SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

GEODESICS, GEODESIC CURVATURE, GEODESIC PARALLELS, GEODESIC COORDINATES, GEODESIC TORSION, GAUSS-BONNET THEOREM

Geodesic. Intuitively, a geodesic is the shortest arc between two points on a surface. If we stretch a rubber band between two points on a convex surface, the rubber band will take the path of the geodesic. See Fig. 1. A geodesic C on a surface S has the properties that at each point of C the principal normal coincides with the normal to S and the geodesic curvature vanishes identically. Conversely, if on a curve C on a surface S, the principal normal coincides with the surface normal at every point, or if the geodesic curvature vanishes identically at every point, the curve is a geodesic. If a straight line lies on a surface, then the line is a geodesic of the surface.

Although defining a geodesic as the shortest arc between two points on a surface gives the main idea of a geodesic, there is a problem with it as a definition. Not every geodesic is a shortest path in the large, as can be seen by noting that on the surface of a sphere every arc of a great circle is a geodesic even though an arc will be the shortest path between two points only if that arc is not greater than a semicircle. From this example we see that a geodesic can be a closed curve. Because of this difficulty a geodesic is often defined as an arc C on a surface S at each point of which the principal normal coincides with the normal to S — or an arc at every point of which the geodesic curvature vanishes identically.

Theorem 1. If two surfaces are tangent along a curve that is a geodesic on one of them, then this curve will be a geodesic on the other.

Reason? At each point along the curve the principal normals coincide with the surface normals.

For an illustration of this theorem see Fig. 2. A straight plane strip of paper with a median line drawn on it is placed in different positions on a cylinder. One can see from this example that a geodesic line on a cylinder can be one of three things: a straight line, a circle or a helix.

Geodesic curvature. Let C be a curve on a surface S. The geodesic curvature of C at a given point P is defined as the curvature, at P, of the orthogonal projection of C onto the plane Q tangent to S at point P. See Fig. 3, where C* is the projection of C onto the tangent plane Q. The geodesic curvature of C at P is defined then as the curvature of C* at P.

For the motivation that led to this definition let us note that if C represents an arc of minimum length between two points A and B on a surface S and point P lies on the arc then the projection of C onto the tangent plane at point P should be a straight line and the geodesic curvature of C in that situation should be counted as zero. Thus as candidates for the arcs of minimum length we are led to consider those curves where the curvature of the orthogonal projection of the curve onto the tangent plane is zero.

Formula for computing geodesic
curvature. Given a curve C: u = u(s), v =
v(s) on a surface S:
where s is arc
length. The curvature vector of C at point P
is defined as the vector k =dt/ds where t is the
tangent vector t =
. Let N be the unit
surface normal at point P, T be the unit tangent
vector to C at point P and U be a unit vector in
the tangent plane Q defined by U = N
T
creating the orthonormal triad shown in Fig. 4. Vector k is orthogonal to T, lying in the N-U
plane. Then the geodesic curvature vector k_{g} is the component of k along vector U i.e.

k_{g} = (k•U)U

Theorem 2. The geodesic curvature vector k_{g} of a curve C at P is the vector projection of
the curvature vector k of C at P onto the tangent plane at P.

One can write the vector k as

k = k_{g} + k_{n}

where k_{g} is the component of k along the U axis and k_{n} is its component along the N axis. The
component k_{n} along the N axis is the normal curvature of the surface S at P in the direction of
tangent T i.e. the curvature of the normal section defined by N and T.

Let the magnitude of the vector k_{g} be denoted by k_{g} i.e. k_{g} = |k_{g}| . Then

k_{g} = k_{g}U .

The scalar k_{g} is called the geodesic curvature of C at P.

Let us denote the magnitude of the vector k by k, i.e. k = |k|. Then the curvature k of curve C at
point P is related to the geodesic curvature k_{g} at P by

k_{g} = k cos θ

where θ is the angle between the osculating plane of C and the tangent plane Q.

Theorem 3. If point P on curve C of surface S is represented by the position vector
then
k_{g} is given by the following box products

k_{g} = [T k N]

or

where N is the normal to the surface at P and T is the unit tangent to C at P.

This theorem follows from k_{g} = k∙U = k∙(N
T).

