In differential geometry, conjugate points or focal points [1] [2] are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing.

LECTURE 16: CONJUGATE AND CUT POINTS 1. Conjugate Points Let (M;g) be Riemannian and : [a;b] !Ma geodesic. Then by de nition, exp p ((t a)_ (a)) = (t): We know that exp p is a di eomorphism near...

The object point and image point of a lens system are said to be conjugate points. Since all the light paths from the object to the image are reversible, it follows that if the object were placed where the image is, an image would be formed at the original object position.

In optics, a conjugate plane or conjugate focal plane of a given plane P, is the plane P′ such that points on P are imaged on P′. [1] If an object is moved to the point occupied by its image, then the moved object's new image will appear at the point where the object originated.

2016年2月9日 · Let $p$ be some point on $c$, and $c_t$ be the arc of $c$ between $p$ and a point at the distance $t$. If $t$ is small, the index is still $0$, and if $t=t_0$ is the length of $c$ the index is $1$, so somewhere in between there is a conjugate point to $p$

4 CONJUGATE POINTS AND COMPARISON THEOREMS So, if ˇ= p , there exists a jsuch that I(V j;V j) <0. This means that Iis no longer positive on V. According to Theorem 0.7, (0) must have a conjugate point along . One of the basic facts in Riemannian geometry is Rauch’s Theorem. Let M and Mfbe riemannian manifolds, and let : [0; ] !M, ~: [0 ...

By definition, if $p,q$ are conjugate along some geodesic $\al$, there exsits a nonzero Jacobi field along $\ga$ that vanishes at $p,q$. This means there is some variation $\ga(t,s)$ of $\al$ ($\ga_0=\al$) where $J(t)= \frac{\partial \ga}{\partial s}(t,0)$.

2018年2月9日 · 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. A simple example of conjugate points are the north and south poles of a sphere (endowed with the usual metric of constant curvature) — every meridian is a geodesic segment having the poles as ...

Let x ∈ M, a point y ∈ Mis called a conjugate point ofx, if there exists a nontrivial Jacobi field J along a minimal geodesic linking x and y such that J vanishes at x and y. Conjugate Points. Let γ be a unit-speed geodesic starting at any point p of the sphere Σ of radius r.

At least within geometrical optics, there is then a one-to-one correspondence between points in the two planes: One point in one of the planes is mapped onto a certain point on the other plane, and vice versa, as shown in Figure 1. The two planes are then called conjugate to each other.