Open access peer-reviewed chapter

Obtaining the Main Curvatures of Orientable Hypersurfaces, Based on Differential Geometry

Written By

Emmanoel Ferreira

Submitted: 02 January 2023 Reviewed: 08 January 2023 Published: 27 February 2023

DOI: 10.5772/intechopen.1001073

From the Edited Volume

Topology - Recent Advances and Applications

Paul Bracken

Chapter metrics overview

36 Chapter Downloads

View Full Metrics

Abstract

Most of the literature on differential geometry presents the coefficients of the first and second fundamental form to simplify the calculation of the curvatures on a surface and also to obtain other important information, such as the area of a surface. In this text, as the interest lies on the generalization of the idea of surface (hypersurface), such simplification by the use of these coefficients was not possible, in view of the complexity of the mathematical operations involved in the calculation of curvatures (if n = number of variables in the implicit function of the surface>3), opting to use the linear operator −DNp. To facilitate the calculation of the normal vector, the surface was described as the graph of a differentiable function f: Rn−1 → R. The example of the sphere thoroughly illustrates the procedure of calculating the main curvatures of a surface in the space R3, and such procedure is extensive to the space Rn. Some examples from the engineering area in which the variables of the implicit functions of the hypersurfaces are random were solved, and the results of the main curvatures, calculated by the proposed procedure, were exact and coincident with those provided in the literature.

Keywords

  • differential geometry
  • main curvatures
  • parametrization of a regular surface
  • orientable surface
  • implicit function

1. Introduction

This work focuses on providing the principal curvatures of hypersurfaces, from the normal vector, considering that much of the literature on the subject provides the results of these curvatures for the space R3 and allows for the calculation of the principal curvatures in space Rn via the linear operator −DNp. This procedure will be exemplified analytically for a surface in R3 (sphere). Furthermore, examples of hypersurfaces used in second-order reliability analysis (e.g., see [1]) of engineering problems with main curvatures results provided by MATLAB software (e.g., see [2]) will be presented.

The applications of the second-order reliability method (SORM), in engineering problems (e.g., see [1, 3, 4, 5, 6, 7, 8, 9, 10, 11]), in recent years, suggest a relevant interest in the SORM, and thus, there is much room for new research in this area, the main challenge being to calculate the main curvatures of the limit state surface, which involves a lot of mathematical complexity and computational effort.

The proposed procedure extends to the space Rn, providing exact values for the main curvatures of any orientable hypersurface with deterministic variables, as in mathematics, or random variables, and in reliability analysis problems in engineering.

Advertisement

2. Procedure to obtain the main curvatures by differential geometry

In this section, the procedure is established, by differential geometry, to calculate the main curvatures of the surface at the p point.

2.1 Surfaces in R3

To get a better understanding, in this subsection, the main curvatures at the p point of a surface in R3 are calculated by differential geometry, and in the Subsection 2.2, the generalization of this process to the space Rn is performed.

2.1.1 Parameterization of a regular surface in R3

The graph of an equation of the form F(x,y,z) = 0, where F is a differentiable function and its partial derivatives do not cancel each other, simultaneously, at any p point, such that F(p) = 0, is an example of a regular surface in R3. It is verified that the graph of a differentiable function f: R2 → R is also an example of regular surface.

More generally, a subset S of R3 is called regular surface if, for each p point S, there is an open neighborhood VR3 of p, an open UR2 and a bijection φ: U → V∩S, where φ is the form φ(u,v) = {x(u,v), y(u,v), z(u,v)}, with the properties described as follows (e.g., see [12]):

  1. φ is Class C, i.e., φ has continuous partial derivatives of all orders at the p point;

  2. φ is a homeomorphism (i.e., its inverse is continuous); and.

  3. for any point qU, the Jacobian matrix of φ has rank two. The referred matrix has rank two, which means that the image of the linear transformation obtained has dimension two, i.e., eliminating a line, conveniently chosen, the resulting 2×2 matrix has a determinant different from zero. The Jacobian matrix, in this case, has dimensions 3×2, represented by.

