analytical geometry – how do I find the equation to an ellipse centered at any point?

It is difficult for me to learn that ellipse, I use The elements of the coordinate geometry by S.L. alone and his derivation of the equation into an ellipse goes like this Here and Here, As you can see, when he says so Let C be the origin and that simplifies everything and because this is just that we get a nice shape like this $$ frac {x ^ 2} {a ^ 2} + frac {y ^ 2} {b ^ 2} = 1 $$ I want to know how to find the equation for an ellipse in the center $ C $ should be with $ (h, k) $,

Here is my attempt: look at the coordinates of $ C $ his $ (h, k) $ and the coordinates of $ S $ his $ (p, q) $, since $ CS $ is $ a ~ e $, that's why $$ (p-h) ^ 2 + (q-k) ^ 2 = a ^ 2 ~ e ^ 2 $$
and similar coordinates of $ Z $ can be accepted $ (c, d) $ and again $$ (c-h) ^ 2 + (d-k) ^ 2 = 1 $$, Let us now turn to our example point $ P (x, y) $, by defining the ellipse: $$ SP ^ 2 = e ^ 2 PM ^ 2 $$
$$ (p-x) ^ 2 + (q-y) ^ 2 = e ^ 2 NZ ^ 2 $$ and now the problem comes, I can not write $ N $ how $ (x, 0) $ there $ ZCZ & # 39; $ is no longer that $ x- $Axis. Why only extracts that we get this problem? It was so easy with circles and parabolas, but when I became acquainted with Ellipse I had problems. You, the equation of the parabola $ y ^ 2 = 4ax $ Tell us that the opening is positive $ x- $Axis and it lies on $ x- $Axis and $ x ^ 2 = 4ay $ just tell us everything is on now $ y- $ Axis, but in the case of ellipses, we have to determine by comparison what the eclipse would look like $ a $ and $ b $ (half the major and minor axis lengths). If we are given an equation like $$ frac {x ^ 2} {a ^ 2} + frac {y ^ 2} {b ^ 2} = 1 $$ and said that the latus rectum would then find which formula we should use $ frac {2b ^ 2} {a} $ or $ frac {2a ^ 2} {b} $ (If it comes into question, it will only be given $ a $ and $ b $ and not said which is bigger).

Finally, I apologize for posting the pictures and not the thing written in latex, Well, I did it so that nothing should be missing, as there is a possibility of looking at something less worth mentioning when writing, but in reality it could be worthy. So literally the best thing is to avoid misunderstandings.

Many thanks.

Riemannian geometry – a consequence of the Ambrose-Singer theorem on holonomy

Consider $ nabla $ a connection in a vector bundle over a smooth distributor $ M $Consider a local frame $ sigma = ( sigma_1, sigma_2, …, sigma_m) $ on a contractable open sentence $ U subset M $ and calculate the curvature matrix $ Omega $ in relation to this framework $ theta_k $ a framework of $ 2 $ forms in $ TU. $ Take a look at the matrices $ S_ {k} $(with real smooth functions as entries) defined by the equation
$$
Omega = sum_k theta_k S_k. $$

It follows from Ambrose-Singer that the matrices $ S_k (p) $ span the lie algebra of $ Hol_p ^ o ( nabla) $ to the $ p in U? $ If so and if that $ S_k $ are symmetrically symmetrical at the same time, then that does not follow $ Hol_p ^ o ( nabla) $ is a subset of $ O (n)? $

Agal Algebraic Geometry – Can a non-singular curve be embedded in a singular surface?

To let $ (X, x) $ be an isolated, normal Gorenstein surface singularity. To let $ C subset X $ to be a curve in $ X $ and go through the singular point $ x $ the surface. Is it possible for the curve $ C $ is not singular, d. H. are there any examples, though $ C $ is not singular?

Ag.algebraic geometry – essentially Steinness of the projective manifold

As we all know, the projective manifold is essentially a stone manifold. Here we use the definition as follows: A Kahler manifold Y is essentially called a stone if there is an analytic hypersurface $ V subseteq Y $so that
$ Y setminus V $ is stone.

