A Historical Walk along the Idea of Curvature, from Newton … · A Historical Walk along the Idea of Curvature, from Newton to Gauss Passing from Euler Gabriele Bardini and Gian

International Mathematical Forum, Vol. 11, 2016, no. 6, 259 - 278HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/imf.2016.6223

A Historical Walk along the Idea of Curvature,

from Newton to Gauss Passing from Euler

Gabriele Bardini and Gian Mario Gianella

Dipartimento di Matematica, Universita di TorinoVia Carlo Alberto 10, 10123 Torino, Italy

Copyright © 2016 Gabriele Bardini and Gian Mario Gianella. This article is distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribu-

tion, and reproduction in any medium, provided the original work is properly cited.

Abstract

From Newton ”Method of fluxions” to Gauss’ Theorema Egregium,the notion of curvature notably evolved from its appliance to a curvein a plane, to the one about a surface in space. Euler was the firstmathematician who tried to define the curvature of a surface, althoughemploying the curvature of a curve by comparing a plane section of thesurface and a circumference. Later, Gauss changed Euler’s definition toovercome its limitations by defining it through the comparison betweenthe linear element of the surface and the spherical one.

Keywords: Euler, curvature, surface

1 The curvature of a curve in a plane and the

osculating circle

1671 Newton completes the ”Method of fluxions” (published posthumouslyin 1736) [11]: in it, he derives a formula to determine the curvature of acurve at a point, using for the first time the cinematic notion of derivative(fluxion). He sets several axioms:

• The straight line has not curvature.

• The curvature of a circumference is a constant inversely propor-tional to its radius (k = 1/R).

260 Gabriele Bardini and Gian Mario Gianella

• If a circumference is tangent to a curve at a point on the concaveside, so that another one cannot be inserted in the contact angle(it will be the highest possible tangent circumference), then it willhave the same curvature of the curve in the point of tangency.

• The curvature centre of a curve at a point is the centre of thecircumference whose radius is perpendicular to the curve, with samecurvature and tangent to the curve in the point. To find it, one hasto look for the intersection of the normal to the curve, on the concaveside, at the point and in another infinitesimally close. Therefore,the curvature of a curve at a point will be the inverse ofthe radius of the highest tangent circumference in it. Theformula derivedd by Newton is

r =(1 + z2)3/2

z=

1

k; z =

y

x; x = 1 , (1)

where the apex · is the fluxion’s (derivative) symbol.

1686 Leibniz, in the ”Acta Eruditorum” [10], defines the osculating circle of acurve at a point as the circumference secant the curve in four points, tothe limit coincident between them at the point of tangency, and whosecurvature is equal to the one of the curve at the point.

1691 Jakob Bernoulli, in the ”Acta Eruditorum” [2], rectifies Leibniz state-ment demonstrating that 3 secant points are sufficient to define the os-culating circle. In the meantime, his brother Johann, in the ”LectionesMatematicae” [1], resultes a theorem to find the radius of curvature as

r =(dx2 + dy2)3/2

−dxd2y=

1

k; d2y = d(dy) ; dx2 = 0 , (2)

in which d2x = 0 because of the cartesian coordinate frame used.

1758 Abraham G. Kastner, in his book ”Anfangsgrunde der Mathematik”(”Foundations of Mathematics”) [9], defines the curvature of a curve ata point as the ratio of the angle ∆Θ subtended between two tangents tothe curve at two points, in the limit of their coincidence, and the lengthof the arc of the curve between them ∆s. One gets that

lim∆x→0

∆Θ

∆s=dΘ

ds. (3)

In the limit of coincidence between the two extremes of the arc, s =dy

dxis the derivative’s value: therefore

ds = d

(dy

dx

)=d2y dx− d2x dy

(dx)2. (4)

A historical walk along the idea of curvature 261

If d2x = 0 then ds =d2y dx

(dx)2: replacing in latter eq. (2), the new one is

r = −dx(1 + s2)3/2

ds=

1

k. (5)

Leonhard Euler, Johann Bernoulli’s doctoral student and entertainer ofa long correspondence with Kastner, used eq. (5) in his 1763 paper”Recherches sur la courbure de surfaces” [5]. According to a recentdemonstration, let T (s) be the unit vector tangent to the parametriccurve α(s), parametrized through the arc length s: then T (s + ds) −T (s) = NdΘ, with N the orthogonal unit vector and T (s + ds) andT (s) tangent unit vectors to α(s) as shown in Fig. 1.

