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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*:

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.

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 du^{2}. 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 k_{1} and k_{2} 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 k_{1} = k_{2} 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 k_{1} . Then the normal curvature k_{n}(φ) in direction φ is given by

k_{n}(φ) = k_{1} cos^{2}φ + k_{2} sin^{2}φ

[On examination of this formula we note that k_{n}(φ) = k_{1} at φ
= 0 and φ = π and k_{n}(φ) = k_{2} at φ = π/2 and φ = 3π/2. See
Fig. 3]

The directions corresponding to the minimum and maximum
values of curvature are called the principal directions of the
surface. The values k_{1} and k_{2} are called the principal
curvatures of the surface.

Values of k_{1} and k_{2} and surface type.

k_{1} and k_{2} have same sign. If k_{1} and k_{2} 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.

k_{1} and k_{2} have opposite signs. If k_{1} and k_{2} 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 k_{1} and k_{2} is equal to zero. If one of the numbers k_{1} and k_{2} 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.

k_{1} = k_{2} = 0. If k_{1} = k_{2} = 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

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 k_{1} and k_{2} 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 k_{1} and k_{2}.

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

or

8) (E + 2Fλ + Gλ^{2})(M + Nλ) - (L + 2Mλ + Nλ^{2})(F + Gλ) = 0

Upon expansion and rearrangement 8) becomes

9) (FN - GM)λ^{2} + (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 λ_{1} and λ_{2. } One can then substitute the two values λ_{1} and λ_{2} into 6) to obtain the principal
curvatures k_{1} and k_{2} . One can also find the principal curvatures k_{1} and k_{2} using the following
theorem.

Theorem 2. The principal curvatures k_{1} and k_{2} are given by the quadratic equation

10) (EG - F^{2})κ^{2} - (EN + GL - 2FM)κ + (LN - M^{2}) = 0

Solving 10) using the quadratic formula gives the two principal curvatures k_{1} and k_{2} .

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 du^{2} + dv^{2}
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 λ_{1} and λ_{2} at a point P can be written in
terms of the variables du and dv as

12) (EM - FL)du^{2} + (EN - GL)dudv + (FN - GM)dv^{2} = 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)du^{2} + (EN - GL)dudv + (FN - GM)dv^{2} = 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)du^{2} + (EN - GL)dudv + (FN - GM)dv^{2} = 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 C^{2} 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 C^{2} 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 k_{1} = L/E and k_{2} = 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 k_{1} = L/E and k_{2} = 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 k_{1} and k_{2} are
the principal curvatures of the point the
mean curvature is

K_{av} = ½ ( k_{1} + k_{2}) .

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 k_{1}
and k_{2} are the principal curvatures of the point the mean curvature is

K = k_{1}k_{2}

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 ( k_{1} and k_{2} 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 ( k_{1} and k_{2} 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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