The D a i s of Shado s at a d elo the T opi
of Ca e
Di it ios S. De d i os
E e itus P ofesso , “ hool of A hite tu e a d U a Desig , U i e sit of
Ka sas, La e e, Ka sas, U“A.
I ‘eside e at O o d Bea h, Flo ida, U“A.
Co ta t:
f-jf@ea thli k. et
Ta le of Co te ts
A st a t
I t odu tio
Shado s a d the Te pe ate Zo e
A ief e ie of the lite atu e o shado s: the autho ’s o k A e ief e ie of the lite atu e o shado s: othe sou es Su a of ke poi ts ega di g o oliths’ ast-off shado s
Shado s at a d elo the T opi of Ca e o the No the He isphe e
The asi ifu atio The a ious phase t a sitio s
The goldilo ks effe t i the te pe ate zo e
Figu e . The st i t e ui ale es a o g azi uth pla e, lo k a d ale da . At all poi ts o Ea th,
the due N-“ a is, a d the due E-W a is a e as follo s: the - i azi uth a d id
ight-oo o the N-“ a is ; a d a - p dail hou s, - i azi uth i the E-W a is . Off
these poi ts, the e ui ale es ollapse fo pla es hi h a e ot e uato ial pa allel that is to the
Ea th s e uato ial pla e . A gle is the “u e “olsti e su ise azi uth at this spe ifi lo atio ,
ith -=ω. A is BE is the “u e “olsti e su ise to Wi te “olsti e su set a is; a d a is CD
is the Wi te “olsti e su ise to the “u e “olsti e su set a is. Poi t F is the due East su ise
U li ited ho izo a d pe fe tl flat g ou d, a d thei lo al o ditio s that is the adius of the i le i Figu e , a e i po ta t issues hi h ha e ee looked at so e detail i [ . ] a d [ . ]. i spe ifi , the su je t of the ho izo is iti al i the dis ussio ega di g oth the Ea
th-elated Ast o o , as ell as a site s A heolog a d A hite tu e. It is so, ostl e ause of the g ou d a o alies a d othe topog aphi al a d geog aphi al featu es tou hed upo i oth [ . ] a d [ . ] he e the su je t of fuzzi ess i shado s has ee add essed. I the latte efe e e, it as de o st ated i fa t p o ed that the e is t a i fi ite le gth i shado s fo a g o o s fi ite height. I additio , the Ea th s u atu e adds the ef a tio fa to , see [ . ] o
ef a tio , i shado le gths al ulatio s.
These a es of Figu e o tai fu da e tal e ui ale es as dis ussed i [ . ], the azi uth- lo
k-ale da e ui k-ale es. At a lo atio , the No th-“outh , - , id ight – oo
e ui ale e, a d the due East-West , - , a d the a – p e ui ale es e ist o
atte the altitude of the lo atio o the Ea th s su fa e, o the pla e of efe e e elati e to the Ea th s E uato ial Pla e, i.e. its slope o tilt elati e to that pla e. Ho e e , these e ui ale es do ot i ge e al e ist fo poi ts off these a ks, u less spe ial o ditio s hold fo the pla e of efe e e. A fi al poi t elated to these topi s: at a poi t o the Ea th s su fa e, a d o atte the g ou ds o the pla e of efe e e s slope elati e to the Ea th s E uato ial Pla e , the shado of the g o o passes th ough the e a t sa e poi t o the azi uth at a fi ed hou of the da , ut its le gth a ies depe di g o the da of the ea .
Off the poi ts he e the e is a e a t e ui ale e et ee azi uths, lo ks a d ale da s the e a e o ple elatio ships hi h o e t the th ee a ia les. These athe ati al elatio ships ill ot p eo up ou a al sis he e. It suffi es to sa that, app o i atel , a lo atio s azi uth
fo the isi g “u at “u e “olsti e ei g a o -li ea fu tio of latitude a e
app o i ated fo p a ti al pu poses, so that it a e assu ed to e a fe deg ees off that
deg ee of app o i atio ei g a fu tio of latitude the diffe e e et ee a d the
Figu e . The le gths of a g o o s shado s lo ated at the o igi poi t O at so e lo atio ith
a azi uth = -ω of the “u e “olsti e su ise fo a lo atio a o e the T opi of Ca e .
“ou e: the autho , ef. [ . ].
“hado s top e d sha p o fuzz , i.e., the -fu tio s of [ . ] see Figu e , t a e ell-shaped
ot si usoid u es, at all poi ts o the su fa e of the Ea th, No th of the N latitude
the so- alled T opi of Ca e . It ust e oted that the -fu tio s do ot fo a o ti uous
Figu e . The Celestial E uato ial Pla e CEP , the Ea th s a is of otatio , a d the T opi of Ca e . “ he ati all , the lo atio s of the Ea th at fou poi ts i the Ea th s o it a ou d the “u poi t “ a e sho , togethe ith the pa t of the Ea th s su fa e ei g illu i ated
su a s: at Wi te “olsti e ight , Ve al E ui o e te -left , “u e “olsti e left a d
Autu al E ui o e te - ight . “ou e: the autho .