Figure 1: Vectors T(s), T(s+ds), T(s+ds)-T(s)

One has that

N =T (s+ ds)− T (s)

dΘ= T ′

ds

dΘ=T ′

k, (6)

thus T ′(s) = k(s)N (s). Moreover, the arc length s is the length, on thecurve, between one inceptive fixed extreme A and a point P so that

s =

∫ t

A

‖α′(u)‖du ⇒ ds

dt= ‖α′(u)‖ . (7)

Going ahead, one gets

T ′(s) =dT (s(t))

dt

dt

ds=

T ′(t)

‖α′(t)‖, (8a)

T ′(t) = α′(t)T ′(s) , (8b)

T ′(t) = ‖α′(t)‖k[s(t)]N [s(t)] = ‖α′(t)‖k(t)N (t) , (8c)

T ′(t) ·N (t) = ‖α′(t)‖k(t)N (t) ·N (t) = ‖α′(t)‖k(t) (8d)

and, in conclusion

k(t) =T ′(t) ·N (t)

‖α′(t)‖. (9)


To keep on with the demonstration, it is necessary to remember

T ′(t) =(x′′(t), y′′(t))√

x′2 + y′2+d

dt

(1√

x′2 + y′2

)(x′(t), y′(t)) , (10a)

N (t) =(−y′(t), x′(t))√

x′2 + y′2, ‖α′(t)‖ =

√x′2 + y′2. (10b)

Therefore, one finally gets

k(t) =−x′′(t)y′(t) + x′(t)y′′(t)

(√x′2 + y′2)3/2

=Jα′(t) ·α′′(t)‖α′(t)‖3

, (11)

where Jα′(t) is the orthogonal complement of α′(t) .

Eq. (11) is an EXTRINSIC definition of curvature, obtained by immers-ing the curve in a larger set (plane), whose elements are used to defineit. Now, one would have to verify two equivalences, the first one betweeneq. (11) and eq. (1), the second one between eq. (11) and eq. (5), in orderto check the consistency between these different definitions of curvatureof a curve at a point. Indeed, in the first case one can show that

−x′′(t)y′(t)(x′2 + y′2)3

=x′2 d

dt( y

′

x′)

x′3(1 + y′2

x′2)32

=z′

x′(1 + z2)32

=z′

(1 + z2)32

= kN . (12)

The value in eq. (11) is then equivalent to Newton’s one given in eq. (1).In the second case, one can show that (from (4))

−ds = −d(dy

dx) =

d2ydx−d2xdy

(dx)2 =

=(dt)3

(dx)2

[−ddt

(dy

dt)dx

dt+d

dt(dx

dt)dy

dt

]=

dt

x′2(y′′x′ − x′′y′), (13)

s =dy

dx=

dydtdxdt

=y′

x′. (14)

Replacing the resulting values of s and ds in eq. (5), one finally gets

kE =dt(y′′x′ − x′′y′)dx x′2(x′2+y′2)

32

x′3

=y′′x′ − x′′y′

x′3(x′2+y′2)32

x′3

= k. (15)

The value in eq. (11) is then also equivalent to Euler’s one given ineq. (5). Lastly, one would have to verify the osculating circle propertiesby demonstrating the following theorem:


Let α(t) be a differentiable plane curve, defined in an interval (a,b),a < t1 < t2 < t3 < b, and let C(t1,2,3) be a circle passing through the3 distinct points α(t1,2,3) not aligned. Lastly k(t0) 6= 0 and t0 ∈ (a, b).Then, C = limt1,2,3→t0 C(t1,2,3) is the osculating circle of α(t) in t0.

Demonstration) Let p be the centre of C(t1,2,3) and f(t) be a function(a, b) → < such that f(t) = ‖α(t)− p(t1,2,3)‖2 , i.e. it restores the os-culating radius’ value in α(t). Therefore, f ′(t) = 2α′(t) · (α(t) − p)and f ′′(t) = 2α(t) · (α(t) − p) + ‖α′(t)‖2: since f(t) is a differentiablefunction and f(t1) = f(t2) = f(t3), then, according to Rolle’s theo-rem, there are u1 ∈ (t1, t2), u2 ∈ (t2, t3), v ∈ (u1, u2)such thatf ′(u1) =f ′(u2) = f ′′(v) = 0. Thus, limt1,2,3→t0 f

′(t0) = 2α′(t0) · (α(t0 − p)) = 0

and limt1,2,3→t0 f′′(t0) = 2α′′(t0) · (α(t0 − p)) + 2 + ‖α′(t0)‖2 = 0, are

both verified in α(t0)− p = −1k(t0)

Jα′(t0)‖α′(t0‖ (according to eq. (11)) : there-

fore C has a radius parallel to the orthogonal complement to the tangentunit vector Jα′(t0)

‖α′(t0)‖ , initial point α(t0), terminal point p and magnitude

1/k(t0). Then, C is the osculating circle of in t0.

2 The curvature of a surface in space accord-

ing to Euler

1763 Euler presents ”Recherches sur la courbure des surfaces” (p. in 1767) [5].

In it, he states that one cannot determine the curvature of a surface in spaceby comparison (in analogy to the plane case) with a sphere whose curvatureis measurable as equivalent to that of its great circles. This is not possible,because infinite different curves pass through every point on the surface: theyare sections of the surface with a plan for every possible direction; in additionto that, for each of these curves, there are infinite possible planes of intersection(infinite planes pass through a straight line). Thus, endless determinations arenecessary in order to discover an accurate value of k. It is yet a comparisonbetween curves, not between surfaces . For this purpose, one considersonly the infinite normal sections to the surface at a point, containing the normalto the surface at a point (as for the osculating circle whose point/centre radiuslies on the normal to the curve at a point): then, all the considered curvatureradii will be directed along the normal to the surface at a point. This set ofcurvature radii provides an accurate value of the curvature at a point. Thisdefinition is extrinsic as the previous one whence it is derived. Euler developedhis theory essentially through 4 steps:

• the study of curvature radius of a section of a generic secant plane;


• the development of the special case of a normal section;

• the comparison between the curvatures of the normal sections throughthe mutual inclinations of the planes;

• the determination of the superficial curvature at a point.