Now my question is:

For all non-zero global holomorphic sections $ s $ Can we take the effective divisor from each holomorphic line bundle? $ s ^ {- 1} (0) $ how $ V $ in the definition of essentially stone? Is namely $ Y setminus s ^ {- 1} (0) $ a Steinkr├╝mmer in this case? Or,$ s $ must be a non-zero global holomorphic section $ textbf {positive} $ holomorphic line bundle?

mg.metric geometry – Converting a 2D plane into a 3D surface

How can I project a 2D plane, such as an array of MxN dimensions, into a 3D surface? Currently, I am working on a research project that aims to design 2D grids and project them over a 3D surface, such as a hemisphere or a piramide, so that we can create them in 3D without designing them in 3D to have to . I've read about map projection, but I'm not sure if it will help me. Is there another mathematical method you can imagine?

dg.differential geometry – conforming parameterization that matches at the boundary

I am interested in the following question:
Consider two compliant immersion $ u_1 $ and $ u_2 $ from the disk $ overline { mathbb {D}} $ to $ mathbb {R} ^ 3 $ such that they (injectively) parameterize the same curve $ partial mathbb {D} $ and so that the normal ones also agree, i.
$$ vec {n} _1 circ u_1 = vec {n} _2 circ u_2 $$
from where $ vec {n} _i $ is the Gauss card from $ u_i ( overline { mathbb {D}}) $,

Can I find a conforming diffeomorphism of the disc? $ phi $ so that
$$ u_1 circ phi = u_2 hbox {on} partial mathbb {D}. $$

The question naturally arises when you consider the plateau problem for Willmore surfaces. They prescribe the limit and the normal limit. You can then assign the compliant Gauss card to each solution $ Y $as made by Bryant here, this is a (compliant) harmonic map $ mathbb {D} $ de Sitter room. I would like to know if there is any way to convert the plateau boundary condition when plunging into a Dirichlet condition on the compliant Gaussian map, that is, up to a compliant reparametrization to match the two compliant Gaussian maps point by point at the boundary compliant immersion with the same limit data.

Riemannian geometry – local isometry implies the cover of the map: non-empty borderline case

The following sentence is well known in the literature:

To let $ M $ and $ N $ Be Riemannian manifold and let $ f: M to N $ be a local isometry. If $ M $ is complete and $ N $ is then connected $ f $ is a covering card.

My question is: Does the same sentence apply if we accept that? $ M $ and $ N $ Are there any Riemannian manifolds now?

P .: This leads me to another question: how do we define the completeness of manifolds with boundaries?

Differential geometry – Relationship between the spatial derivative of a flow and its generating vector field

To let $ M $ Be a smooth variety and leave $ X $ a compact supported vector field $ M $, To let $ psi: M to M $ be that $ 1 $-Time flow associated with $ X $,

Is there a reasonable way to express yourself? $ d psi $ in terms of $ X $?

I'm referring to the "Spatially" derivative $ d psi_p: T_pM to T_ { psi (p)} M $, not for time derivation of $ t $Flow, which is given directly by $ X $ ($ frac {d} {dt} psi_t (p) = X ( psi_ {t} (p) $)

Ag.algebraic Geometry – Can the automorphism group in families of complex projective varieties vary too much?

In a family of smooth projective curves over a reduced complex scheme of finite type, the list of isomorphism classes of the automorphism groups of the fibers is finite. This results from the Hurwitz binding and the constancy of the genus on each connected component of the base.

Is it possible to prove a similar result for families of higher-dimensional varieties?

Ag.algebraic geometry – Pullback of a local system is constant

I have a finite eternal morphism $ f Colon Z Right Arrow X $ and I define the bundle of the crowd $ mathcal {F} _Z $ on $ X _ { acute {e} tale} $ how:
$$ mathcal {F} _Z (U) = Hom_X (U, Z) $$
This is a locally constant sheaf. I want to prove that the retreat of this sheaf continues $ Z _ { acute}} $ is constant. Basically, I want to prove that for every ├ętale card $ g colon U rightarrow Z $ the number of the card $ g \ colon U rightarrow Z $ so that $ g & # 39; circ f = g circ f $ is $ deg (f) $,