Liouville’s formula for geodesic curvature. Given: The curves C_{1} and C_{2} of an
orthogonal system on a surface are so directed that at each point the directed angle from the
directed curve C_{1} to the directed curve C_{2} is π/2 (e.g. curves C_{1} and C_{2} could be u- and v-coordinate curves of an orthogonal system). Then the geodesic curvature of an arbitrary directed
curve C: u = u(s), v = v(s) is then given by the formula

k_{g} = k_{1} cos θ + k_{2} sin θ + dθ/ds

where θ is the directed angle at an arbitrary point P of C from the directed curve C_{1} through P to
the directed curve C, k_{1} is the geodesic curvature of curve C_{1}, k_{2} is the geodesic curvature of
curve C_{2}, and s is arc length.

Geodesics and asymptotic lines. A curve is an asymptotic line if and only if at each
point of the curve k_{n} = 0; a curve is a geodesic if and only if at each point of the curve k_{g} = 0. A
asymptotic line is either a straight line or a curve along which the osculating plane and the
tangent plane coincide; a geodesic is either a straight line or a curve along which the osculating
plane is perpendicular to the tangent plane.

Theorem 4. The geodesic curvature k_{g} of a curve is equal numerically to the ordinary
curvature k at every point of the curve if and only if the curve is an asymptotic line.

The geodesic curvature k_{g} of a curve which is not a straight line is equal numerically to the
ordinary curvature k at every point of the curve if and only if the osculating plane of the curve at
each point is the tangent plane to the surface at the point. As an example consider a circle in a
plane, which meets the requirements for being an asymptotic line whose osculating plane at each
point is the tangent plane.

Beltrami’s formula for geodesic curvature. Given a curve C: u = u(s), v = v(s) on a surface S: where s is arc length. Beltrami’s formula for the geodesic curvature at point P of the curve is:

where the Γ_{ij}^{k} are the Christoffel symbols of the second kind.

In contrast to k_{n} which depends on both the first and second fundamental coefficients, the
geodesic curvature k_{g} depends only on the first fundamental coefficients E, F and G (and their
derivatives) and is thus an intrinsic property of the surface. This is evident in the above formula
since the functions Γ_{ij}^{k} are functions of the first fundamental coefficients E, F and G only.

Geodesic curvature of the coordinate curves. We can compute the values of du/ds and dv/ds in 1) above for the u- and v-coordinate curves from the first fundamental quadratic form. For the u-coordinate curves where v = constant, dv/ds = 0 and du/ds = 1/ . Along the v-coordinate curves where u = constant, du/ds = 0 and dv/ds = 1/ . Substituting into 1) we get

For the case where the coordinate curves are orthogonal, F = 0, Γ_{11}^{2} = -½E_{v}/G, Γ_{22}^{1} = -½G_{u}/E.
Thus

Theorem 5. The u-curves on a surface are geodesics if and only if Γ_{11}^{2} = 0; and the v-curves if and only if Γ_{22}^{1} = 0.

Proof. This theorem follows directly from 2) above.

Theorem 6. If the coordinate curves form an orthogonal system, the u-curves are geodesics when and only when E is a function of u alone, and the v-curves are geodesics when and only when G is a function of v alone.

Proof. This theorem follows directly from 3) above. For the u-curves, if E is a function of u
alone, then E_{v} = 0 and k_{g} = 0. For the v-curves, if G is a function of v alone, then G_{u} = 0 and k_{g} =
0.

Corollary. If both the u-curves and the v-curves are geodesics then E and G are, respectively, functions U(u) and V(v), of u and v.

Theorem 7. If there exists on a surface an orthogonal system of geodesics, the system is a developable or a plane.

Examples of systems of orthogonal geodesics.

1. Let α be the set of all lines in the plane directed in any specified direction. Let β be the set of all lines in the plane perpendicular to the α lines. See Fig. 5. Then the two families of lines constitute an orthogonal system of geodesics.

2. Let α be the set of all lines in the surface of a cylinder parallel to the axis of revolution. Let β be the set of all circles formed as plane sections by planes cutting through the cylinder perpendicular to the axis of revolution. See Fig. 6. Then the two families of lines constitute an orthogonal system of geodesics.

Differential equations of the geodesics.

Theorem 8. 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. Then a directed curve C on S represented parametrically in terms of arc length s by

u = u(s)

v = v(s)

is a geodesic if and only if u(s) and v(s) satisfy the following differential equations:

Using equations 4) and an existence theorem from the theory of differential equations one can prove the following:

Theorem 9. In the neighborhood of a point P on a surface of class 3 there exists one and only one geodesic through P in any given direction.