2.1 Curvature radius of a section of a secant plane

Figure 2: Z = (x, y, z = f(x, y)) belongs to the surface, EF segment ofthe straight line of intersection between the generic secant plane and xy, TYnormal segment to EF , led from Y = (x, y, 0) projection of Z on xy.

p = dz

dx

q = dzdy

⇒

dp = ∂p

∂xdx+ ∂p

∂ydy

dq = ∂q∂xdx+ ∂q

∂ydy

. (16)

As shown in Fig. 2, the secant plane of the surface at Z is

z = αy − βx+ γ, (17)

so that

dz = αdy − βdx = qdy + pdx (18)

and thusdy

dx=β + p

α− q. (19)

Moreover, the straight line EF is y = βαx − γ

α, thus |AE| = γ

β(A is the axis’

origin) and tan(FEC) = βα

. EF is also normal to TZ, which is the intersectionbetween the secant plane and another normal to it passing through Y Z.(ET = t, TZ = u) is the new couple of coordinates on the secant


plane, whose origin is T , whereby the curvature radius is deter-minable. (AT = x, TY = y) are the old coordinates, so that

t = x−|AE|cos(FEC)

+ [y − (x− |AE|) tan(FEC)] sin(FEC)

u = [y − (x− |AE|) tan(FEC)] cos(FEC)⇒ (20a)

t = αx+βy√

α2+β2− αy

β√α2+β2

u =

√α2+β2+1(αx−βx+γ)√

α2+β2

. (20b)

Setting

s =du

dt, r = −dt(1 + s2)

32

ds(21)

and replacing u and t from eq. (20) into eq. (21), one obtains, after somealgebraic and differential manipulations, the required formula

r = − (α2 + β2 − 2αq + 2βp+ (αp+ βq)2 + p2 + q2)32√

α2 + β2 + 1[(α− q)2 ∂p

∂x+ (β + p)2 ∂q

∂y+ 2(α− q)(β + p)∂p

∂y

] . (22)

2.2 Special case of a normal section

Figure 3: EF segment of the straight line of intersection between the normalsection to the surface and xy, Y Z is the z coordinate of Z = (x, y, z = f(x, y)),Z, Y and N belong to a secant plane λ perpendicular to xy and parallel to theordinate axis, Z, Y and M belong to a secant plane µ perpendicular to xy andparallel to the abscissa axis.

M is the intersection between the normal to the surface section in Z andthe x-axis, thus YM is the subnormal of Z in ν, while N is the intersection


between the normal to the surface section in Z and the y-axis. Thus Y N isthe subnormal of Z in µ: thus, according to the subnormal equation

YM = ∂z∂xz = pz

Y N = ∂z∂yz = qz

. (23)

Through M one draws the straight line parallel to the y-axis and through N theone parallel to the x-axis, which intersect at the point P : then, the segment ZPwill be normal to both considered sections and finally to the surface at the pointZ. Indeed, in ν, the tangent to the section in Z is ( ∂z

∂x, 0, 1) and thus the normal

to the section in Z is (1, 0,− ∂z∂x

), while in µ the tangent is (0,−∂z∂y, 1) and thus

the normal is (0, 1,−∂z∂y

). Hence the vector passing through Z with direction

ZP is (1, 1,− ∂z∂x− ∂z

∂y), which is also normal to the tangent to the surface at Z

( ∂z∂x, ∂z∂y, 1). So P ∈ EF . As in eq. (17), ζ = PEL ⇒ tan(ζ) = β

α: thus

Y T = z[p sin(ζ)− q cos(ζ)] ⇒ (24a)

tan(Y TZ) =1

p sin(ζ)− q cos(ζ). (24b)

Then, α = cos(ζ)

p sin(ζ)−q cos(ζ)

β = sin(ζ)p sin(ζ)−q cos(ζ)

. (25)

Substituting the latter in (22), one obtains

r =−(1 + p2 + q2)

32 (1 + (psinζ − qcosζ)2)

λ ∂p∂x

+ η ∂q∂y

+ ψ ∂p∂y

, (26a)λ = [(1 + q2) cos(ζ)− pq sin(ζ)]

2

η = [(1 + p2) sin(ζ)− pq cos(ζ)]2

ψ = 2[(1 + q2) cos(ζ)− pq sin(ζ)][(1 + p2) sin(ζ)− pq cos(ζ)]

.(26b)

2.3 Comparison between curvature radii of the normalsections through planes’ mutual inclinations

Among the normal sections, the MAIN one is defined as the one whose inter-section with xy passes through the projection Y of Z on xy. By construction,

Y T = 0⇒

sin(ζ) = q√p2+q2

cos(ζ) = p√p2+q2

. Then, considering (26a), one gets

r =−(1 + p2 + q2)

32 (p2 + q2)

p2 ∂p∂x

+ q2 ∂q∂y

+ 2pq ∂p∂y

. (27)


Between a generic normal section, intersecting the plane xy in EP ,and the main one, containing Y , Z and P , there is an angle φ: byconstruction, it is equivalent to Y RS (Y S normal to Y P , R intersection of thenormal plane to xy, passing through Y S, with ZP ).

