How to Check Pre Image is Continuous

Ray Optics

Avijit Lahiri , in Basic Optics, 2016

3.8.1.5 Entrance window, exit window, field of view

Among the preimages of all the stops in the system under consideration, there will be one that subtends the minimum angle (θ ₀) at the center (N) of the entrance pupil. This is referred to as the entrance window, and the stop itself is termed the field stop, since it determines the field of view of the system—that is, the angular width (θ ₀) of the area of the object plane, rays from which get admittance into it. The postimage of the field stop (ie, the image of the entrance window formed by the optical system as a whole) is termed the 'exit window.' In Fig. 3.32, S₂ depicts the field stop, while E and F are the entrance and the exit windows, respectively.

The entrance window and the entrance pupil are both located in the object space, though the entrance window need not be located to the left of the entrance pupil. Similarly, the exit pupil and the exit window are both located in the image space.

We turn our attention now to the instruments proper. The telescope (see Section 3.8.3) as also the microscope (see Section 3.8.4), is made up of an objective and an eyepiece, or ocular, which constitute the principal optical components of the system. The objective forms a real image of the object to be viewed, which is then magnified by the eyepiece to be either recorded on an appropriate device or viewed by the eye. While telescope and microscope objectives differ widely in their design principles, the eyepieces are of similar construction. The following section includes a brief outline of a number of eyepieces in common use in telescopes and microscopes.

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Linear Transformations from a Geometric Viewpoint

J. Douglas Carroll , Paul E. Green , in Mathematical Tools for Applied Multivariate Analysis, 1997

4.3.3 Applying the Concept of an Inverse to Point Transformations

Suppose we now return to the problem of finding the preimage vector x, given the image vector x ^* and the transformation matrix

$\text{T}=\left[\begin{array}{cc}0.90& 0.44\\ 0.60& 0.80\end{array}\right]$

As recalled from Section 4.2.3, ${\text{x}}^{*}=\left[\begin{array}{l}1.78\\ 2.20\end{array}\right]$ shown in Fig. 4.7. Now we wish to find x which, of course, we already know to be $\text{x}=\left[\begin{array}{l}1\\ 2\end{array}\right]$ . The starting equation is x^* = Tx, and we wish to solve for x.

By taking advantage of the fact that T ⁻¹ T = I, we can find x from a matrix transformation that premultiplies both sides of x^* = Tx by T ⁻¹. Thus

${\text{T}}^{-1}{\text{x}}^{*}={\text{T}}^{-1}\text{Tx}\text{=}\text{Ix}\text{=}\text{x}$

In terms of the specific problem of interest, we need to perform the following calculations. First, we obtain the determinant of T:

$\left|\text{T}\right|=0.9\left(0.8\right)-0.6\left(0.44\right)=0.46$

Based on the simple definition of the adjoint in the 2 × 2 case, we find

$\text{adj}\left(\text{T}\right)=\left[\begin{array}{cc}\hfill t_{22}& \hfill -t_{12}\\ \hfill -t_{21}& \hfill t_{11}\end{array}\right]=\left[\begin{array}{cc}\hfill 0.80& \hfill -0.44\\ \hfill -0.60& \hfill 0.90\end{array}\right]$

Having found both |T| and adj(T), we compute T ⁻¹ as follows:

${\text{T}}^{-1}=\frac{1}{0.46}\left[\begin{array}{cc}\hfill 0.80& \hfill -0.44\\ \hfill -0.60& \hfill 0.90\end{array}\right]=\left[\begin{array}{cc}\hfill 1.74& \hfill -0.96\\ \hfill -1.30& \hfill 1.96\end{array}\right]$

It now remains to show that

$\begin{array}{ccc}& {\text{T}}^{-1}& {\text{x}}^{*}\\ \text{x}=\left[\begin{array}{l}1\\ 2\end{array}\right]=& \left[\begin{array}{cc}\hfill 1.74& \hfill -0.96\\ \hfill -1.30& \hfill 1.96\end{array}\right]& \left[\begin{array}{l}1.78\\ 2.20\end{array}\right]\end{array}$

which is, indeed, the case.