Theorem 10. Let S: be a simple surface element of class 2 such that E = E(u), F = 0, and G = G(u). Then

1. The v-curves are geodesics.

2. The u-curve v = v_{0} is a geodesic if and only if G_{u}(u_{0}) = 0.

3. A curve of the form is a geodesic if and only if

where C is a constant.

*************************************************

Geodesic parallels, geodesic coordinates

*************************************************

Def. Trajectory. A curve which cuts all curves (or surfaces) of a given family at the same angle. An orthogonal trajectory is a curve which cuts all the members of a given family of curves (or surfaces) at right angles.

James/James. Mathematics Dictionary.

Geodesic parallels. The orthogonal trajectories of a family of lines in the plane have the following property: the segments cut from the lines by any two of them are all equal. The orthogonal trajectories of a family of geodesics on an arbitrary surface have the same property. Any two of them cut equal segments from the geodesics. Conversely, if any two orthogonal trajectories of a family of curves on a surface cut equal segments from the curves of the family, the curves of the family are geodesics on the surface.

Theorem 11. A necessary and sufficient condition that the segments cut from the curves of a family of curves on a surface by two arbitrarily chosen orthogonal trajectories of the family be all equal is that the curves of the family be geodesics on the surface.

Def. Geodesic parallels. The orthogonal trajectories of a family of geodesics are known as geodesic parallels. They are called parallels because each two of them are equally distant and, geodesic parallels because the distances in question are measured along geodesics.

Example of geodesic parallels. An example of geodesic parallels are the parallel circles that are traced out on a surface of revolution by the individual points of the plane curve that is rotated to generate the surface. The parallels of latitude on a sphere are geodesic parallels.

By Theorem 7, a family of geodesic parallels can consist of geodesics only if the surface is a developable or a plane. Normally geodesic parallels are not geodesics.

Geodesic parallels on a surface. Given a smooth curve C_{0} on a surface S, there exists a
unique family of geodesics on S intersecting C_{0} orthogonally. If segments of equal length s be
measured along the geodesics from C_{0}, then the locus of their end points is an orthogonal
trajectory C_{s} of the geodesics. The curves C_{s} are geodesic parallels.

James/James. Mathematics Dictionary.

Geodesic coordinates. A simple surface element where the coordinate curves are orthogonal and one of the families of coordinate curves are geodesics is called a set of geodesic coordinates.

Def. Geodesic parameters (coordinates). Parameters u, v for a surface S such
that the curves u = const. are the members of a family of geodesic parallels, while the curves v =
const. = v_{0} are members of the corresponding orthogonal family of geodesics, of length u_{2} - u_{1}
between the points (u_{1}, v_{0}) and (u_{2}, v_{0}). A necessary and sufficient condition that u, v be geodesic
parameters is that the first fundamental form of S reduce to ds^{2} = du^{2} + G dv^{2}.

James/James. Mathematics Dictionary.

From Theorem 6 we know that a necessary and sufficient condition for the u-curves to be geodesics and the v-curves to be geodesic parallels orthogonal to them is that E = U(u) and F = 0. This means that the first fundamental form has the form

5) ds^{2} = U(u)du^{2} + Gdv^{2}.

If we then make a substitution

5) becomes

Geodesic polar coordinates. These are geodesic parameters u, v for a surface S,
except that the curves u = const. = u_{0}, instead of being geodesic parallels, are concentric geodesic
circles, of radius u_{0}, and center, or pole, P corresponding to u = 0; the curves v = v_{0} are the
geodesic radii; and for each v_{0}, v_{0} is the angle at P between the tangents to v = 0 and v = v_{0}.
Necessary and sufficient conditions that u, v be geodesic polar coordinates are that the first
fundamental quadratic form of S reduce to ds^{2} = du_{2} + μ^{2}dv^{2}, η
0, and that at u = 0 we have η =
0 and
=1. All points on u = 0 are singular points corresponding to P.

James/James. Mathematics Dictionary.

Geodesic circle on a surface. If equal lengths are laid off from a point P of a surface S along the geodesics through P on S, the locus of the end points is an orthogonal trajectory of the geodesics. The locus of end points is called a geodesic circle with center at P and radius r. The “radius” r is a geodesic radius; it is the geodesic distance on the surface from the “center” P to the circle. See “geodesic polar coordinates”.

James/James. Mathematics Dictionary.