Figure 4: Mutual inclination between normal sections

Y S = Y R tan(φ) = Y P sin(Y PZ) tan(φ) = z√p2 + q2

Y Z

ZPtan(φ) =

=z√p2 + q2

z√

1 + p2 + q2tan(φ) =

√p2 + q2√

1 + p2 + q2tan(φ), (28a)

PS =√Y P 2 + Y S2 =

√√√√(z√p2 + q2)

2+

[z√p2 + q2√

1 + p2 + q2tan(φ)

]2

=

= z

√(p2 + q2)(1 + p2 + q2)tan2(φ)

1 + p2 + q2(28b)

From (28), it follows

tan(EPY ) =Y S

Y P=

tan(φ)√1 + p2 + q2

⇒ (29a)

tan(EPM) = tan(Y PM − EPY ) =p√

1 + p2 + q2 − q tan(φ)

q√

1 + p2 + q2 + p tan(φ)=

= cot(PEL) =cos(ζ)

sin(ζ). (29b)

Thus, one obtainssin(ζ) =

q√

1+p2+q2+p tan(φ)√[1+p2+q2+ptan2(φ)](p2+q2)

cos(ζ) =p√

1+p2+q2−q tan(φ)√[1+p2+q2+ptan2(φ)](p2+q2)

, (30)


and moreover, by substituting in (26a), one can write

r =V

P (cos(φ))2 +Q(sin(φ))2 + 2R cos(φ) sin(φ), (31a)

u =√

(1 + p2 + q2, V = −(1 + p2 + q2)(p2 + q2)3

P = p2 ∂p∂x

+ q2 ∂q∂y

+ 2pq∂p∂y

Q = u2(q2 ∂p∂x

+ p2 ∂q∂y− 2pq∂p∂y)

R = u[pq(− ∂p∂x

+ ∂q∂y

) + (p2 + q2)∂p∂y]

. (31b)

Finally, through further manipulations on (31a), one getsL = P+Q

2V,M = P

2V, N = 2R−Q

2V

r = 1L+M cos(2φ)+N sin(2φ)

. (32)

The curvature radius depends on φ and no longer on ζ as in (26a).

2.4 Determination of superficial curvature at a point

In order to find the value of the superficial curvature at the point Z, one hasto notice that L,M,N are functions of p, q, ∂p

∂x, ∂p∂y, ∂q∂x

and ∂q∂y

calculated in Z.Thus, their values are common to all the normal sections passing through Z.Therefore rZ depends ONLY on φ.

r′(φ) = 2[MZ sin(2φ)−NZ cos(2φ)]

[LZ+MZ cos(2φ)+NZ sin(2φ)]2= 0 if φ = arctan

(NZ

2MZ+ lπ

2

), l ∈ Z⇒

φMAX = arctan

(NZ

2MZ

), φmin = arctan

(NZ

2MZ

+π

2

). (33)

Other extreme points are coincident to those or opposite on the same direc-tion (on the same normal section): thus the directions of maximum andminimum curvature are normal to each other. From (33), one gets

cos(2φMAX) = MZ2−NZ

2

MZ2+NZ

2 , sin(2φMAX) = 2MZNZ

MZ2+NZ

2

cos(2φmin) = NZ2−MZ

2

MZ2+NZ

2 , sin(2φmin) = −2MZNZ

MZ2+NZ

2

, (34a)

f = rZ,MAX , g = rZ,min, (34b)f = MZ2+NZ

2

LZ(MZ2+NZ

2)2+MZ(MZ

2−NZ2)

2+2MZNZ

2

g = MZ2+NZ

2

LZ(MZ2+NZ

2)2−MZ(MZ

2−NZ2)

2−2MZNZ2

. (34c)

Now, one introduces a new frame of reference on the tangent plane of thesurface at Z: on it, one rotates the coordinate axes to put them onto thedirections of extreme curvature radii. φ′MAX = 0, φ′min = π

2⇒ NZ = 0. Thus


(34c) becomes f = MZ

2

LZMZ2+MZ

3

g = MZ2

LZMZ2−MZ

3

⇒

L = f+g

2fg

M = f−g2fg

. (35)

By substituting in (32), one obtains

rZ =2fg

(f + g) + (g − f) cos(2φ). (36)

The value of the curvature radius depends on its maximum f , on itsminimum g and on φ.

