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Developable Surfaces: Their History and Application

Authors:

Abstract

Developable surfaces form a very small subset of all possible surfaces and were for centuries studied only in passing, but the discovery of differential calculus in the seventeenth century meant that their properties could be studied in greater depth. Here we show that the generating principles of developable surfaces were also at the core of their study by Monge. In a historical context, from the beginning of the study of developable surfaces, to the contributions Monge made to the field, it can be seen that the nature of developable surfaces is closely related to the spatial intuition and treatment of space as defined by Monge through his descriptive geometry, which played a major role in developing an international language of geometrical communication for architecture and engineering. The use of developable surfaces in the architecture of Frank Gehry is mentioned, in particular in relation to his fascination with ‘movement’ and its role in architectural design.
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c
h
oo
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o
f
E
du
cation
B
at
h
S
p
a University
N
e
w
ton Par
k
B
ath, BA2 9BD, United Kin
g
dom
s
nezana@mat
h
sis
g
oo
df
oryou.com
K
eywor
d
s:
d
eve
lo
p
a
bl
e sur
f
aces,
Gaspard Mon
g
e, descriptive
g
eometry, ruled surfaces, Fran
k
Gehr
y
, Leonhard Euler
Resea
r
c
h
D
D
e
e
v
v
v
e
v
e
e
l
l
o
o
p
p
a
a
b
b
l
l
e
e
S
S
u
u
r
r
f
f
a
f
a
ff
a
ff
a
f
f
a
f
f
c
c
e
e
s
s
:
:
T
T
h
h
e
e
i
i
r
r
H
H
i
i
s
s
t
t
o
o
r
r
y
y
y
r
r
y
y
a
a
n
n
d
d
A
A
p
p
p
p
p
p
l
l
i
i
c
c
a
a
t
t
i
i
o
o
n
n
Presente
d
at Nexus 2010: Re
l
ations
h
i
p
s Between Arc
h
itecture
and Mathematics, Porto, 13-15
J
une 2010.
AA
bb
ss
tt
rr
aa
cc
tt
..
D
eve
l
opa
bl
e sur
f
aces
f
orm a
v
ery sma
ll
su
b
set o
f
a
ll
p
ossi
bl
e sur
f
aces an
d
were
fo
r
centuries stu
d
ie
d
on
l
y in
p
assing,
b
ut t
h
e
d
iscovery o
f
d
i
ff
erentia
l
ca
l
cu
l
us in t
h
e
s
eventeenth centur
y
meant tha
t
their
p
ro
p
erties could be
s
tudied in
g
reater depth. Here we show that the
g
eneratin
g
p
rinci
p
les of develo
p
able surfaces
w
ere also at the core of their
s
tu
d
y
b
y Mon
g
e. In a
h
istorica
l
context,
f
rom t
h
e
b
e
g
innin
g
of the study of develo
p
able surfaces, to the contributions
M
on
g
e made to the field, it can be seen that the nature o
f
d
eve
l
o
p
a
bl
e sur
f
aces is c
l
ose
ly
r
e
l
ate
d
to t
h
e s
p
atia
l
intuition
an
d
treatment o
f
space as
d
e
f
ine
d
b
y Monge t
h
roug
h
h
is
d
escriptive geometry, w
h
ic
h
p
l
ay
e
d
a major ro
l
e in
d
eve
l
opin
g
an internationa
l
l
anguage o
f
geometrica
l
communication
f
o
r
architecture and en
g
ineerin
g
. The use of developable surfaces
i
n the architecture of Frank Gehry is mentioned, in
p
articula
r
i
n relation to his fascination
wi
th ‘movement’ an
d
its role in
architectural desi
g
n.
I
I
n
n
t
t
r
r
o
o
d
d
u
u
c
c
t
t
i
i
o
o
n
n
Contem
p
orary use of develo
p
able surfaces and their
g
eometric manipulation with or
w
it
h
out t
h
e use o
f
computer so
f
tware
h
as
b
een increasing
l
y we
ll
d
ocumente
d
(see in
p
articu
l
ar [Liu et a
l
. 2006] an
d
t
h
e
b
i
bl
iog
r
a
p
h
y t
h
erein). T
h
e app
l
ication o
f
d
eve
l
opa
bl
e
s
urfaces is wide ran
g
in
g
– from ship-buldin
g
to manufacturin
g
of clothin
g
– as they are
s
uita
bl
e to t
h
e mo
d
e
ll
ing o
f
sur
f
aces w
h
ic
h
can
b
e ma
d
e out o
f
l
eat
h
er, paper,
f
i
b
re, an
d
s
heet metal. Some of the most bea
u
tif
u
l works
o
f modern architecture b
y
architects such
as Hans Ho
ll
ein, Fran
k
O. Ge
h
r
y
an
d
San
t
i
a
g
o Ca
l
atrava use t
h
e properties o
f
d
eve
l
opa
bl
e sur
f
aces, yet t
h
e
h
istory o
f
t
h
is ty
p
e o
f
sur
f
ace is not we
ll
k
nown; t
h
is is t
h
e
task the
p
resent
p
a
p
er is set to achieve.
D
eve
l
opa
bl
e sur
f
aces
f
orm
a
very sma
ll
su
b
set o
f
a
ll
possi
bl
e sur
f
aces:
f
or centuries
cylinders and cones were believed to be the only ones, until studies in the ei
g
hteenth
century proved that the tan
g
ent surfaces bel
o
ng
to the same mathematical family (see in
p
articu
l
ar [Eu
l
er 1772] an
d
[Monge 1780, 1785]).