J=∂x∂u∂x∂v∂y∂u∂y∂vz∂uz∂vE1

In these conditions, it is said that φ is a parameterization for S, as illustrated in Figure 1:

Figure 1.

Parameterization of a regular surface. Source: Adapted from Ref. [13].

A regular surface SR3 is orientable, if and only if there is a differentiable field N: S R3 of normal vectors in S, according to Ref. [13].

2.1.2 Curvatures of a surface in R3

Being S an orientable surface, the Gauss application is the field of normal vectors N: S → S2, where S2R3 is the sphere of radius 1 and center at origin. N is a differentiable application and its derivative −DNp: TpS → TpS is an endomorphism (i.e., a linear transformation T:U → V, being U=V), where TpS is the space (plane) tangent to S surface at the p = φ(u,v) point. From the definition of derivative (rule of the chain), highlighted by Ref. [14], one has to.

Nu=DNφuvφuE2

and

Nv=DNφuvφv

where φu and φv are partial derivates of parameterization φ(u,v), i.e., they are tangent vectors that generate the plane TpS.

The vectors N and φu are orthogonal, as well as N and φv. Derivating the scalar products <φu,N > = 0 and < φv,N > = 0, it was concluded that −DNp is an self-adjunct linear application of TpS in TpS. Thus, according to Ref. [14], the eigenvalues k1(p) and k2(p) of the linear operator (−DNp) are named main curvatures of S at the point p, and the orthogonal directions defined in TpS by eigenvalues k1(p) and k2(p) are named main directions.

2.1.3 Normal curvature

Being α:(a,b) → S be a curve parameterized by arc length. The normal curvature of α in α (s) is the component of α"(s) according to the normal to S at this point and is given by kn (α,s) =<α"(s), N Ο α(s)> (inner product between α"(s) and N, with N applied in the point α(s)) and illustrated as Figure 2. If the curve was not parameterized by arc length, the formula of the normal curvature, according to Ref. [12], becomes.

Figure 2.

Normal curvature at point p. Source: The author.

knαt=1α't2<α"t,NΟαt>E3

According to Ref. [12], the maximum and minimum values of the normal curvatures of the normal sections at p are the main curvatures of the surface in point p, as illustrated in Figure 3.

Figure 3.

Normal curvatures when α is a normal section in point p. Source: The author.

2.2 Surfaces in Rn

Most of the literature on differential geometry shows the coefficients of the first and second fundamental form to simplify the calculation of the curvatures in a surface in R3 and also to obtain other information, such as a surface area. Herein, as the interest is the generalization of the surface idea (hypersurface), such simplification using these coefficients could not be done, due to the complexity of mathematical operations involved in calculating the curvatures when n > 3, and hence, in Subsection 2.1, it was opted to use the linear operator −DNp.

To simplify calculation of normal vector to surface, it is described as the graph of a differentiable function f: Rn−1 → R.

2.2.1 Parameterization of the surface g(V) = 0

A parameterization for the surface in this vicinity can be given by.

φpV1V2Vn1=V1V2Vn1fV1V2Vn1E4

The function f(V), with V=V1V2Vn1Rn1, is obtained by explicitness for any variable of the vector V of the function g(V) = 0, where V=V1,V2Vn1VnRn. Considering, for example, the explicitness of the last variable of g(V) = 0; it has Vn=fV1V2Vn1.

2.2.2 Obtaining the vectors tangent to the surface g(V) = 0

The vectors tangent, which correspond to partial derivatives of Eq. (4), is calculated in point p according to.

