Surface curvature: normal, total and mean curvature, Euler�s theorem, Meusnier�s theorem, umbilical point, Rodrigues� Formula, lines of curvature

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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.

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:

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 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 k_n(α). For each value of α there is a curvature associated with that particular normal section. This curvature k_n(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

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

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

Proof.

The curvature of a normal section, k_n, is positive if the surface unit normal is pointed in the direction of concavity of the surface, negative if the unit normal 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

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

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

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.

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 k_n(φ) in direction φ is given by

k_n(φ) = k₁ cos²φ + k₂ sin²φ

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

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.

k₁ and k₂ have same sign. If k₁ and k₂ have the same sign, the sign of k_n(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 k_n(φ) changes twice so that near the point the surface has a saddle-shaped form as shown in Fig. 5.

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) .

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 k_L of the curve L at point P is related to the curvature k_N 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

Proof.

Vector form of Meusnier’s Theorem. Let C_L be any curve passing through a point P on a surface S. Let be the unit normal to the surface at point P, be the unit tangent to curve C_L at P and be the curvature of C_L at P. Let be the curvature at P of the normal section passing through the unit normal and the tangent . Then and are related by

where θ is the angle between and and .

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

where λ = dv/du . We wish to find those values of λ at which the function k_n(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 k_n(λ) to have a maxima or minima at a point is that at that point d k_n(λ) /dλ = 0. Using the usual formula for computing the derivative of a quotient we obtain

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 λ_2. 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 - F²)κ² - (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 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,

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: be a simple surface element and let 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 = -κ d

where and .

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.

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

K_av = ½ ( k₁ + k₂) .

The mean curvature at a point P is given by

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

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.