T
here are therefore three ty
p
es of develo
p
ab
l
e
surfaces (excludin
g
a fourth type, the
pl
anar sur
f
ace):
1.
s
urfaces in which
g
eneratin
g
lines are t
a
ng
ents of a space curve: this type of
s
urface is spanned by a set of strai
g
ht l
i
n
es tan
g
ential to a space curve, which is
ca
ll
e
d
t
h
e e
d
ge o
f
regression (
f
ig. 1a);
2.
s
urfaces which can be described as a
g
en
e
r
alised ‘cone’ where all
g
eneratin
g
lines
run through a fixed point, the apex or
vertex of the surface (fig. 1b);
r
3
.
s
urfaces which
,
in the same manner
,
can
b
e described as a
g
eneralised cylinder,
w
here all generating lines are parallel, swept by a set of mutually parallel lines
(
f
ig. 1c).
Nexus Networ
k
Journa
l
13 (2011) 701–71
4
N
exus
N
etwork
J
ourna
l
V
ol
.
13
,
N
o
.
3
,
2011
70
1
DOI 10.1007/s0000
4
-011-0087-z;
p
u
b
lishe
d
online
2
2 Novem
b
er 2011
©
2011 Kim Wi
ll
iams Boo
k
s, Turi
n
70
2
S
nežana Lawrence
Develo
p
a
b
le Sur
f
aces: Their Histor
y
an
d
A
pp
licatio
n
I
t should be mentioned that the developable surfaces are also those which contain
e
l
ements o
f
any o
f
t
h
e a
b
ove mentione
d
g
ener
a
l
cases o
f
d
eve
l
opa
bl
e sur
f
aces, as
l
on
g
as
t
hey can be flattened onto a
p
lanar surface,
w
ithout creasing, tearing or stretching (fig.
1)
.
F
i
g
. 1. The three kinds of devel
o
p
able surfaces: a, left) tan
g
entia
l
;
b, centre) conical; c, ri
g
ht)
cy
lindrical. Curves in bold are d
i
r
ectrix or base cruves; strai
g
ht
l
ines in bol
d
a
r
e
d
irectors or
g
eneratin
g
lines (curves)
Developable surfaces are a special kind of
ruled surfaces: the
y
have a Gaussian
f
c
urvature equa
l
to 0,
1
and can be ma
pp
ed onto the
p
lane surface without distortion of
curves: an
y
curve
f
rom suc
h
a sur
f
ace
d
rawn o
n
to t
h
e
fl
at
pl
ane remains t
h
e same. In t
h
is
context, it is important to remem
b
er t
h
e
f
o
llo
wing property o
f
Gaussian curvature: i
f
t
h
e
s
ur
f
ace is su
bj
ecte
d
to an isometric tra
n
sformation (or more plainly bending), the
Gaussian curvature at any point o
f
t
h
e sur
f
ace wi
ll
remain invariant
.
2
The Ga
u
ssian
curvature is in
f
act
d
etermine
d
b
y t
h
e inner
m
etric o
f
a sur
f
ace
,
t
h
e
re
f
ore a
ll
t
h
e
l
engt
h
s
an
d
ang
l
es on t
h
e sur
f
ace remain invar
i
ant un
d
er
b
en
d
ing, a property immense
ly
i
mportant in usin
g
developable surfaces in manufacturin
g
. As a consequence, developable
s
ur
f
aces,
h
aving t
h
e Gaussian curvature equa
l
to zero, t
h
e same as p
l
ane sur
f
aces, can
b
e
obtained from unstretchable materials without fear of extendin
g
or tearin
g
, but b
y
t
rans
f
ormin
g
a p
l
ane t
h
rou
gh
f
o
ld
i
n
g
,
b
en
d
in
g
,
gl
uin
g
or ro
ll
in
g
.
The use of developable surfaces in contem
p
orary arc
h
itecture,
f
or examp
l
e in t
h
e
w
ork of Gehr
y
,
3
to which we will return later on in this
p
a
p
er, have been, more recently,
m
a
d
e possi
bl
e
b
y t
h
e
d
eve
l
opment o
f
computer so
f
tware w
h
ic
h
,
g
iven user-speci
f
ie
d
t
h
ree-
d
imensiona
l
b
oun
d
ary curves, generates a smoot
h
d
eve
l
opa
bl
e sur
f
ace.
I
n our usual world of three dimensions
,
t
he one in which we make architect
u
ral
o
b
jects, a
ll
d
eve
l
opa
bl
e sur
f
aces are ru
l
e
d
.
F
i
g
. 2. A hyperbolic paraboloid is a ruled, but not develo
p
able, surface
N
exus
N
etwork
J
ourna
l
– V
ol
V
V
.13
,
N
o. 3
,
2011 7
03
T
he r
u
le
d
s
u
rface is s
u
ch that it contains a
t
least one family of strai
g
ht lines. Ruled
s
urfaces are
g
enerated by
a
directrix
or
x
base curve
– the curve along which the ruling,
e
director, or
g
eneratin
g
curve moves. The
rulings
or
s
directors
of ruled surfaces are straight
s
l
ines. Although all developable surfaces (
i
n
three dimensions
)
are ruled, not all ruled
s
ur
f
aces are
d
eve
l
o
p
a
bl
e. For exam
pl
e, t
h
e
h
y
p
e
rb
o
l
ic
p
ara
b
o
l
oi
d
is not
d
eve
l
o
p
a
bl
e
b
ut it
i
s a ruled surface (fi
g
. 2).
T
o summarise: t
h
e
d
eve
l
opa
bl
e sur
f
aces are cy
l
in
d
ers, cones, an
d
tangent
d
eve
l
opa
bl
e
s
urfaces, or a com
p
osition of these. The ta
ng
ential developable surfaces can be bes
t
d
escri
b
e
d
an
d
vis
u
a
l
ise
d
as s
u
r
f
aces
f
orme
d
b
y moving a straig
h
t
l
ine in space (t
h
e
d
irector as exp
l
aine
d
a
b
ove) a
l
ong a
d
irectrix. I
f
you imagine a generating
l
ine
d
escri
b
in
g
a tan
g
ential surface, then a
l
l
points on that
g
eneratin
g
line share a common tan
g
en
t
pl
ane.