We can also verify that, within rounding error, TT ⁻¹ = T ⁻¹ T = I. Finally, we should state that if |T| is zero, then 1/|T| is not defined, and the (regular) inverse of T does not exist. In this case the matrix T is said to be singular. Otherwise, as is the case here, it is called nonsingular.

A nonsingular matrix A, then, is one in which

$\begin{array}{c}\left|\text{A}\right|\ne 0\end{array}$

Nonsingularity is very important to the topic of matrix inversion since every nonsingular matrix has an inverse; moreover, only nonsingular matrices have (regular) inverses.

Now that we have found out how to compute a matrix inverse and solve the equation

$\text{x}={\text{T}}^{-1}{\text{x}}^{*}$

we should also state a property involving the inverse of the product of two (or more) matrices.

Given the product of two or more conformable matrices, T₁T₂ ⋯ T _s , the inverse of that product equals the product of the separate inverses in reverse order:

$\begin{array}{c}{\left({\text{T}}_1{\text{T}}_2\cdots {\text{T}}_s\right)}^{-1}={\text{T}}_s^{-1}\cdots {\text{T}}_2^{-1}{\text{T}}_1^{-1}\end{array}$

Notice that this property is similar to the property involving the transpose of the product of two or more matrices.

Having discussed some introductory aspects of matrix inversion, we return to the topic of vector transformation, but now in the context of changing basis vectors for the case of general linear transformations. As it turns out, the concept of matrix inverse is also needed here.

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Geometric Continuity

Jörg Peters , in Handbook of Computer Aided Geometric Design, 2002

8.3.2 Lemma

If p and g are polynomials, then, up to a common factor, λ, μ and v in (1) are polynomials of degree no larger than respectively

$\begin{array}{l}\mathrm{deg}ree(D_{e^{\perp }}g)+\mathrm{deg}ree\left(D_{e^{\perp }}p\right),\mathrm{deg}ree\left(D_{e}g\right)+\mathrm{deg}ree\left(D_{e^{\perp }}p\right)and\\ \mathrm{deg}ree\left(D_{e}g\right)+\mathrm{deg}ree\left(D_{e^{\perp }}g\right).\end{array}$

Proof Due to regularity of g along the boundary, the pre-image of the boundary is covered by overlapping intervals U such that for each U there are two components i, j ∈ {x, y, z} with det $M_{ij}\ne 0,M_{ij}:=\left[\begin{array}{l}\begin{array}{cc}{D}_e{g}^{[i]}& D_{e^{\perp }}{g}^{[i]}\end{array}\\ \begin{array}{cc}{D}_e{g}^{[j]}& D_{e^{\perp }}{g}^{[j]}\end{array}\end{array}\right]$ g ^[j] the j th component of g. Therefore we can apply Cramer's rule to

$M_{ij}=\left[\begin{array}{l}\begin{array}{c}{D}_{e^{\perp }}{p}^{[i]}\end{array}\\ \begin{array}{c}{D}_{e^{\perp }}{p}^{[j]}\end{array}\end{array}\right]$

Figure 8.12. A global periodic parametrization.

and obtain

$\left[\begin{array}{l}\lambda \\ -\mu \\ v\end{array}\right]=\frac{v}{\det M_{ij}}\left[\begin{array}{l}-\det \left[\begin{array}{l}{D}_{e^{\perp }}{p}^{[i]}{D}_{e^{\perp }}{g}^{[i]}\\ D_{e^{\perp }}{p}^{[j]}{D}_{e^{\perp }}{g}^{[j]}\end{array}\right]\\ \det \left[\begin{array}{l}{D}_e{g}^{[i]}{D}_{e^{\perp }}{p}^{[i]}\\ D_e{g}^{[j]}{D}_{e^{\perp }}{p}^{[j]}\end{array}\right]\\ \det \left[\begin{array}{l}{D}_{e^{\perp }}{g}^{[i]}{D}_e{g}^{[i]}\\ D_{e^{\perp }}{g}^{[j]}{D}_e{g}^{[j]}\end{array}\right]\end{array}\right].$

The degree of λ, μ and v is bounded by the degrees of the determinants since the common factor v/ det M _ij can be eliminated in the constraints. Since det M _ij vanishes at most at isolated points, the degree bound can be extended from U to the whole interval.