φV1p=100fV1p=100∂fp∂V1
φVip=010fVip=010∂fp∂Vi;1<i<n1E5
φVn1p=0001fVn1p=001∂fp∂Vn1

2.2.3 Obtaining the normal vector to the surface g(V) = 0 and its partial derivatives

The normal vector at point p is calculated by extending the equation shown in Ref. [13] for this vector, i.e.,

Np=fV1pfV2pfVn1p1fV1p2+fV2p2++fVn1p2+1).E6

The partial derivatives of the normal vector are obtained by

Nvjp=NpVj,j=1,2,,n1E7

2.2.4 Obtaining the main curvatures of the surface g(V) = 0

Once performed the calculation of the normal vector and its partial derivatives Nvj, they can be written as a linear combination of the vectors,φV1, … ,φVn1, of the tangent plane, obtaining the matrix (M) of the linear operator −DNp, whose eigenvalues are the main curvatures. Extending the equation shown by Ref. [14] for the referred linear operator, it has

DNpφVj=Nvj,j=1,2,,n1E8

thus:

Nv1=N1,1.φV1++N1,n1.φVn1Nvn1=Nn1,1.φV1++Nn1,n1.φVn1E9

and

M=N1,1..N1,n1N2,1N2,2......Nn1,1Nn1,2.Nn1,n1E10

2.3 Example solved analytically by the proposed procedure

Compute the main curvatures, at point p 02222, of a sphere (S) of radius equal to 1 and center at the origin, illustrated through Figure 4, whose equation is given by g(V)=V12 +V22 +V32 −1 = 0.

Figure 4.

Sphere (S) of radius equal to 1 and center at the origin. Source: The author.

By explaining, for example, the last variable (V3) of g(V) = 0 (it could have been chosen the variables V1 or V2) this sphere can be obtained through the function f: R2 → R defined by

V3=fV1V2=1V12V22

thus generating the parameterization

φV1V2=V1V21V12V22

The partial derivatives of fV1V2 are

fv1V1V2=∂fV1V2V1=V11V12V22

and

fv2V1V2=∂fV1V2V2=V21V12V22

According to Ref. [13]

NV1V2=fV1fV21fV12+fV22+1E11

defines the normal vectors to the surface.

Using Eq. (11), the result is for the normal vector

NV1V2=V1V21V12V22

Performing the partial derivatives of the normal vector in relation to V1 and V2, we have

Nv1V1V2=10V11V12V22

and

Nv2V1V2=01V21V12V22

Considering the point p02222 that corresponds, in the parameterization φV1V2, to the point q=φ022 and applying the equations obtained at that point, we have

Nv1022=100

and

Nv2022=011

Following this procedure, the vectors that form the basis of the tangent plane TpS are determined. At point p, we have

φV1V1V2=10V11V12V22

and

φV2V1V2=01V21V12V22

because

φV1V1V2=φV1V2V1

and

φV2V1V2=φV1V2V2

At the point considered, we have V1=0eV2=22, so

φV1022=100

and

φV2022=011

To determine the linear operator matrix (−DNp), one must apply this operator to the vectors φV1. and φV2 and write the results as a linear combination of φV1 e φV2

As reported by Ref. [14], one has that

DNpφv1=Nv1andDNpφv2=Nv2E12

Thus, one can write −Nv1 and Nv2 as a linear combination of the vectors φV1andφV2. From the information obtained above, we have, at the point considered,

Nv1100=1φV1+0φV2

and

Nv2011=0φV1+1φV2

Therefore, the matrix of the linear operator is

M=1001

whose determinant, which is the Gaussian curvature, is K = 1 and whose eigenvalues, which are the main curvatures, are k1(p)=1 and k2(p)= 1. The main curvatures values are exact according to the Ref. [15].

2.4 Examples solved by the proposed procedure via MATLAB software

Four examples were solved, being two from the mathematics area, aiming to validate the proposed procedure, and the last two from the engineering area.

2.4.1 The main curvature of a curve (parabola of degree 2)

Determine the main curvature of the parabola in Figure 5 at the point (1,1).

Figure 5.

Graphical representation of a plane curve—2nd degree parabola. Source: The author.

The result of the main curvature is shown in Table 1 and coincides with that presented in Ref. [12].

Implicit functionPointki
g(V) = u12−u2 = 0(1,1)k1 = 0.1788

Table 1.

Main curvature of the parabola of Figure 5 at the point (1,1).

