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SURFACE CURVATURE: NORMAL, TOTAL AND MEAN CURVATURE, EULER’S THEOREM, MEUSNIER’S THEOREM, UMBILICAL POINT, RODRIGUES’ FORMULA, LINES OF CURVATURE


Curvature in a plane.


ole.gif

Theorem 1. Given a plane curve C shown in Fig.1. Let T be the tangent to the curve at point P, Q be a point on the curve near P and let QM be a line perpendicular to the tangent. Let h = QM and l = PM as shown in the figure. Let k be the curvature of the curve at point P. Then we have the following relationship between k, l and h: 


ole1.gif  



Proof.


Thus we see that the curvature describes the rate at which the curve leaves the tangent.




Def. Plane section (of a surface). The intersection of a plane and a surface i.e. the curve defined by the intersection of a plane and a surface. For example, the plane section formed by the intersection of a sphere and a plane is a circle.


Def. Normal section (of a surface). Let us construct a normal to a surface at a point P. Then the curve that is described on the surface by any plane passing through the normal (i.e. containing the normal) is called a normal section of the surface. In other words a normal section is a plane section formed by a plane containing a normal to the surface.



Curvature of a surface at a point. Let us construct a unit normal ole2.gif and a tangent plane Q at some given point P on some surface S and consider the curves that are formed on the surface by planes passing through P containing the normal i.e. the various normal sections passing through point P. Each normal section passing through P possesses a particular curvature at point P. We can specify a particular normal section by use of a polar coordinate system constructed on the tangent plane, origin at point P, polar axis as some arbitrarily chosen initial ray in the tangent plane, and an angle α measured counterclockwise from the polar axis to the plane of the normal section. The curvature at point P in direction α is thus given as the function kn(α). For each value of α there is a curvature associated with that particular normal section. This curvature kn(α) is called the normal curvature of the surface S at point P in the direction α..


Let S be a simple surface element defined by the one-to-one mapping


            x = x(u, v)

            y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space. Let ole3.gif


             ole4.gif


be the position vector to point P on the surface and let ole5.gif be the unit normal to the surface at point P. Then the normal curvature at point P is given by


ole6.gif


ole7.gif

where E, F, G, L, M, N are the fundamental coefficients of the first and second order.

                                                                                    

Proof.


The curvature of a normal section, kn, is positive if the surface unit normal ole8.gif is pointed in the direction of concavity of the surface, negative if the unit normal ole9.gif is pointed in the direction of convexity. Fig. 2 shows a situation in which the surface normal is pointed in the direction of concavity. In this case the normal curvature would be positive at point P. The direction of the surface unit normal is arbitrary, a matter of choice.

 

Formula 2) above can be re-written in the following way


ole10.gif


simply by dividing the numerator and denominator by du2. In this form it is obvious that kn is a function of the ratio dv/du. If we let tan θ = dv/du then 3) becomes

                                                                                                

ole11.gif



Setting tan θ = sin θ / cos θ gives the more symmetric form



ole12.gif



 

 A surface may be curved in many ways and consequently one might think that the dependence of the curvature k on the angle α might be arbitrary. In fact this is not so. The following theorem is due to Euler.


Euler’s theorem. At each point P on a surface there exist two particular directions such that

 

1. They are mutually perpendicular.


ole13.gif

2. The curvatures k1 and k2 of the normal sections in these directions are the smallest and the largest values of the curvatures of all normal sections at point P (in the particular case where k1 = k2 the curvature of all sections is the same, as for example, on a sphere).


3. Let φ be the angle, in the tangent plane, measured counterclockwise from the direction of minimum curvature k1 . Then the normal curvature kn(φ) in direction φ is given by


            kn(φ) = k1 cos2φ + k2 sin2φ


[On examination of this formula we note that kn(φ) = k1 at φ = 0 and φ = π and kn(φ) = k2 at φ = π/2 and φ = 3π/2. See Fig. 3]

ole14.gif

Proof.                                                                           


The directions corresponding to the minimum and maximum values of curvature are called the principal directions of the surface. The values k1 and k2 are called the principal curvatures of the surface.

                                                                                     


Values of k1 and k2 and surface type.



ole15.gif

k1 and k2 have same sign. If k1 and k2 have the same sign, the sign of kn(φ) is constant and the surface near the point has the form shown in Fig. 4.