T
T
h
h
e
e
h
h
i
i
s
s
t
t
o
o
r
r
y
y
y
r
r
y
y
o
o
f
f
d
d
e
e
v
v
e
e
l
l
o
o
p
p
a
a
b
b
l
l
e
e
s
s
u
u
r
r
f
f
a
f
a
ff
a
ff
a
f
f
a
f
f
c
c
e
e
s
s
T
he history of develo
p
able surfaces can be traced as far back as Aristotle
(
384-322
B
.C.
)
: in
De Anima
(I, 4) he states that ‘a line by it
a
s
motion
p
roduces a surface’ [Aristotle
2
004: 409], a
l
t
h
ou
gh
t
h
is
d
oes not mean t
h
at it was Aristot
l
e w
h
o
d
e
f
ine
d
or name
d
t
h
e
r
u
l
e
d
or
d
eve
l
opa
bl
e sur
f
aces. Nevert
h
e
l
ess, t
h
e statement
h
a
d
pro
f
oun
d
in
fl
uence on
p
erceivin
g
the
g
eneration of surfaces throu
g
h movement. More than twenty centuries
later, Monge (1746-1818) used the principle of generating surfaces in a task he was given
while he worke
d
at the Mézière
s
4
as a draftsmen in the 1760s
[
Taton 1951; Sakarovitch
1
989, 1995, 1997
]
, w
h
ic
h
l
e
d
to
h
is
d
isco
v
e
ry o
f
a tec
h
nique w
h
ic
h
l
ater
g
aine
d
t
h
e
n
ame o
f
d
escriptive geometry’. We wi
ll
retur
n
to
h
is invention,
b
ut
f
irst
l
et us
l
oo
k
at
h
ow similar treatment of motion was used in practical
g
eometry before him.
5
A
f
irst examp
l
e is Wi
ll
iam Hawney, a minor
e
ig
h
teent
h
-century aut
h
or on surveying,
who in 1717 described the cylinder as a surfac
e ‘ro
ll
e
d
over a p
l
ane so t
h
at a
ll
its points
are brou
g
ht into coincidence with the plane’.
6
In 1737 Amédée Fran
ç
ois Frézier (1682
-
1
773) a
l
so consi
d
ere
d
t
h
e ro
ll
in
g
o
f
t
h
e p
l
a
n
e to
f
orm a circu
l
ar c
yl
in
d
er an
d
cone (see
[Frézier 1737-39]),
b
ut
h
e
d
i
d
not genera
l
is
e
on t
h
e mat
h
ematica
l
properties o
f
t
h
is
p
rocess, nor did he distin
g
uish between developable and ruled surfaces.
Al
most
h
a
lf
a century
l
ater, Eu
l
er (1707-1783) an
d
Monge
b
ecame intereste
d
in
develo
p
able surfaces and used differential calculus to study their
p
ro
p
erties. Only in 1886
h
owever, was the term “differential
g
eometry” coined. It was used for the first time by
I
ta
l
ian mat
h
ematician Luigi Bianc
h
i in
h
is text
b
oo
k
Lezioni di geometria differenziale
(Pisa, 1886). T
h
e investigati
o
n
s o
f
Eu
l
er an
d
Monge t
h
ere
f
o
r
e
prece
d
e
d
t
h
e
b
eginning o
f
a stu
d
y o
f
d
i
ff
erentia
l
geometry, an
d
initiate
d
i
nvestigations in t
h
e
f
ie
ld
o
f
d
eve
l
opa
bl
e
su
rfaces.
B
e
f
ore we get t
h
ere, a remin
d
er nee
d
s to
b
e ma
d
e a
b
out t
h
e nature o
f
t
h
e stu
d
y o
f
change through differential calculus. In the
s
eventeenth century, inde
p
endently of each
ot
h
er
,
7
Isaac Newton
(
1643-1727
)
and Gottfried Leibniz
(
1646-1716
)
discovered
ca
l
cu
l
us, w
h
ic
h
d
ea
l
s wit
h
t
h
e stu
d
y o
f
c
h
ange. Newton use
d
ca
l
cu
l
us to
d
etermine an
expression
f
or t
h
e curvature o
f
p
l
ane curve
s
. As t
h
e stu
d
y o
f
tangentia
l
sur
f
aces is
m
entioned throu
g
hout this paper, it is worth notin
g
the comparison of the study o
f
curves in two
d
imensions t
h
roug
h
d
i
ff
erentia
l
c
a
l
cu
l
us. For examp
l
e,
f
in
d
ing a tangent to
a curve at any point involves seekin
g
the first d
e
r
ivative of that c
u
rve
a
nd usin
g
this result
to find an equation of a tangent at
a
p
articular
p
oint to the curve.
t
70
4
S
nežana Lawrence
Develo
p
a
b
le Sur
f
aces: Their Histor
y
an
d
A
pp
licatio
n
E
u
l
er, w
h
o was
by
t
h
at time
bl
in
d
, w
r
ote a
p
a
p
er entit
l
e
d
“De so
l
i
d
is
q
uorum
s
uperficiem in planum explicare licet” (On s
o
l
i
d
s whose s
u
rface can be
u
nfol
d
e
d
onto
a
pl
ane
,
E
4
19
)
[1772] in which his perception of space is clearly shown to be that of
id
enti
f
ying an
d
d
escri
b
ing sur
f
aces as
b
oun
d
aries o
f
so
l
i
d
s, not as co
ll
ection o
f
so
l
i
d
s
.