The characterizations of geometric continuity in terms of geometric invariants (tangents, curvatures) are characterizations of continuity by covariant derivatives [53].

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Linear Transformations

Stephen Andrilli , David Hecker , in Elementary Linear Algebra (Fourth Edition), 2010

Functions

If you are not familiar with the terms domain, codomain, range, image, and pre-image in the context of functions, read Appendix B before proceeding. The following example illustrates some of these terms:

Example 1

Let $f:{ℳ}_{23}\to {ℳ}_{22}$ be given by

$f\left(\left[\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right]\right)=\left[\begin{array}{cc}a& b\\ 0& 0\end{array}\right].$

Then f is a function that maps one vector space to another. The domain of f is ${ℳ}_{23}$ , the codomain of f is ${ℳ}_{22}$ , and the range of f is the set of all 2 × 2 matrices with second row entries equal to zero. The image of $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ under f is $\left[\begin{array}{cc}1& 2\\ 0& 0\end{array}\right]$ . The matrix $\left[\begin{array}{rrr}\hfill 1& \hfill 2& \hfill 10\\ \hfill 11& \hfill 12& \hfill 13\end{array}\right]$ is one of the pre-images of $\left[\begin{array}{cc}1& 2\\ 0& 0\end{array}\right]$ under f. Also, the image under f of the set S of all matrices of the form $\left[\begin{array}{ccc}7& \ast & \ast \\ \ast & \ast & \ast \end{array}\right]$ (where "*" represents any real number) is the set f(S) containing all matrices of the form $\left[\begin{array}{cc}7& \ast \\ 0& 0\end{array}\right]$ . Finally, the pre-image under f of the set T of all matrices of the form $\left[\begin{array}{cc}a& a+2\\ 0& 0\end{array}\right]$ is the set f ⁻¹(T) consisting of all matrices of the form $\left[\begin{array}{ccc}a& a+2& \ast \\ \ast & \ast & \ast \end{array}\right]$ .

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Hamiltonian Group Actions

L.C. Jeffrey , in Encyclopedia of Mathematical Physics, 2006

The Moment Polytope

Given a compact symplectic manifold M equipped with the Hamiltonian action of a torus T, we see that there is an associated polytope P, the "moment polytope." The fibers of the moment map μ are preserved by the action of T, so the value of μ parametrizes a family $\left\{M_t\right\}$ of symplectic quotients. By Theorem 1 the moment polytope is the convex hull of the images of the fixed-point set under the moment map.

By Proposition 1, we see that the moment polytope is decomposed according to the stabilizers of points in the preimage, and the critical values of the moment map are the images ${\mu }_T\left(W_j\right)$ of the fixed-point sets $W_j$ of one-parameter subgroups $S_j$ of T. These critical values form hyperplanes ("walls") which subdivide the moment polytope: the complement of the walls is a collection of open regions consisting of regular values of the moment map.

Example 12

The group SU(3) has maximal torus $T\cong \text{U}{\left(1\right)}^2$ . We identify $\mathit{g}*$ with g via the bi-invariant inner product (i.e., the Killing form) on g , and thus identify $\mathit{t}*$ with t . For $\lambda \in \mathit{t}$ , the Weyl group images of λ are the six vertices of a hexagon: the "walls" in the moment polytope for the action of T on the coadjoint orbit ${\mathcal{O}}_{\lambda }$ arising from the action of G on $\mathit{g}*$ through $\lambda \in \mathit{t}*$ are the edges of the hexagon (exterior walls) and the three lines connecting opposite vertices (interior walls).