2.4.2 Main curvatures of a torus

Determine the main curvatures of the torus in Figure 6 at point (7.1568, 7.1568, and 2.1214) with implicit function

Figure 6.

Torus surface. Source: The author.

gV=98+u2+v22z=0E13

where

V=uvzE14

The results of the main curvatures are shown in Table 2 and agree with those presented in Ref. [15].

Implicit functionPointki
g(V) = 98+u2+v22 −z = 0(7.1568, 7.1568, 2.1214)k1 = −0.0698
k2 = −0.3333

Table 2.

Main curvatures at four points on the torus of Figure 6.

2.4.3 Main curvatures of a hyperparaboloid in standard normal space

For limit state obtained from Ref. [16], the limit state surface function of a hyperparaboloid, composed of its reliability index β and its main curvatures kj, i.e.,

gV=β+0,5i=19kjXj2X10=0E15

where β=3, k1 = 0.22, k2 = 0.23, k3 = 0.24, k4 = 0.25, k5 = 0.26, k6 = 0.27, k7 = 0.28, k8 = 0.29, and k9 = 0.30.

and

V=X1X2X3X4X5X6X7X8X9X10E16

The characteristics of the (mutually independent) random variables are summarized in Table 3.

VariableDistributionMeanStandard deviation
X1, , X10Standard Normal01

Table 3.

Statistical characteristics of random variables.

Determine the main curvatures of the hyperparaboloid, whose implicit function corresponds to Eq. (15), at point (0, 0, 0, 0, 0, 0, 0, 0, 0, 3).

The results obtained are presented in Table 4.

Implicit functionkj
g((V) = β + 0,5i=19kj Xj2 − X10 = 0k1 = 0.22
k2 = 0.23
k3 = 0.24
k4 = 0.25
k5 = 0.26
k6 = 0.27
k7 = 0.28
k8 = 0.29
k9 = 0.30

Table 4.

Main curvatures of the limit state surface at point p.

The results of the main curvatures are shown in Table 4 and agree with those presented in Ref. [16].

2.4.4 Main curvatures of a hypersurface: Case of bearing capacity of a shallow footing

In the Ref. [1], reliability analysis was performed to verify the structural safety of a shallow foundation. This is a foundation supported on a homogeneous layer of silty sand subjected to the loading illustrated in Figure 7.

Figure 7.

Shallow footing with rectangular base of width B and length L supported on a silty sand layer. Source: Ref. [10].

with B = 5.0 m, L = 25.0 m, D = 1.8 m, and h = 2.5 m. The limit state is ruled by the limit state surface function, whose equation is

GU=qultq=0E17

where qult is the vertical resistance to tipping, and q is the applied vertical pressure. The vector U represents the random variables—soil cohesion (c’), soil friction angle (φ’), soil unit weight (γ), horizontal distributed load (PH), and vertical distributed load (PV), which are considered to be normally distributed.

qult=cNcscic+γDNqsqiq+0,5γBNγsγiγE18

where Nq = e (π tan φ’) tan2[45o + (φ’/2)], Nc = (Nq − 1) cot φ’, Nγ = 2 (Nq − 1) tan φ’ are the (dimensionless) resistance factors, which depend on the soil friction angle, sq = 1 + (B′ / L’) sen φ’, sc = (sq Nq − 1) / (Nq − 1), and sγ = 1−0.3 (B′ / L’), which are also dimensionless, are factors related to the shape of the base (B × L) and the eccentricity of the loads, B′ = B − 2 eB, L’ = L, eB = h (PH / PV). Similarly, iq = {1 − [PH / (PV + B’L’ c’ cot φ’)]}m, ic = iq − [(1−iq) / Nc tan φ’], and iγ = {1 − [PH /(PV + B′ L’ c’ cot φ’)]}m + 1 are dimensionless correction factors responsible for the slope of the resulting load and m = [2+ (B′ / L’)] / [1+ (B′ / L’)]. Employing this approach, failure will occur when the applied vertical pressure, q, which can be calculated as

q=PV/BE19

becomes greater than the value of the overturning resistance, qult, calculated according to Eq. (18).