 

k1 and k2 have opposite signs. If k1 and k2 have opposite signs it can be shown that as φ passes through values in the interval from 0 to π the sign of kn(φ) changes twice so that near the point the surface has a saddle-shaped form as shown in Fig. 5.


ole16.gif

One of the numbers k1 and k2 is equal to zero. If one of the numbers k1 and k2 is equal to zero, the curvature always has the same sign, except for the one value of φ for which it vanishes. This occurs, for example, for every point on a cylinder. See Fig. 6. In the general case the surface near such a point has a form close to that of a cylinder.

                                                                                                

k1 = k2 = 0. If k1 = k2 = 0 all normal sections have zero curvature. In the vicinity of such a point the surface is especially close to its tangent plane. Such points are called flat points and the properties of a surface near a flat point may be very complicated. An example of such a point is point M shown in Fig. 7 (this surface is called a monkey saddle) .



ole17.gif

Meusnier’s theorem. Let us now consider the plane section L formed by an arbitrary plane Q passing through some point P on a surface S (i.e. a plane Q not, in general, passing through the normal). See Fig. 8. Let the angle between plane Q and the normal be θ, as shown in the figure. Meusnier showed that the curvature kL of the curve L at point P is related to the curvature kN of the normal section of the same direction (i.e. plane Q and the plane of the normal section intersect the tangent plane in the same straight line) at P by the formula

 

             ole18.gif

 

Proof.



Vector form of Meusnier’s Theorem. Let CL be any curve passing through a point P on a surface S. Let ole19.gif be the unit normal to the surface at point P, ole20.gif be the unit tangent to curve CL at P and ole21.gif be the curvature of CL at P. Let ole22.gif be the curvature at P of the normal section passing through the unit normal ole23.gif and the tangent ole24.gif . Then ole25.gif and ole26.gif are related by


             ole27.gif


where θ is the angle between and ole28.gif and ole29.gif .

ole30.gif

Thus, if we know the principal curvatures k1 and k2 for a particular point P on a surface, the curvature of any curve passing through P is defined by the direction of its tangent at P and the angle between its osculating plane and the normal to the surface. One can therefore say that the character of the curvature of a surface at a given point is completely defined by the two numbers k1 and k2.



Computing the principal directions and curvatures at a point P. Given a point P on a surface S, the directions at which the normal curvature at P attains its minimum and maximum values can be computed as follows. Let the normal curvature at P be given as


ole31.gif


where λ = dv/du . We wish to find those values of λ at which the function kn(λ) has its minimum and maximum values. We are thus faced with a problem of finding the maxima and minima of a function. A necessary condition for the function kn(λ) to have a maxima or minima at a point is that at that point d kn(λ) /dλ = 0. Using the usual formula for computing the derivative of a quotient we obtain


             ole32.gif


or


 

8)        (E + 2Fλ + Gλ2)(M + Nλ) - (L + 2Mλ + Nλ2)(F + Gλ) = 0


Upon expansion and rearrangement 8) becomes

 

9)        (FN - GM)λ2 + (EN - GL)λ + EM - FL = 0


One can then solve 9) for its two roots using the quadratic formula thus obtaining the two principal directions λ1 and λ2. One can then substitute the two values λ1 and λ2 into 6) to obtain the principal curvatures k1 and k2 . One can also find the principal curvatures k1 and k2 using the following theorem.


Theorem 2. The principal curvatures k1 and k2 are given by the quadratic equation

 

10)      (EG - F22 - (EN + GL - 2FM)κ + (LN - M2) = 0


Proof.


Solving 10) using the quadratic formula gives the two principal curvatures k1 and k2 .



Theorem 3. A real number κ is a principal curvature at P in the direction dv/du if and only if κ, du and dv satisfy


11)      (L - κE)du + (M - κF)dv = 0

            (M - κF)du + (N - κG)dv = 0


where du2 + dv2 ole33.gif 0.



Umbilical point on a surface. A point of a surface S which is either a circular point or a planar point of S. A point of S is an umbilical point of S if, and only if, its first and second fundamental quadratic forms are proportional. The normal curvature of S is the same in all directions on S at an umbilical point of S. All points on a sphere or plane are umbilical points. The points where an ellipsoid of revolution cuts the axis of revolution are umbilical points.

Syn. Umbilic point.

                                                                                                James & James. Mathematics Dictionary.