3 The development of a surface in space on a

plane

1772 Euler publishes ”De solidis quorum superficiem in planum explicarelicet” [4].

Starting from the property, known in elementary geometry, of cones and cilin-ders (not of spheres) to be unfolded (”developed”) on a plane without dis-tortions, Euler wants to establish sufficient conditions under which a genericsurface has this feature. He expands its analysis in three steps:

I. the search for a solution obtained from analytical principles

II. the search for a solution obtained from geometrical principles

III. the use of the second one in the first one.

The second part of this article and the third one will not be dealt there becausethey are irrelevant for paper’s purpose.

3.1 General definition of surface development obtainedfrom analytical principles

Let Z be a point of a surface in space whose coordinates are AX = x, XY = y,Y Z = z and let V a point in the development plane such that Z ≡ V . Be-cause V is a point on a plane, it has a couple a orthogonal coordinates OT = t,TV = u and Euler writes ”it is evident the triad of the previous coordinates(x, y, z) somehow must depend on the couple (t, u)” (introducing in mathe-matics history the idea of parametric surfaces). The differential equations for


Figure 5: Coordinates of Zpoint belonging to the surface

Figure 6: Development on aplane

x(t, u), y(t, u), z(t, u) aredx = ldt+ λdu, dy = mdt+ µdu, dz = ndt+ νdu

l =∂x

∂t,m =

∂y

∂t, n =

∂z

∂t, λ =

∂x

∂u, µ =

∂y

∂u, ν =

∂z

∂u

(37)

Let v, v′ two points indefinitely near to V such that their coordinates are(t, u + du), (t + dt, u): because the development is an isometry, the triangleV vv′ must be equivalent to the triangle Zzz′, with z and z′ respectively coun-terimages of v and v′ on the surface. Thus

Zz = V v = du, Zz′ = V v′ = dt, zz′ = vv′ =√du2 + dt2 (38)

It follows z = Z + du, z′ = Z + dt and then

z = (x+ λdu, y + µdu, z + µdu), z′ = (x+ ldt, y +mdt, z + ndt) (39)

From (39), using Euclidean formula of distance beetwen two points on a plane

Zz = du√λ2 + µ2 + ν2, Zz′ = dt

√l2 +m2 + n2, (40a)

zz′ =√du2(λ2 + µ2 + ν2) + dt2(l2 +m2 + n2)− 2dudt(λl + µm+ νn) (40b)

One finally discovers the six conditions under which a surface in space is devel-opable on a plane by comparison the equations (38) with the equations (40)and from the irrelevance of derivation order in partial derivatives in (37):

∂l

∂u=∂λ

∂t,∂m

∂u=∂µ

∂t,∂n

∂u=∂ν

∂t,

λ2 + µ2 + ν2 = 1, l2 +m2 + n2 = 1, λl + µm+ νn = 0

(41)


TWO SIMPLE EXAMPLES:

• The cylindrical surfaceLet ξ : (t, u) → (cos(t), sin(t), u) be the parametrization of a cylindricalsurface: in this case all the six conditions are satisfied.

• The spherical surfaceLet γ : (t, u) → (sin t cosu, sin t sinu, cos t) be the parametrization of aunit spherical surface with radius r: in this case, condition λ2

γ+µ2γ+ν2

γ =sin2 t 6= 1 if t 6= π

2∨ 3π

2is not satisfied and thus γ is not developable.

In this work, Euler starts to use intrinsic (not dependent on shape) propertiesof surfaces, manages to determine a general definition of linear element of asurface in space ( (40b)) but only for the developable ones.

4 The curvature of a surface in space accord-

ing to Gauss

1827 Gauss publishes ”Disquisitiones generales circa superficies curvas” [6].

In this paper he investigates the surface curvature by the comparison betweenthe studied surface and the spherical one. His assumption is then differentfrom the Eulerian one, because Euler used the notion of curvature of a curvein a plane on the sections of the studied surface. To this purpose, he introducesa correspondence between points of the surface in space and points of unitaryradius sphere: for a point P (x, y, z) of the surface σ, there is a point P ′(X, Y, Z)of unitary sphere Σ whose coordinates are equivalent to the direction cosinesof the unit vector OP (it is still an extrinsic approach). The MEASURE OFCURVATURE k of the surface at a point is the ratio among the sphericalsurface element dΣ, centred in P ′, and the surface element dσ, centred in P(local measure): its SIGN will depend on the projection of dσ on dΣ;. If itmaintains its orientation in relation to the normal outgoing from the externalface of dΣ, then k is positive, otherwise it is negative.Let zdσ and zdΣ be the projections of surface elements on xy: then zdΣ

zdσ=

dΣdσ

= k and the ratio among projections is still k. Let A,B,C ∈ dσ, projectedon xy, whose projections are Axy = (x, y), Bxy = (x + dx, y + dy), Cxy = (x +δx, y+ δy). Let A′, B′, C ′ ∈ dΣ, projected on xy, whose projections are A′xy =(X, Y ), B′xy = (X+dX, Y +dY ), C ′xy = (X+δX, Y +δY ). Let A′, B′, C ′ be theprojections respectively of A,B,C on dΣ. Since the cross product magnitude isequivalent to parallelogram area subtended by two vectors, the value of surfaceelements will be the cross product magnitude of distance vectors of two points