At “u e “olsti e, a poi t o the Ea th s E uato splits the -hou pe iod spe di g the
-hou seg e t u de da light elo the CEP a d the othe half a o e it. At Autu al E ui o , the sa e poi t o the Ea th s E uato t a es a path si ila to that fo the Ve al E ui o e ept o e e se. Whe eas, at Wi te “olsti e, a poi t o the Ea th s E uato spe ds the -hou da light seg e t a o e the CEP, a d elo it at the -hou seg e t at ight. I the te po al
a ual d a i s, as the Ea th o its the “u , i te ediate o ditio s p e ail.
I ase at Wi te Solsti e, a d at su ise, the shado is as ptoti to the se i-a is that goes
th ough the poi t of su set at “u e “olsti e C o its azi uth spe ifi to e e lo atio , gi e
its latitude , a d the o igi O, i.e., the se i-a is CO ; he eas at su set, the o olith s shado e o es as ptoti to the se i-a is OB hi h joi s the poi t azi uth of the “u e “olsti e su ise, i.e., poi t B at that lo atio , ith the o igi O.
Fi all , i ase at Su e Solsti e, at su ise the shado is as ptoti to the se i-a is EO
hi h goes th ough the poi t he e the Wi te “olsti e su set o u s at that latitude, poi t E,
a d the o igi , O; a d at su set, the shado e o es as ptoti to se i-a is DO , hi h joi s
Shado s at a d elo the t opi of Ca e o the No the He isphe e
The asi ifu atio
I Figu e , a e ui ale t diag a to that i Figu e is sho . Although Figu e as d a so that the ualitati e featu es of shado s fo latitudes g eate tha the T opi of Ca e TC e e d a , the ualitati e featu es d a i Figu e appl to lo atio s o the Ea th s su fa e
et ee the latitude a d the E uato latitude . At the outset, it ust e oted
that, all latitudes a o e the TC th oughout the -hou pe iod of a Ea th da a e fou d a o e
the CEP. A poi t o the TC tou hes the CEP at p at Ve al E ui o , oo at “u e
“olsti e, a at Autu al E ui o , a d at id ight o Wi te “olsti e, see Figu e .
I Figu e , the o ditio s ide tif i g the a o-d a i s ithi a da , a d o e a ea of the le gths a d lo atio s of shado s at a poi t o the Ea th s E uato a e sho . At E ui o ,
Ve al o Autu al, the shado s ast off the g o o , the fu tio , fall o the due
East-West a is, the st le ei g at the o igi O at poi t A at lo al oo ti e. Up till oo ti e, the st le s shado o es f o poi t G to a ds O; he eas past oo ti e, it o es f o O to a ds
poi t F. At Wi te “olsti e, the st le s shado – fu tio - t a es a ell-shaped u e ith a
a i u lo al oo ti e i i u at poi t A . E ui ale tl , at “u e “olsti e, the i o
i age fu tio attai s a a i u i i u at lo al oo ti e at poi t A , su h that
le gths OA is e ual to OA .
Figu e .a. The te po al d a i s of shado s le gths fo poi ts o the Ea th s latitudes elo
the T opi of Ca e . The -fu tio is o as ptoti ithout a i fle tio poi t as it
Figu e . The d a i s of shado s at the Ea th s E uato . Both a d fu tio o tai i fle tio poi ts the a e ell-shaped t pe u es . “ou e: the autho .
The a ious phase t a sitio s
I p e ious pape s, deali g ith poi ts o the Ea th s su fa e a o e the T opi of Ca e , Figu e , the ge e all appli a le te po al dis o ti uit i ol i g the E ui o - elated t a sitio f o a ell-shaped u e ha gi g to a st i tl o e o e as dis ussed. No , o e ela o atio o the a ious spatial a d te po al dis o ti uities ill e offe ed, o e i g all poi ts o the Ea th s su fa e i the No the He isphe e .
Poi t of dis o ti uous eha io L as ide tified ea lie . Whe e is the the poi t L o the Ea th s
su fa e he e the phase t a sitio PT o u s, he e the fu tio fo poi ts elo
, i.e., elo TC o tai s a i fle tio poi t? It is he e poi t A * o espo ds to a
g atitude is e te ded to the as ell. “pe ial e tio ust e ade to P ofesso Te e e Meade fo his o t i utio , a d espe iall his o k o the shado s at D o eg a d “to ehe ge.
But ost i po ta t a d dee to this autho has ee the o e tha ea s of e ou age e t a d suppo t he has e ei ed f o his ife Cathe i e a d thei daughte s Daph e-I is a d Ale ia-A te is. Thei o ti ui g assista e a d u de sta di g fo all those lo g hou s he spe t o doi g
esea h, he he ould ha e allotted ti e ith the , this autho ill al a s e deepl app e iati e.