8
Eu
l
er’s a
pp
roac
h
inc
l
u
d
es
d
i
ff
erentia
l
tre
a
tment com
b
ine
d
wit
h
g
eometrica
l
. First
h
e
l
ooks at any surface and assumes that a lim
i
ting value from this surface will be its
derivative: this he obtains throu
g
h selectin
g
a
n
infinitesimally small ri
g
ht trian
g
le on the
s
ur
f
ace an
d
see
k
ing its re
l
ations
h
ip to a congr
u
ent triang
l
e in a p
l
ane. At t
h
is point Eu
l
er
i
s
g
eneralisin
g
a problem via a system of d
i
f
ferential e
q
uations, but he then
p
roceeds to
s
ee
k
its so
l
ution via
g
eometrica
l
treatme
n
t. E
ul
er
d
raws t
h
ree
l
ines on a s
h
eet an
d
d
escri
b
es t
h
em simp
l
y as
Aa
,
Bb
,
C
c(
f
i
g
. 3).
F
i
g
. 3. Euler’s
g
eneralisation be
gi
n
s from three strai
g
ht lines on a sheet of paper, and the bendin
g
of the paper alon
g
those lines
H
e then proceeds to investi
g
ate what happens when this sheet is bent alon
g
a strai
g
ht
l
ine;
h
owever t
h
at
h
a
pp
ens, it is a
l
ways
p
ossi
bl
e to conceive o
f
a so
l
i
d
w
h
ic
h
f
its t
h
at
b
en
t
sh
eet. From t
h
is it
f
o
ll
ows, Eu
l
er conc
l
u
d
es t
h
at,
b
esi
d
es prismatic an
d
pyrami
d
a
l
b
o
d
ies,
t
here are an
y
number of other kinds of bodies which ma
y
be covered in this manner b
y
s
uc
h
a s
h
eet, an
d
w
h
ose sur
f
aces may
a
c
cor
d
ing
l
y
b
e un
f
o
ld
e
d
upon a p
l
ane:
L
et us now increase to in
f
inity t
h
ose
l
ines
Aa
,
Bb
,
C
c
,
etc. so t
h
at ou
r
s
olid ac
q
uires a surface everywher
e
curved, as our
p
roblem demands
accordin
g
to the law of continuity. A
n
d now it a
pp
ears at once, that the
s
ur
f
ace o
f
suc
h
b
o
d
ies s
h
ou
ld
b
e so c
o
n
stitute
d
t
h
at
f
rom any point in it at
l
east one strai
g
ht line may be drawn which lies wholly on this surface;
a
l
t
h
oug
h
t
h
is con
d
ition a
l
one
d
oes
n
ot ex
h
aust t
h
e requirements o
f
our
p
roblem, for it is necessary also that any two proximate strai
g
ht lines lie in
the same plane and therefore meet unle
s
s they are
p
arallel (translation b
y
Fl
orian Cajori [1929]).
9
Th
e enormit
y
o
f
Eu
l
er’s contri
b
ution in t
h
i
s res
p
ect is sim
pl
e to esta
bl
is
h
. He
exten
d
e
d
t
h
e stu
d
y o
f
sur
f
aces to
d
eve
l
opa
bl
e
s
ur
f
aces
b
y posing a simp
l
e question a
b
ou
t
w
w
h
h
h
w
h
i
i
c
c
h
h
k
k
i
i
n
n
d
d
o
o
f
f
s
s
u
u
r
r
f
f
a
f
a
ff
a
ff
a
f
f
a
f
f
c
c
e
e
s
s
c
c
a
a
n
n
b
b
e
e
d
d
e
e
v
v
v
e
v
e
e
l
l
o
o
p
p
e
e
d
d
i
i
n
n
t
t
o
o
a
a
p
p
l
l
a
a
n
n
e
e
:
Notissima est proprietas cylindri et co
ni, qua neorum superficiem in planum
explicare licet atque adeo haec proprietas
ad omnia corpora cylindrical et conica
extenditur, quorum bases figuram habeant quamcunque; contra vero sphaera hac
proprietate destituitur, quum eius supe
rficies nullo modo in planum explicari
neque superficie plana obduci queat; ex
quo nascitur quaestio aeque curiosa ac
N
exus
N
etwork
J
ourna
l
– V
ol
V
V
.13
,
N
o. 3
,
2011 70
5
notatu digna, vtrum praeter conos et cy
lindros alia quoque corporum genera
existant, quorum superficiem itidem in pl
anum explicare liceat nec ne? quam ob
rem in hac differtatione sequens considerare constitui Problema:
I
nuenire aequationem genera
l
em pro omni
b
us so
l
i
d
is, quorum super
f
iciem in
planum explicare licet,
cuius solutionem variis modis sum agressurus.
[Eu
l
er 1772: 3].
E
u
l
er procee
d
s to
l
oo
k
at t
h
e intersections
of
t
h
e pairs o
f
l
ines: i
f
connecte
d
t
h
ese
i
ntersections t
h
emse
l
ves wou
ld
f
orm a twiste
d
s
u
rf
ace o
f
d
ou
bl
e curvature, a sur
f
ace, as it
w
ere, of a ‘hi
g
her’ de
g
ree than the one fro
m
which we started (of sin
g
le curvature,
a
d
eve
l
opa
bl
e sur
f
ace). Eu
l
er t
h
en esta
bl
is
h
e
d
t
h
e ana
l
ytica
l
re
l
ations
h
ip
b
etween t
h
is
twisted curve and the
p
oints on the develo
pa
b
le surface. His ex
p
osition was based on
t
h
ree aspects o
f
t
h
e stu
d
y o
f
suc
h
sur
f
aces: t
h
e stu
d
y o
f
d
eve
l
opa
bl
e sur
f
aces
b
y t
h
e means
o
f
ana
l
ytica
l
princip
l
es; t
h
eir stu
d
y
b
y t
h
e m
e
ans o
f
geometrica
l
princip
l
es; an
d
f
ina
ll
y t
h
e
s
tudy in which the second is a
pp
lied to the study
of the first. It is interesting to note that,
a
l
t
h
oug
h
Eu
l
er’s opus inc
l
u
d
es some 832 origina
l
wor
k
s,
10
nevertheless this paper of
E
u
l
ier is consi
d
ere
d
b
y some to
b
e
h
is very
b
est mat
h
ematica
l
p
iece.