Bonnet’s formula for geodesic curvature. Let a curve C on a surface S: be defined by an equation of the form

Then Bonnet’s formula for the geodesic curvature at point P of the curve is:

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

Geodesic torsion.

Torsion of a geodesic. The torsion of a geodesic passing through point P of a surface in the direction of the unit tangent vector T is given by

where is the unit surface normal at point P and is the position vector at point P, or, equivalently, by

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

We note that the condition that the geodesic torsion τ_{g} = 0 along a curve is tantamount to the
condition that the curve is a line of curvature. A curve is a line of curvature if and only if at each
point on the curve the direction of its tangent satisfies

9) (EM - FL)du^{2} + (EN - GL)dudv + (FN - GM)dv^{2} = 0 .

The left member of 9) is identical to the numerator in 8) so if τ_{g} = 0 along a curve the curve is a
line of curvature.

Theorem 12. The geodesic torsion of a curve is identically zero if and only if the curve is a line of curvature.

Theorem 13. A geodesic, which is not a straight line, is a plane curve if and only if it is a line of curvature.

Def. Geodesic torsion of a surface at a point in a given direction . The
expression for τ_{g} in 7) above is similar to the expression for normal curvature in that it depends
only on the point P and on the direction dv/du at P. The quantity τ_{g} defined by 7) is called the
geodesic torsion of the surface at P in the direction dv/du. Though the torsion at P of the
geodesic which issues from P in a given direction fails to exist when the geodesic is a straight
line, the geodesic torsion always exists because 7) defines it for every direction at P, regardless of
the nature of the geodesic in the direction.

Theorem 14. The geodesic torsion at a point P on a surface is related to the normal
curvatures k_{1} and k_{2} at P by

10) τ_{g} = ½ (k_{2} - k_{1}) sin 2θ

where θ is the angle, in the tangent plane, measured counterclockwise from the direction of
minimum curvature k_{1} .

At an umbilic, the geodesic torsion is zero in every direction. If we exclude this case, 9) reveals that the geodesic torsion is zero in the two principal directions and takes on its extreme values in the two perpendicular directions that bisect the angles between the principal directions.

Since sin 2(θ + π/2) = - sin 2θ and sin 2(-θ) = - sin 2θ we have the following:

Theorem 15. 1. The geodesic torsions in two perpendicular directions, θ and θ + π/2, are negatives of each another. 2. The geodesic torsions in two directions +β and -β, as measured from a principal direction, are negatives of each other.

In particular, the two extreme values of the geodesic torsion at a point are negatives of each another. Also, the geodesic torsions in the two asymptotic directions are negatives of one another.

Def. Geodesic torsion of a curve on a surface. By the geodesic torsion of a curve C (whether curve C is a geodesic or not) at a point P is meant the geodesic torsion of the surface at P in the direction which C has at P. Thus all curves through P that have the same direction at P have the same geodesic torsion at P.

Theorem 16. The relationship between the geodesic torsion τ_{g} of a curve C : u = u(s), v =
v(s) at a point P on a surface S and the ordinary torsion τ of C at point P is given by

11) τ_{g} = dα/ds - τ

where α is the directed angle from the unit surface normal N to the principal normal n to C.

We note that the angle α represents the angle between the osculating plane of C and the tangent plane to the surface.

Theorem 17. The ordinary torsion and the geodesic torsion of a curve C, which is not a straight line, are identically equal if and only if the osculating plane of C makes a constant angle with the tangent plane to the surface.

It follows from 11) that if τ_{g} = 0, then τ = 0 when and only when dα/ds = 0. In other words:

Theorem 18. A necessary and sufficient condition that a line of curvature, other than a straight line, be a plane curve is that its osculating plane always makes the same angle with the tangent plane to the surface.

The Gauss-Bonnet Theorem, published by Bonnet in 1848, is an application of Green’s theorem to the integral of geodesic curvature.

Gauss-Bonnet Theorem. If the Gaussian
curvature K of a surface is continuous in a simply
connected region R bounded by a closed curve C
composed of k smooth arcs making at the vertices
exterior angles θ_{1}, θ_{2}, ... ,θ_{k}, then

where k_{g} represents the geodesic curvature of the arcs. See Fig. 7.

References.

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

2. Graustein. Differential Geometry.

3. Lipschutz. Differential Geometry. Chapter 11.

4. James/James. Mathematics Dictionary.

5. Struik. Lectures on Classical Differential Geometry.

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

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

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

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

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

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

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

[ Home ] [ Up ] [ Info ] [ Mail ]