The characteristics of the random variables, which are correlated as indicated below in the matrix Ω, are summarized in Table 5.

DistributionVariableMean (μ)Standard deviation(σ)
Normalc′ (KPa)154.5
Normalφ′ (o)255
Normalγ (KN/m3)202
NormalPH (KN/m)40040
NormalPV (KN/m)80080

Table 5.

Statistical characteristics of random variables.

Correlation matrix (Ω)

c′φ′γPHPV
c′ (KPa)1−0.5000
φ′ (o)−0.510.500
γ (KN/m3)00.5100
PH (KN/m)00010.5
PV (KN/m)0000.51

L=Cholesky factorizationΩTE20

Lower triangular matrix (L) obtained through Cholesky factorization of the matrix Ω.

10000−0.50.86600000.5770.81600000100000.50.866

The vector V, shown in Eq. (21), was obtained from Ref. [1], where it underwent an orthogonal transformation to make the random variables c’, φ’, γ, PH, and PV, which were correlated, independent.

V=c'=4.5V1+15φ'=0.043633231299858V1+0.075574973509759V2+0.436332312998582γ=1.154700538379252V2+1.632993161855452V3+20PH=40V4+400PV=40V4+69.282032302755098V5+800E21

In Ref. [1], the Cholesky factorization was also performed according to Eq. (20) in order to find the point of interest (V*) for determining the main curvatures (see Table 6).

VariableU*V’* = (U* − μ) / σV* = L−1 V’*
c′ (KPa)14.915−0.019−0.019
φ′ (o)18.497−1.301−1.514
γ (KN/m3)17.934−1.033−0.195
PH (KN/m)422.6000.5650.565
PV (KN/m)808.4000.105−0.205

Table 6.

Obtaining the point of interest to perform the reliability analysis.

Now is applied the proposed procedure to calculate the principal curvatures of the hypersurface at point p = V*. As recommended in the proposed procedure, any random variable of the surface g(V) = 0, that is explicitness, can be chosen. The surface g(V) = 0 will be obtained by replacing the independent variables of the vector V, according to Eq. (17), that is

gV=cNcscic+γDNqsqiq+0,5γBNγsγiγPV/B=0E22

the random variables of Eq. (22) are replaced by the random variables of the vector V of Eq. (21). In order to determine the function f(V), the variable V3, is chosen, and then, according to what was preconceived, is made explicit. After making V3 = f(V) only need to determine the main curvatures of the hypersurface f(V) at point p = V* (see Table 7).

ki
−0.0886
−0.0574
−0.0039
0.0090

Table 7.

Main curvatures of limit state surface at point V*.

The results of the limit state surface main curvatures are accurate and in agreement with the values presented by the Ref. [10].

Advertisement

3. Conclusions

Five examples were solved, being three (one with an analytical solution and two via MATLAB software) from the mathematics area, aiming to validate the proposed procedure, and the last two (via MATLAB software) from the engineering area, whose principal curvatures needed to be calculated to perform the second-order reliability analysis via second-order reliability method by differential geometry (SORM DG), as recommended by Ref. [1]. However, the focus here is on generalizing the calculation of the principal curvatures at any point p, this was demonstrated through the solution of the aforementioned examples, since the procedure calculated the main curvature of a plane curve, whose implicit function has n = 2 (Example 2.4. 1—parabola of the second degree), the main curvatures of surfaces with implicit functions with n = 3 (Example 2.3—Sphere and Example 2.4.2—Torus) and the main curvatures of hypersurfaces with implicit functions with n > 3 (Example 2.4.3—Hyperparaboloid and Example 2.44—Shallow footing) as initially established. The procedure can be applied to any surface (or hypersurface) that is orientable and whose implicit function (g(V) = 0) has at least one variable that can be made explicit, so that the function f(V) can be obtained. This procedure, presented in this chapter, has been successfully applied to engineering problems requiring the calculation of the main curvatures of hypersurfaces at a point of interest, as illustrated in the simpler Example 2.43 and the more complex Example 2.44.