in respect to the origin (P or P ′). Finally,

zdΣ =

∣∣∣∣∣∣∣i j k

dX dY 0

δX δY 0

∣∣∣∣∣∣∣ = dXδY − dY δX

zdσ =

∣∣∣∣∣∣∣i j k

dx dx 0

δx δy 0

∣∣∣∣∣∣∣ = dxδy − dyδx

⇒ k =dXδY − dY δX

dxδy − dyδx. (42)

5 The determination of curvature measure

In order to result a formula for curvature from the differential relation (42),three different ways to define a surface in space (thus three different differentialrelations (x, y)↔ (X, Y )) are studied:

I. W (x, y, z) = 0 IMPLICIT,

II. (p, q)→ (x(p, q), y(p, q), z(p, q)) PARAMETRIC,

III. z = z(x, y) EXPLICIT.

Let (dx, dy, dz) be the distance vector between two points on σ at limit coin-cident: it is therefore normal to (X, Y, Z) (unit vector of direction cosines in(x, y, z)) so that subsequently

Xdx+ Y dy + Zdz = 0

X2 + Y 2 + Z2 = 1. (43)

The first one will not be dealt here because it is irrelevant for paper’s purpose.

5.1 Determination of k by third definition of surface

Gauss first investigats the third definition z = z(x, y), reformulated in orderto develop the reasoning:

S(x, y, z) = z(x, y)− z = 0⇒ ds =∂S

∂xdx+

∂S

∂ydy +

∂S

∂zdz = 0. (44)

Thus, from (43) one derives∂S∂x

= ∂z∂x

= t∂S∂y

= ∂z∂y

= u∂S∂z

= −(∂z∂z

) = −1

⇒

X = −t√

1+t2+u2

Y = −u√1+t2+u2

Z = 1√1+t2+u2

, (45)


so that dX = ∂X

∂xdx+ ∂X

∂ydy, dY = ∂Y

∂xdx+ ∂Y

∂ydy

δX = ∂X∂xδx+ ∂X

∂yδy, δY = ∂Y

∂xδx+ ∂Y

∂yδy

⇒ (46a)

dXδY − dY δX =

(∂X

∂x

∂Y

∂y− ∂X

∂y

∂Y

∂x

)(dxδy − dyδx) ⇒ (46b)

k =∂X

∂x

∂Y

∂y− ∂X

∂y

∂Y

∂x. (46c)

Setting

T =∂2z

∂x2=∂t

∂x, U =

∂2z

∂x∂y=∂t

∂y=∂u

∂x, V =

∂2z

∂y2=∂u

∂y, (47)

and after several algebraic and differential manipulations performed on (45)and (47), one finally finds the curvature formula

k =TV − U2

(1 + t2 + u2)2 . (48)

Gauss keep on writing that, if xy is tangent to the surface at the studied pointz0, then in Iεz0 ∇z|z0 = (t|z0 , uz0) = 0: therefore kz0 = Tz0Vz0 − U2

z0. Moreover,

if one puts the origin axis in z0, then (Ω negligible, H Hessian of z)

z = z|z0 +∇z|z0 · ((x, y)− z0) +((x, y)− z0) ·H|z0 · ((x, y)− z0)T

2+ Ω =

=Tz0x

2

2+ Uz0xy +

Vz0y2

2+ Ω. (49)

If one applies an axis rotation of Θ such that tan(2Θ) =2Uz0

Tz0−Vz0, then

z =T ′z02

+V ′z02

+ Ω ⇒ (50a)

k′z0 = T ′z0V′z0. (50b)

From (50a) one understands (remembering (11)) that T ′z0 and V ′z0 are thecurvature values in z0 along the surface restrictions in respect to new axesx′, y′:

y′ = 0⇒ z ≈T ′z02x′

2 ⇒ τ(x′) = (x′,T ′z02x′

2)⇒ k′z0,y′ = T ′z0 ,

x′ = 0⇒ z ≈V ′z02y′

2 ⇒ υ(y′) = (y′,V ′z02y′

2)⇒ k′z0,x′ = V ′z0 .


One considers a surface restriction in respect to a generic direction subtendingan angle ρ with new abscissas: by using polar coordinates one can write

z ≈[T ′z02

cos2(ρ) +V ′z02

sin2(ρ)

]r2 ⇒ ι(r) =

(r,

[T ′z02

cos2(ρ) +V ′z02

sin2(ρ)

]r2

),

and thus

k′z0,ρ = T ′z0 cos2(ρ) + V ′z0 sin2(ρ) ⇒ (51)

dk′z0,ρdρ

= (V ′z0 − T′z0

) sin(2ρ) = 0 ifρ = 0,π

2⇒ k′z0,ρ(0) = T ′z0 , k

′z0,ρ

(π

2) = V ′z0 .

