Website owner: James Miller
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 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
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, kn, 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 kn 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 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]
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 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.
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 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
Vector form of Meusnier’s Theorem. Let CL 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 CL at P and be the curvature of CL 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 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
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 - F2)κ2 - (EN + GL - 2FM)κ + (LN - M2) = 0
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
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
Kav = 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
1. James & James. Mathematics Dictionary.
2. Mathematics, Its Contents, Methods and Meaning. Vol. II, Chapter VII.
3. Lipschutz. Differential Geometry. Chapter 9.
The Way of Truth and Life
God's message to the world
Jesus Christ and His Teachings
Words of Wisdom
Way of enlightenment, wisdom, and understanding
Way of true Christianity
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
The teaching is:
On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
America has lost her way
The really big sins
Theory on the Formation of Character
You are what you eat
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
These things go together
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
The True Christian
What is true Christianity?
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
On responding to wrongs
Real Christian Faith
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
Sizing up people
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?