11
M
M
o
o
n
n
g
g
e
e
s
s
s
s
t
t
u
u
d
d
y
y
y
d
y
o
o
f
f
d
d
e
e
v
v
v
e
v
e
e
l
l
o
o
p
p
a
a
b
b
l
l
e
e
s
s
u
u
r
r
f
f
a
f
a
ff
a
ff
a
f
f
a
f
f
c
c
e
e
s
s
a
a
n
n
d
d
t
t
h
h
e
e
i
i
n
n
v
v
e
e
n
n
t
t
i
i
o
o
n
n
o
o
f
f
D
D
e
e
s
s
c
c
r
r
i
i
p
p
t
t
i
i
v
v
e
e
G
G
e
e
o
o
m
m
e
e
t
t
r
r
y
y
y
r
r
y
y
I
ndependently from Euler, and at almost the same time, a youn
g
drau
g
htsman
g
l
impse
d
a new way o
f
imagining spatia
l
re
la
t
ions t
h
roug
h
princip
l
es o
f
generation o
f
geometrica
l
entities
b
y movement. Monge t
h
e
r
e
f
ore stu
d
ie
d
d
eve
l
opa
bl
e sur
f
aces,
h
avin
g
s
tarted from an entirely different view
p
oint
.
In
,
or around
,
1764
,
a
nd while workin
g
at
the l’
É
cole Royale du Génie de Mézières, Monge was given the task of determining the
n
ecessary hei
g
ht of an outer wall in a desi
g
n of fortification (fi
g
. 4).
Fig. 4. Monge’s work on determining the height of
the fortification wall incl
ud
e
d
constr
u
ction o
f
f
develo
p
able surfaces
70
6
S
nežana Lawrence
Develo
p
a
b
le Sur
f
aces: Their Histor
y
an
d
A
pp
licatio
n
Up
to t
h
at time, t
h
ere were two met
h
o
d
s most wi
d
e
l
y use
d
f
or t
h
is
p
ro
bl
em. One
i
nvolved determining numerous view poi
n
t
s from the terrain; the triangles thus
determined by the view point, a point of the edge of the fortification, and the height o
f
t
h
e wa
ll
su
ff
icient to o
ff
er e
ff
ective
p
rotectio
n
,
were a
ll
necessary measurements; a
l
en
g
t
hy
em
p
irica
l
p
rocess was invo
l
ve
d
. T
h
e ot
h
er met
h
o
d
, w
h
ic
h
was em
pl
oye
d
in t
h
e sc
h
oo
l
in
M
ézières, was based on lon
g
calculations w
i
t
h the hei
g
hts of each crucial point bein
g
m
easured directl
y
on the terrain and noted on a plan. Instead of using these methods,
wh
ic
h
wou
ld
h
ave ta
k
en a wee
k
to yie
ld
t
h
e
f
ina
l
resu
l
t, Monge
f
oun
d
a way to
f
inis
h
h
is
w
ork in two days. Mon
g
e conceived of a new technique, which he called
Géometrie
Descriptive
and which was naturall
y
related to the methods widel
y
used at the time. The
m
et
h
o
d
was examine
d
wit
h
t
h
e
h
i
ghl
y criti
c
a
l
eye o
f
h
is su
p
erinten
d
ents w
h
o
h
a
d
t
h
eir
s
uspicions a
b
out Monge’s spee
d
in reso
l
vin
g
th
e pro
bl
em; w
h
en proper
l
y un
d
erstoo
d
, it
w
as ruled a militar
y
secret.
T
o so
l
ve t
h
e pro
bl
em
h
e was
f
ace
d
wit
h
, Monge
d
etermine
d
a p
l
ane tangent to t
h
e
terrain. T
h
is p
l
ane is
d
etermine
d
b
y a point an
d
a
l
ine: poin
t
A
l
ies in t
h
e groun
d
p
l
ane
of the
p
lan of fortification (fortification desi
g
ns were usually predetermined and their
p
lans were strictly
g
eometrical), line
q
is drawn as a
p
er
p
endicular to line p
(
p
((
is a tan
g
ent
to
t
), which is the contour line of the terrain).
t
M
on
g
e further considered conical su
r
faces which
u
se
d
lines s
u
ch as pas
g
eneratrixes
an
d
p
oints suc
h
as
A
as centres to determine the height of the fortification (the thickened
A
l
ine perpendicular to the plane in which the fortification plan rests).
Once it
b
ecame a mi
l
itary secret, Descriptive Geometry saw
l
itt
l
e
l
ig
h
t unti
l
t
h
e
r
eform of the educational s
y
stem of Franc
e
,
which took place durin
g
the Revolution.