According to what found, T ′z0 and V ′z0 are the extreme values related to theextreme points (respectively minimum and maximum) of angle ρ. Gauss itselfwrites: ”These conclusions contain almost all what that the famous Eulerrealized about the curvature of curve surfaces”. In fact, according to (36),one gets

rZ =2fg

(f + g) + (g − f) cos(2φ)=

fg

f sin2(φ) + g cos2(φ)=

1sin2(φ)g

+ cos2(φ)f

⇒

kz =

(sin2(φ)

g+

cos2(φ)

f

)= k′MAX sin2(φ) + k′min cos2(φ). (52)

Since φ has the same definition of ρ, (36) is then equivalent to (51).He concludes this part of his work:”THE MEASURE OF CURVATURE AT ANY POINT WHATEVER OFTHE SURFACE IS EQUAL TO A FRACTION WHOSE NUMERATOR ISUNITY, AND WHOSE DENOMINATOR IS THE PRODUCT OF THE TWOEXTREME RADII OF CURVATURE OF THE SECTIONS BY NORMALPLANES”.Checking the Gaussian reasoning through a modern perspective, one noticesthat the angle Θ such that tan(2Θ) =

2Uz0

Tz0−Vz0(used for the rotation applied

on the tangent plane of the surface at z0) is the one relative to a rotationmatrix between the canonical basis and Hessian unit eigenvectors one. Indeed,

H(z)|z0 =

[Tz0 Uz0Uz0 Vz=

], det(H(z)|z0−λI) = 0 ⇔ λ1,2 =

Tz0+Vz0±√

(Tz0−Vz0 )2+4U2z0

2

and the eigenvectors arev1 =

(Tz0−Vz0−

√(Tz0+Vz0 )2+4U2

z0

2Uz0, 1

)= (v11, v12)

v2 =

(Tz0−Vz0+

√(Tz0+Vz0 )2+4U2

z0

2Uz0, 1

)= (v21, v22)

. The matrix equation for


the change of basis

v11√v211+v212

v12√v211+v212

v21√v221+v212

v22√v221+v222

=

[cos(Ξ) − sin(Ξ)sin(Ξ) cos(Ξ)

]·[1 00 1

]is

solved by the solution Θ given above.

5.2 Determination of k by second definition of surface

One introduces the variables

a =∂x

∂p, a′ =

∂x

∂q, α =

∂2x

∂p2, α′ =

∂2x

∂p∂q, α′′ =

∂2x

∂q2, (53a)

b =∂y

∂p, b′ =

∂y

∂q, β =

∂2y

∂p2, β′ =

∂2y

∂p∂q, β′′ =

∂2y

∂q2, (53b)

c =∂z

∂p, c′ =

∂z

∂q, γ =

∂2z

∂p2, γ′ =

∂2z

∂p∂q, γ′′ =

∂2z

∂q2, (53c)

A = bc′ − cb′, B = ca′ − ac′, C = ab′ − ba′, (53d)

D = αA+ βB + γC,D′ = α′A+ β′B + γ′C,D′′ = α′′A+ β′′B + γ′′C.(53e)

a, b, c, a′, b′, c′ in (53) are respectively equivalent to l,m, n, λ, µ, ν in (37).

It is necessary to perform several algebraic and differential manipulations inorder to reformulate (48) with the variables just written: at last it will be

k =DD′′ −D′2

(A2 +B2 + C2)2 . (54)

To keep on, one has to introduce the new variables

m = aα + bβ + cγ,m′ = aα′ + bβ′ + cγ′,m′′ = aα′′ + bβ′′ + cγ′′, (55a)

n = a′α + b′β + c′γ, n′ = a′α′ + b′β′ + c′γ′, n′′ = a′α′′ + b′β′′ + c′γ′′(55b)

and

E = a2 + b2 + c2, F = aa′ + bb′ + cc′, G = a′2

+ b′2

+ c′2. (56)

Remebering (37),

E = l2 +m2 + n2, F = lλ+mµ+ nν,G = λ2 + µ2 + ν2 (57)


By performing another long sequence of algebraic and differential manipula-tions (54) is reformulated, through the variables (55) and (56), in

k =αα′′ + ββ′′ + γγ′′ − α′2 − β′2 − γ′2

EG− F 2+

+(n′2 − nn′′)E + (nm′′ − 2m′n′ +mn′′)F + (m′2 −mm′′)G

EG− F 2. (58)

At last, (58) is reformulated only through E, F, G and their partial derivativeswith respect to p and q. According to eqs (55), (56) and (53), one gets

n =∂F

∂p− 1

2

∂E

∂q, n′ =

1

2

∂G

∂p, n′′ =

1

2

∂G

∂q, (59a)

m =∂E

∂p− 1

2

∂E

∂q,m′ =

1

2

∂G

∂p,m′′ =

∂F

∂q− 1

2

∂G

∂p, (59b)