The mathematical procedure proposed here for computing main curvatures has proven to be very advantageous in the area of reliability analysis, including not only in terms of accuracy but also computational efficiency (e.g., see [1]).

In the area of mathematics, the procedure contributes, since it performs the calculation of the principal curvatures of plane curves and any orientable surface in space Rn, taking into account that in the literature of mathematics it is more common to calculate these curvatures up to space R3.

Advertisement

Conflict of interest

The author declares no conflict of interest.

References

  1. 1. Ferreira E, Freitas M, Rocha J, Sisquini G. SORM DG—An efficient SORM based on differential geometry. REM International Engineering Journal. 2019;589:60-672. DOI: 10.1590/0370-44672018720171
  2. 2. Lee H. Programming and Engineering Computing with MATLAB 2021. 1st ed. Mission: SDC Publications; 2021. p. 532
  3. 3. Cho S. Probabilistic stability analysis of slopes using the ANN-based response surface. Computers and Geotechnics. 2009;36:787-797. DOI: 10.1016/j.compgeo.2009.01.003
  4. 4. Lü Q, Low B. Probabilistic analysis of underground rock excavatios using response surface method and SORM. Computers and Geotechnics. 2011;2011:1008-1021. DOI: 10.1016/j.compgeo.2011.07.003
  5. 5. Lü Q, Sun H-Y, Low B. Reliability analysis of ground–support interaction in circulartunnels using the response surface method. International Journal of Rocks Mechanics and Minig Sciences. 2011;48:1329-1343. DOI: 10.1016/j.ijrmms.2011.09.020
  6. 6. Lü Q, Chan L, Low B. Probabilistic evaluation of ground–support interaction for deep rock excavation using artificial neural network and uniform design. Tunneling and Undreground Space Technology. 2012;32:1-18. DOI: 10.1016/j.tust.2012.04.014
  7. 7. Chan L, Low B. Practical second-order reliability analysis applied to foundation engineering. International Journal for Numerical and Analytical Methods in Geomechanics. 2012;36:1387-1409. DOI: 10.1002/NAG.1057
  8. 8. Zeng P, Jimenez R. An approximation to the reliability of series geotechnical systems using a linearization approach. Computers and Geotechnics. 2014;62:304-309. DOI: 10.1016/J.COMPGEO.2014.08.007
  9. 9. Zeng P, Jimenez R, Jurado-Piña R. System reliability analysis of layered soil slopes using fully specified slip surfaces and genetic algorithms. Engineering Geology. 2015;193:106-117. DOI: 10.1016/J.ENGGEO.2015.04.026
  10. 10. Zeng P, Jimenez R, Li T. An efficient quase-Newton approximation-based SORM to esimate the reliability analysis of geotechnical problems. Computers and Geotechnics. 2016;76:33-42. DOI: 10.1016/J.COMPGEO.2016.02.003
  11. 11. Zeng P, Li T, Jimenez R, Freng X, Chen Y. Extension of quase-Newton approximation-based SORM for series system reliability analysis of geotechnical problems. Engineering with Computers. 2017;34:215-224. DOI: 10.1007/s00366-017-0536-8
  12. 12. Rodrigues P. Introdução às curvas e superficies. 1st ed. Niterói: EdUFF; 2001. p. 236
  13. 13. Carmo M. Differential Geometry of Curves and Surfaces: Revised and Updated. 2nd ed. Mineola: Dover Publications; 2016. p. 528
  14. 14. Araújo P. Geometria diferencial. 1st ed. Rio de Janeiro: IMPA; 1998
  15. 15. Gray A. Modern Differential Geometry. 3rd ed. Boca Raton: CRC Press; 2006. p. 1053
  16. 16. Kiureghian A, De Stefano M. An Efficient Algorithm for Second-Order Reliability Analysis. Report no. UCB/SEMM – 90/20. University of California; 1990

Written By

Emmanoel Ferreira

Submitted: 02 January 2023 Reviewed: 08 January 2023 Published: 27 February 2023