H
owever, Monge’s
f
irst pu
bl
ications were on
d
eve
l
opa
bl
e sur
f
aces: t
h
ese came years
b
efore his technique of descriptive
g
eo
m
e
try could be made
p
ublic. His
p
a
p
er,
Mémoire
sur les développées, les rayons de courbure et les différents genres d’in
À
n
n
exions des courbes
À
à double courbure
[Monge 1785; written in 1771, but published 1785] gave his theory
e
o
f
d
eve
l
o
p
a
bl
e sur
f
aces in an a
b
stract
a
n
d
p
ure
l
y mat
h
ematica
l
manner. T
h
e
p
a
p
er
“contains a
b
roa
d
exposition o
f
t
h
e w
h
o
l
e
di
ff
erentia
l
geometry o
f
space curves” [Strui
k
1
933: 105], introducin
g
the rectifyin
g
developable, and describes such crucial terms in
th
e stu
d
y o
f
d
eve
l
opa
bl
e sur
f
aces as norma
l
pl
ane, ra
d
ius o
f
f
irst curvature, an
d
t
h
e
osculatin
g
sphere [Reich 2007].
I
t is now known that Mon
g
e read Euler’s paper (E419) only after this, his own (and
f
irst) pu
bl
ication on
d
eve
l
opa
bl
e sur
f
aces, w
h
i
c
h
ma
d
e
h
im more intereste
d
in t
h
e su
b
jec
t
and u
p
on which he wrote his second
p
a
p
er
Mémoire sur les propriétés de plusieurs genres
de surfaces courbes, particulièrtement sur
celles des surfaces d´eveloppables, avec une
application à la theorie des ombres et des pénombres
[Monge 1780]. In this paper
s
M
on
g
e esta
bl
is
h
es
h
is simp
l
i
f
ication o
f
Eu
l
er’s
f
in
d
in
g
s
a memoir o
f
Mr Eu
l
er … on
d
eve
l
opa
bl
e
sur
f
aces … in w
h
ic
h
t
h
at i
ll
ustrious
Geometer
g
ave the formulas for reco
g
nizin
g
whether or not a
g
iven curved surface
has the property of being able to be mapped to a plane, … I arrived at some
r
esults which seem much sim
p
ler to me and easier to use for the same
p
ur
p
ose
.
12
T
hese ‘simpler’ results are summarized in h
i
s
definition of a developable surface as
one which is “flexible and inextensible, one ma
y conceive of mapping it onto a plane, …
aa
s
o that the way in which it rests on the
pl
ane is without du
p
lication or disru
p
tion of
continuity
[Monge 1780: 383].
N
exus
N
etwork
J
ourna
l
– V
ol
V
V
.13
,
N
o. 3
,
2011 7
07
I
n this, his second paper on the topic, Mon
g
e begins with a curve of double curvature
(a twiste
d
curve –
f
or examp
l
e, a
h
e
l
ix) an
d
d
e
f
ining a point on it. T
h
roug
h
t
h
is point
h
e
draws a
p
lane which is
p
er
p
endicular to a line tan
g
ent to the curve at this point; he does
the same with another point, tangent and p
l
ane that go perpendicularly through this
tangent. The two planes intersect in a line. If
one imagines a curve which goes around the
f
ori
g
inal surface (of double curvature), takes con
s
ecutive
p
oints on it and does the same as
a
b
ove (i.e.,
d
raws t
h
rou
gh
eac
h
point a t
a
ng
ent, an
d
t
h
rou
gh
eac
h
tan
g
ent a norma
l
pl
ane), t
h
en
f
in
d
s t
h
e intersections o
f
t
h
ese
pl
anes, a
d
eve
l
opa
bl
e sur
f
ace wi
ll
h
ave t
h
us
b
een constructed. The
g
eneralisation to w
h
i
ch Mon
g
e arrived at was that, in fact, an
y
curvature of double surface can be enveloped by
a space curve; this tangent curve can be
y
u
sed, as explained, to
g
enerate a devel
o
p
able surface. Mon
g
e’s spatial and Euler’s
analytical insi
g
hts, proved that the tan
g
ent su
r
faces belon
g
to the same family as conical
an
d
cy
l
in
d
rica
l
ones.
F
urther, develo
p
able surfaces are, as will be shown, an inte
g
ral part of the Mon
g
ean
treatment o
f
space. His
d
escriptive geometry w
a
s
b
orn out o
f
an insig
h
t into a possi
b
i
l
it
y
of constructin
g
an ima
g
inary tan
g
ential pla
n
e
to a terrain in or
d
er to solve the real
p
ro
bl
em o
f
f
orti
f
ication
d
esi
g
n. But it was not on
l
y t
h
at
h
is
d
escri
p
tion o
f
t
h
e tec
h
ni
q
ue
i
tse
lf
is a stu
d
y in
d
eve
l
opa
bl
e sur
f
aces; it was t
h
at in
f
act, a
ll
o
b
jects imagine
d
t
h
roug
h
the use of descriptive geometry as given by
Monge are ruled surfaces (and most are
y
d
eve
l
opa
bl
e). Monge
d
escri
b
es a
ll
geometric
a
l
o
b
jects t
h
roug
h
generation: a point is
a
g
eneratrix of a line; similarly any plane is
ge
n
erated by two lines. In descriptive
g
eometr
y
the
p
osition of any element is determined by
i
ts position with respect to the pro
j
ection
planes, of which there are two (for simplicity of
explanation, the horizontal and vertical).
f
T
he
g
eneration of a plane surface could be described by the lines in which the plane in
question intersects t
h
e projection p
l
anes. T
h
ese two
l
ines
d
etermine t
h
e p
l
ane in
f
u
ll
an
d
are called the traces of the
p
lane.
I
n or
d
er to arrive at a
full
er
u
n
d
erstan
d
i
n
g
o
f
Mon
g
e’s conception o
f
space we wi
ll
b
ring one examp
l
e as
f
o
ll
ows. T
h
e traces o
f
t
h
e
p
l
ane on t
h
e projection p
l
anes may a
l
so
b
e re
g
arded as the lines that
g
enerate the plane. It should be now easy to see that those
two traces meet on t
h
e strai
gh
t
l
ine in w
h
ic
h
th
e p
l
anes o
f
pro
j
ection meet eac
h
ot
h
er. In
M
onge’s
d
rawing (
f
ig. 5) t
h
is means t
h
at i
n
t
h
e p
l
ane, w
h
ic
h
h
as
b
een name
d
f
rom its
traces as BAb, those traces, namel
y
the line AB and Ab, meet on the line AC=LM, which
r
epresents t
h
e intersecting
li
n
e o
f
two projection p
l
anes.