αα′′ + ββ′′ + γγ′′ − α′2 − β′2 − γ′2 = −1

2

∂2E

∂q2+∂2F

∂p∂q− 1

2

∂2G

∂p2,(59c)

and thus it becomes

k =

E

[∂E∂q

∂G∂q− ∂F

∂p∂G∂q

+(∂G∂p

)2]

4(EG− F 2)2 +

+F[∂E∂p

∂E∂q− ∂E

∂q∂G∂p− 2∂E

∂q∂F∂q

+ 4∂F∂p

∂F∂q− 2∂F

∂p∂G∂p

]4(EG− F 2)2 +

+

G

[∂E∂p

∂G∂p− 2∂E

∂p∂F∂q

+(∂E∂q

)2]

4(EG− F 2)2 −

[∂2E∂q2− 2 ∂2F

∂p∂q+ ∂2G

∂p2

]2(EG− F 2)

. (60)

5.3 The ”Theorema Egregium”

The relevance of expressing k as a function of only E,F,G (from (56)) residesin the formula of linear element of curved surface. For a parametric surface,

dx = adp+ a′dq

dy = bdp+ b′dq

dz = cdp+ c′dq

⇒ (61)

√(dx)2 + (dy)2 + (dz)2 =

√E(dp)2 + 2Fdpdq +G(dq)2. (62)

Formula (62) is almost equivalent to (40b) (the difference of a minus is dueto the fact the second one is a distance and the first a vector sum).


Gauss writes: ”The analysis developed shows us that for finding the measureof curvature there is no need of finite formulae, which express the coordinatesx, y, z as functions of the indeterminates p and q; but that the general ex-pression for the magnitude of any linear element is sufficient”. Let σ(x, y, z) eσ′(x′, y′, z′) be two surfaces in space such that the first can be DEVELOPEDUPON the second: therefore, ”to each point of the former surface, determinedby the coordinates x, y, z will correspond a definite point of the later one, whosecoordinates are x′, y′, z′. Evidently, x′, y′, z′ can also be regarded as functions ofthe indeterminates p and q and thus for the element

√(dx′)2 + (dy′)2 + (dz′)2

we shall have an expression of the form√E ′(dp)2 + 2F ′(dpdq)2 +G′(dq)2”.

He concludes: ”But from the very notion of the development of one sur-face upon another it is clear that the elements corresponding to one anotheron the two surfaces are necessarily equal. Hence we shall have identicallyE = E ′, F = F ′, G = G′. Thus the previous formula leads of itself to theremarkable theorem (”Egregium theorema”): If a curved surface is devel-oped upon any other surface whatever, the measure of curvature ineach point remains unchanged”.

Overcoming Euler’s work, Gauss manages to define a measurable feature alsofor spherical surfaces and all those not developable on a plane, not only theequivalence between local plane metric and developable surfaces’ ones.

According to modern terminology, the importance of this theorem is relatedto the possibility to express the second fundamental form of a surface entirelythrough the variables of the first one, i.e. the Gaussian curvature k of a sur-face in space depends only on its local metric and not on its shape (how it isparametrized in the external frame of reference from where it is observed). Fi-nally one deduces the passage from an EXTRINSIC definition of curvature toan INTRINSIC one, because the local metric, not dependent on shape (and itsoperator), does not change if surfaces are locally homeomorph between them.

TWO SIMPLE EXAMPLES

• The plane and the cylindrical surfaceLet µ : (p, q)→ (cos(p), sin(p), q) be the local parametrization of a cylin-drical surface. In this case E = G = 1 and F = 0 for the plane (p, q)and for the cylindrical surface: thus µ is a locally isometric to a plane.

• The spherical surface and the planeLet γ : (θ, ζ)→ (r sin θ cos ζ, r sin θ sin ζ, r cos θ) be the local parametriza-tion of a spherical surface with radius r. Its Gaussian curvature isk = r−2, while the plane’s one is 0. The two surfaces are then notlocally isometric.


References

[1] Johann Bernoulli, Lectiones Matematicae, 1691/1692.

[2] Jakob Bernoulli, Acta Eruditorum, 1691.

[3] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Pearson,1976.

[4] L. Euler, De Solidis Quorum Superficiem in Planum Explicare Licet, NoviCommentarii academiae scientiarum Petropolitanae, 16 (1772), 3-34. 1772.

[5] L. Euler, Recherces sur la Courbure des Surfaces, Memoires de l’academiedes sciences de Berlin, 16 (1767), 119-143.

[6] C.F. Gauss, Disquisitiones Generales Circa Superficies Curvas, 1827.

[7] A. Gray, E. Abbena, S. Salomon, Modern Differential Geometry of Curvesand Surfaces with Mathematica, Chapman & Hall, 2006.

[8] S. Greco, P. Valabrega, Algebra Lineare, Levrotto & Bella, 2009.

[9] A.G. Kastner, Anfangsgrunde der Mathematik, 1758.

[10] G.W. Leibniz, Acta Eruditorum, 1686.

[11] I. Newton, Method of Fluxions, 1736.

Received: January 3, 2016; Published: February 26, 2016

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