan(v,phi);
X(n+1,m+1)=r(1);
63
1.5
1
0.5
0
-0.5
-1
-1.5
1
1
0.5
0.5
0
0
Figure 46
1.5
1
0.5
0
-0.5
-1
1
1.5
0.5
1
0
0.5
0
-0.5
-0.5
Figure 47
64
Y(n+1,m+1)=r(2);
Z(n+1,m+1)=r(3);
end
end
mesh(X,Y,Z)
axis('equal')
The result is in Figure 46. As is seen, it is not immediate whether this surface
can be used as shadow caster surface (parts of it might be covered from Sun by
other parts). The time of unavailability is about 23 days and it is certainly too
large.
Timeshifts do not seem to help any, either: Choosing δ = π/4 and the “allowed”
timeshift (see the previous section) ζ = 1, and α + 0.2 ≤ φ ≤ π + α – 0.2, 0 ≤ v ≤ 2π,
one gets Figure 47. Only two surfaces appear to be needed but again they are
very complicated, Matlab even has some difficulties in plotting Figure 47.
Acknowledgement. The author thanks Dr. Armo Pohjavirta for the many
discussions which were very helpful and inspiring.
65
References
[AA] The Astronomical Almanac for the Year 1995. Nautical Almanac Office of
the United States Naval Observatory and Her Majesty’s Nautical Almanac
Office. Washington and London (1994)
[Fr] FREEMAN, J.G.: A Latitude-Independent Sundial. J. Roy. Astron. Soc. Can.
72 No. 2 (1978), 69–80
[HS] HACKMAN, C. & SULLIVAN, D.B.: Resource Letter: TFM-1: Time and Frequency Measurement. Am. J. Phys. 63 No. 4 (1995), 306–317
[Lo] LOSKE, L.M.: Die Sonnenuhren. Kunstwerke der Zeitmessung und ihre Geheimnisse. Springer–Verlag. Berlin (1959)
[Pe] P EITZ , A.: Sonnenuhren 2. Tabellen und Diagramme zur Berechnung.
Callwey–Verlag. Munich (1978)
[Ro] ROHR , R.R.J.: Die Sonnenuhr. Geshichte, Theorie, Funktion. Callwey–
Verlag. Munich (1982)
[Sa] S ADLER, P.M.: An Ancient Time Machine: The Dial of Ahaz. Am. J. Phys.
63 No. 3 (1995), 211–216
[Sch] S CHUMACHER, H.: Sonnenuhren 1. Gestaltung–Konstruktion–Ausführung.
Callwey–Verlag. Munich (1978)
[SP] S CHUMACHER, H. & PEITZ, A.: Sonnenuhren 3. 303 Beispiele aus 12 Ländern. Callwey–Verlag. Munich (1982)
[Se] S EIDELMAN, P.K.: Explanatory Supplement to the Astronomical Almanac.
University Science Books. Mill Valley CA. (1992)
66
ISBN
ISSN
951-722-332-3
1236-9599
Mathematics Software Reports
4.
5.
6.
7.
8.
9.
Multisilta, Jari, TDP-DPS-jakautuneiden järjestelmien simulaattori ,1990.
Piché, Robert, Pohjolainen, Seppo, Virvalo, Tapio, Simulab Model of Hydraulic
Position Servo, 1992.
Ruohonen, Keijo, Laadunvalvonnan ja tarkastusotannan suunnittelu MAPLEohjelmistolla, 1992.
Piché, Robert, MATLAB Programs for a Course in Numerical Analysis, 1994.
Pohjolainen, Seppo, Multisilta, Jari, Antchev, Kostadin, Väljas, Mati,
Hypermedia Learning Environment for Mathematics, 1994.
Ruohonen, Keijo, Designing Sundials by MATLAB and MAPLE, 1995.
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