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# conjugate points

Let $M$ be a manifold on which a notion of geodesic is defined. (For instance, $M$ could be a Riemannian manifold, $M$ could be a manifold with affine connection, or $M$ could be a Finsler space.)

Two distinct points, $P$ and $Q$ of $M$ are said to be conjugate points if there exist two or more distinct geodesic segments having $P$ and $Q$ as endpoints.

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## Comments

## Conjugate vs. Transverse Axes/Radii?

Is the poles being "conjugate points" from which the y-axis of an ellipse/spheroid is considered the conjugate axis/diameter?:

http://books.google.com/books?id=Uk4wAAAAMAAJ&jtp=381

If so, then how can "ConjugateDiametersOfEllipse" be?:

http://images.planetmath.org:8080/cache/objects/9907/l2h/img35.png

Aren't they actually "oblique diameters"?

In "A treatise on analytical geometry", on pg.199, conjugate and transverse axes are noted, regarding oblate and prolate spheroids:

http://books.google.com/books?id=uHcLAAAAYAAJ&pg=PA199

But, back on pg.107, the above concept of "ConjugateDiametersOfEllipse" appears to being discussed

http://books.google.com/books?id=uHcLAAAAYAAJ&pg=PA107

How can that be? Are these two different meanings of "conjugate diameter?

~Kaimbridge~

## Re: Conjugate vs. Transverse Axes/Radii?

The conjugate diameters of ellipse have been defined in

http://planetmath.org/encyclopedia/ConjugateDiametersOfEllipse.html

Similar definitions may be set in hyperbola and parabola.

Jussi

## Incorrect definition

Conjugate points need not have more than one geodesic connecting them.

The correct definition is the existence of a non-trivial Jacobi fields along a geodesic between the two points and vanishes at both points.