Equation 9) above which gives the two principal directions λ1 and λ2 at a point P can be written in terms of the variables du and dv as

 

12)      (EM - FL)du2 + (EN - GL)dudv + (FN - GM)dv2 = 0


or, equivalently,


ole34.gif


The following is an important theorem:


Theorem 4. Let S be a simple surface element defined by the one-to-one mapping


            x = x(u, v)

            y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space where the defining functions x(u, v), y(u, v), z(u, v) have continuous second order derivatives i. e. S is a class 2 surface. Then a direction dv/du is a principal direction at a point P of S if and only if du and dv satisfy

 

14)      (EM - FL)du2 + (EN - GL)dudv + (FN - GM)dv2 = 0

 

At a nonumbilical point the above equation can be shown to factor into two linear equations of the form Adu + Bdv = 0 for the two perpendicular directions.



Rodrigues’ Formula. Let S: ole35.gif be a simple surface element and let ole36.gif be the unit normal to the surface at point P. The direction dv/du is a principal direction at point P on S if and only if for some scalar κ


                        d ole37.gif = -κ d ole38.gif


where ole39.gif and ole40.gif .


When such is the case κ is the principal curvature in the direction dv/du.



Lines of curvature. A curve on a surface whose tangent at each point is along a principal direction is called a line of curvature. In other words, given a curve C on a surface S, if at each point of C its tangent is pointed in a principal direction, C is a line of curvature. Thus a curve is a line of curvature if and only if at each point on the curve the direction of its tangent satisfies


15)      (EM - FL)du2 + (EN - GL)dudv + (FN - GM)dv2 = 0 .


Lines of curvature can be defined as exactly those curves that satisfy this equation. Equation 15) can be regarded as a differential equation for two families of lines of curvature.

ole41.gif

Fig. 9 depicts two orthogonal “line of curvature” systems on a half sphere.


Theorem 5. In the neighborhood of a nonumbilical point on a class 3 surface there exist two orthogonal families of lines of curvature.


If a surface is sufficiently smooth one can introduce a class C2 parametric representation for a surface element in the neighborhood of a nonumbilical point P such that the u and v coordinate curves are themselves the lines of curvature. 


Theorem 6. For every point P on a class C2 surface S there exists a mapping from the uv-plane into the surface, i.e. a parametric representation for the surface, such that the directions of the u and v coordinate curves at P are principal directions.


Theorem 7. The directions of the u and v coordinate curves at a nonumbilical point on a surface element are in the direction of the principal directions if and only if F = M = 0 at the point.


Corollary. The u and v coordinate curves on a surface element without umbilical points are lines of curvature if and only if at every point on the element F = M = 0.


Theorem 8. If the directions of the u and v coordinate curves at a point P on a surface element are principal directions, then the principal curvatures are given by k1 = L/E and k2 = N/G.


Corollary. If the u and v coordinate curves on a surface element are lines of curvature, then at each point the principal curvatures are given by k1 = L/E and k2 = N/G.



Mean curvature. The mean curvature at a point on a surface is the average of the principal curvatures at the point i.e. if k1 and k2 are the principal curvatures of the point the mean curvature is

                                                            

            Kav = ½ ( k1 + k2) .


The mean curvature at a point P is given by

                                                            

             ole42.gif


where E, F, G, L, M, N are the Fundamental Coefficients of the First and Second Order evaluated at point P. Mean curvature is a concept frequently encountered in applications in physics and engineering, often in differential equations.


Syn. Mean normal curvature



Total curvature (or Gaussian curvature). The total curvature (or Gaussian curvature) at a point on a surface is the product of the principal curvatures at that point i.e. if k1 and k2 are the principal curvatures of the point the mean curvature is

 

            K = k1k2


The total curvature at a point P is given by

 

             ole43.gif


where E, F, G, L, M, N are the Fundamental Coefficients of the First and Second Order evaluated at point P.


The sign of the total curvature at a point defines the character of the surface near that point. If K > 0 at a point then the surface in the vicinity of the point has the form of a bowl ( k1 and k2 have the same sign) and if K < 0 at the point then the surface in the vicinity of the point has the form of a saddle ( k1 and k2 have opposite signs).


Syn. Total normal curvature






References.

1. James & James. Mathematics Dictionary.

2. Mathematics, Its Contents, Methods and Meaning. Vol. II, Chapter VII.

3. Lipschutz. Differential Geometry. Chapter 9.

 




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