F
i
g
. 5. Plate III from the 1811 Hachette
é
dition of Mon
g
e’s
Géométrie Descriptive
s
howin
g
the primary operations with two
p
lanes. The system, alt
h
ou
g
h determined by
two planes of pro
j
ection,
i
s actually re
p
resented
onl
y
b
y
their line o
f
i
ntersections
,
AC
=LM on
th
is
d
ia
g
ram
70
8
S
nežana Lawrence
Develo
p
able Sur
f
aces: Their Histor
y
and A
pp
licatio
n
If
we were to consi
d
er a secon
d
pl
ane (t
h
is
pl
ane is
l
a
b
e
ll
e
d
DC
d
) we wou
ld
b
e a
bl
e
to see how this plane meets the first plane,
a
nd their intersection line is EF
,
the first
p
rojection of which is given as Ef and second as eF. We may bear in mind that this is the
p
rocess t
h
at is use
d
in t
h
e construction o
f
any
s
o
l
i
d
b
o
d
y wit
h
pl
ane sur
f
aces
f
or its si
d
es.
T
wo important things that explain, in basic
t
erms, the principles of the technique, are
to
b
e
l
earnt
f
rom t
h
is exam
pl
e. First, e
v
ery t
h
ree-
d
imensiona
l
g
eometrica
l
b
o
d
y can
b
e
d
escri
b
e
d
as a sum o
f
p
l
anes w
h
ic
h
are
g
ener
a
te
d
by
some mo
d
e o
f
motion an
d
intersect
eac
h
ot
h
er. T
h
is means t
h
at t
h
e so
l
i
d
is not
r
egar
d
e
d
as an in
d
epen
d
ent entity in itse
lf
,
b
ut as a
p
roduct of motion and intersectio
n
of its
p
rimary elements. Had we further
e
l
a
b
orate
d
upon t
h
e a
b
ove examp
l
e, an
d
sai
d
t
h
at,
f
or examp
l
e, t
h
e two p
l
anes w
h
ic
h
i
ntersect should be ortho
g
onal to each oth
e
r,
and if we had con
t
i
nued by addin
g
fou
r
f
urt
h
er p
l
anes in a simi
l
ar manner so t
h
at eac
h
o
f
t
h
em is ort
h
o
g
ona
l
to t
h
e one i
t
i
ntersects, an
d
t
h
at t
h
ey s
h
ou
ld
b
e on t
h
e sam
e
d
istance to t
h
e para
ll
e
l
ones on a
ll
si
d
es,
w
e would
g
enerate a cube. In this way the
p
erception of all
g
eometrical entities is
c
h
an
g
e
d
f
rom a co
ll
ection o
f
various
f
orms, to a co
ll
ection o
f
various met
h
o
d
s an
d
p
rocesses by which those forms are bein
g
g
enerated.
M
on
g
e realised that, althou
g
h descriptive
g
eometry had a very practical application in
s
tone-cutting, engineering, an
d
arc
h
itectura
l
d
rawing, t
h
is
t
reatment o
f
space
h
a
d
a
l
so
i
m
p
lications for the study of the
p
ro
p
erties
o
f curves. To Mon
g
e, developable surfaces
w
ere not just mat
h
ematica
l
curiosities w
h
ic
h
arose out o
f
h
is spatia
l
insig
h
t un
d
erpinne
d
b
y t
h
e tec
h
nique o
f
d
escriptive geometry w
h
ic
h
h
e per
f
ecte
d
. In
f
act, t
h
e stu
d
y o
f
sur
f
aces
w
as an inte
g
ral part of his interest i
n
stonecuttin
g
; in the first edition o
f
Descriptive
Geometry
he describes, for example, how the surfaces of the voussoir joints must be
y
d
eve
l
o
p
a
bl
e
…T
h
is is a practica
l
consi
d
eration: i
f
suc
h
a sur
f
ace were not a ru
l
e
d
s
urface
,
it would not be executable s
u
f
ficiently ra
p
idly to be economically
via
bl
e or wit
h
su
ff
icient precision t
o
ensure proper contact
b
etween t
h
e
vo
u
ssoirs. The fact that the s
u
rface
i
s also developable allows
g
reater
p
recision with the use of
p
anels f
o
r
tracin
g
[Mon
g
e 1795b, quoted in
S
a
k
arovitc
h
2009: 1295].
F
i
g
. 6. One of Mon
g
e’s main
w
orks on develo
p
able surfaces,
Feuille d’Analyse
[
1795a
]
B
ut bein
g
practical, however useful, was not the
m
ain achievement of Monge
.
The visual insight and
a
b
i
l
ity to manipu
l
ate o
b
j
e
c
ts
i
n an
i
mag
i
nary space
with his techni
q
ue, meant
t
hat his res
u
lts in three-
d
imensiona
l
d
i
ff
erentia
l
geometry were superior to
those of Euler. The two
p
a
p
ers mentioned earlier
s
ummarising his results were followed by his
Feuilles
d’analyse appliquée à la géométrie
[1795a] (fi
g
. 6).
Th
is
b
oo
k
o
ff
ere
d
a s
y
stematic treatment an
d
t
rans
l
ate
d
properties o
f
curves an
d
sur
f
aces into
a
nalytical lan
g
ua
g
e which could be manipulated
th
roug
h
partia
l
d
i
ff
erentia
l
equations. In t
h
e wor
d
s o
f
M
orris Kline, “Mon
g
e reco
g
nised that a family of
s
urfaces havin
g
a common
g
eometric property or
d
erive
d
b
y t
h
e same met
h
o
d
o
f
generation s
h
ou
ld
s
atisfy a
p
artial differential e
q
uation” [1990: II, 566].
N
exus
N
etwork
J
ourna
l
– V
ol
V
V
.13
,
N
o. 3
,
2011 7
09
M
onge’s treatment of developable surfaces
i
s, as can be glimpsed from these shor
t
insights into his work on the topic, entirely
d
i
ff
erent
f
rom t
h
at o
f
E
ul
er. W
h
i
l
e E
ul
er’s
y
treatment was ‘of a
p
rofoundly analytic s
p
i
rit’ [Taton 1951: 21], Mon
g
e had keen
geometrical intuition which manifested itself
most profoundly in his conception o
f
f
d
escriptive
g
eometry, an
d
w
h
ic
h
ena
bl
e
d
h
im t
o
app
l
y ana
l
ysis to
g
eometry, rat
h
er t
h
an
consider the two disci
p
lines as se
p
arate ways of in
q
uiry.
T
T
h
h
e
e
c
c
o
o
l
l
l
l
a
a
b
b
o
o
r
r
a
a
t
t
i
i
v
v
e
e
f
f
r
ff
r
f
f
r
f
r
f
f
a
a
t
t
e
e
r
r
n
n
i
i
t
t
y
y
y
t
y
y
o
o
f
f
M
M
o
o
n
n
g
g
e
e
a
a
n
n
d
d
G
G
e
e
h
h
r
r
y
y
y
r
r
y
y
S
e
p
arated by centuries and different
p
rof
e
s
sions, it may seem that Mon
g
e and Gehr
y
m
ay not
h
ave anyt
h
in
g
in common apart
f
rom
a
n interest in
d
eve
l
o
p
a
bl
e sur
f
aces. An
d
p
er
h
aps one wou
ld
b
e rig
h
t to
d
isregar
d
an
y
f
urt
h
er ana
l
ysis o
f
t
h
eir common interest.
L
et us however
,
entertain an idea that bot
h
,
in their own times and in their own wa
y
s,
w
ere intereste
d
in t
h
ese sur
f
aces not
b
y c
h
ance,
b
ut
b
ecause t
h
ey em
b
o
d
ie
d
certain i
d
eas
about motion that had a
p
articular slant on creative
p
rocesses that resonated with bot
h
M
on
g
e an
d
Ge
h
ry. In Mon
g
e’s case t
h
is mani
f
es
t
e
d
itse
lf
in
h
is invo
l
vement in a
ll
socia
l,
p
o
l
itica
l
, an
d
e
d
ucationa
l
aspects o
f
h
is
i
nvo
l
vement in
b
ui
ld
ing t
h
e new Repu
bl
ic.
Monge built on the tradition and knowledge of
stonemasons, resulting in his conception
f
o
f
a new, a
ll
-encompassing tec
h
nique w
h
ic
h
wou
ld
serve as a
l
anguage o
f
grap
h
ica
l
communication throu
g
hout the territory i
n
which French educational s
y
stem woul
d
exten
d
its in
fl
uence. T
h
is prove
d
to
b
e a
l
ar
g
e territory, wit
h
in
fl
uences
f
e
l
t up to mo
d
ern
times, and with descriptive geometry stil
l
surviving in many nationa
l
e
d
ucationa
l
sy
s
t
ems.
13
Mon
g
e’s wholehearted involvement wit
h
the ideals of the Enli
g
htenment an
d
the foundin
g
of the revolutionary education
a
l
institutions of the new Republic,
g
aine
d
h
im the title of the Father of
É
cole Po
l
ytec
h
nique (see [Sa
k
arovitc
h
2009]).
T
he teachin
g
of descriptive
g
eometry in this institution was certainly not
a
continuation of the educational tradition of
the Ancien Régime; it was a revolutionary
f
s
ub
j
ect tau
g
ht in a revolutionary way,
i
n the first of revolutionar
y
schools:
A
scholastic disci
p
line which was born in a school, by a school and for
a
s
chool (but maybe one should say in the
É
cole Polytechnique, by the
É
cole Polytechni
q
ue, and for the École Polytechni
q
ue), descri
p
tive
geometry a
ll
ows t
h
e passage
f
rom one process o
f
training
by
apprenticeship in little
g
roups which
w
as characteristic of the schools o
f
t
he Ancien Ré
g
ime, to an education i
n
am
p
hitheatres, with lectures, and
p
ractica
l
exercises, w
h
ic
h
are no
l
on
g
e
r a
dd
resse
d
to 20 stu
d
ents,
b
ut to
400 students. Descriptive geometry
also stems from revolutionar
y
y
m
et
h
o
d
s. A means to teac
h
space in a
n
acce
l
erate
d
way in re
l
ation to t
h
e
f
ormer way of teachin
g
stereotomy, an abstract lan
g
ua
g
e, minimal, rapid
i
n t
h
e or
d
er o
f
steno
g
rap
h
y,
d
escripti
v
e
g
eometry permits a response to t
h
e
u
rgent situation as
f
or t
h
e e
d
ucation o
f
an e
l
ite, w
h
ic
h
was t
h
e case o
f
F
r
a
n
ce
at
t
h
e
m
o
m
e
n
t
o
f
t
h
e
c
r
eat
i
on
of École Polytechni
q
ue [Sakarovitch
1
995: 211
]
.
I
t is
h
ere, in t
h
e context o
f
suc
h
sweeping c
h
anges, t
h
at we move our
f
ocus
b
ac
k