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Topology of differentiable mappings

Kevin Houston , in Handbook of Global Analysis, 2008

Multiple point spaces

We shall assume now that our maps are finite and proper, i.e., each point in the target has a finite number of preimages and the preimage of a compact set is compact); for the moment we shall not assume smoothness of the map, and hence will have a continuous map f : X → Y.

There are many ways of defining multiple point spaces for a finite and proper map. For example, one can define the double point set as the set of points in X where f is not injective. That is, the closure of the set x ∈ X such that there exists y ≠ x such that f(x) = f(y). Alternatively, some authors define the double point set as the image of this set.

We take a third alternative which has a number of advantages. The double point space of a map f is the closure in X ² (= X x X) of the set of pairs (x, y), with x ≠ y, such that f(x) = f(y). This first advantage of this is that often this space is, in some vague sense, less singular than that which the other definitions give. The second advantage, though this may not appear so useful at first sight, is that this space has more symmetry - the group of permutation on 2 objects acts on X ² by permutation of copies.

We can generalize this so that the k ^th multiple point space of a map is the closure of the set of k-tuples of pairwise distinct points having the same image:

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Geometric Function Theory

Ch. Pommerenke , in Handbook of Complex Analysis, 2002

2.2 Prime ends and cluster sets

Now we turn to general simply connected domains.

Theorem 2.3

Let f map $\mathbb{D}$ conformally onto F. Then a curve in F ending at a point of ∂F has as preimage a curve in $\mathbb{D}$ ending at a point of $\mathbb{T}$ . Moreover curves with distinct endpoints on ∂F have preimages with distinct endpoints on the unit circle $\mathbb{T}$ .

More precisely, we are given a halfopen curve Г: w(t), 0 ≤ t < 1, with w(t) ∈ F such that lim _t→1 w(t) ∈ ∂F exists. Then ${\lim }_{t\to 1}{f}^{-1}(w(t))\in \mathbb{T}$ also exists. Note however that the image in F of a smooth curve in $\mathbb{D}$ can oscillate wildly.

A crosscut C of F is a Jordan arc that lies in F except for its (distinct) endpoints that lie on ∂F. A crosscut C divides a simply connected domain into exactly two domains U and V with [105, p. 27]

$F\cap \partial U=F\cap \partial V=F\cap C.$

It follows from Theorem 2.3 that its preimage f ⁻¹(C) is a crosscut of $\mathbb{D}$ .

Let w ₀ be some fixed point of F and let (C _n ) be a sequence of crosscuts of F with w ₀ ∉ C _n . Let V _n be the component of F \ C _n that does not contain w ₀. We say that (C _n ) is a null-chain if

(2.2.1) $\begin{array}{lll}{C}_n\cap C_{n+1}=\varnothing \hfill & V_{n+1}\subset V_n\hfill & \text{for}\text{all}n\hfill \end{array},$

(2.2.2) $\begin{array}{ll}\text{diam}{C}_n\to 0\hfill & \text{as}n\to \infty \hfill \end{array}.$

If F is unbounded then we have to use the spherical diameter. Note that (2.2.2) does not imply that diam V _n → 0.

The null-chains (C _n ) and $(C_n^{\prime })$ are called equivalent if, for each m,

(2.2.3) $\begin{array}{lll}{V}_n\subset V_m^{\prime },\hfill & V_n^{\prime }\subset V_m\hfill & \text{for}n>n_0(m)\hfill \end{array}.$

The equivalence classes of null-chains are called the prime ends of F. This concept was introduced by Carathéodory. It is perhaps a complicated notion but it gives a full and completely geometric description of a complicated situation.

The impression of the prime end p is defined by

(2.2.4) $I(p)={\displaystyle {\displaystyle \underset{n=1}{\overset{\infty }{\cap }}}{{\displaystyle \overline{V}}}_n}.$

The impression is a compact connected subset of ∂F and does not depend on the choice of the null-chain representing the prime end p.

Fig. 4. Three null-chains and a further sequence (at top right) that does not satisfy (2.2.2). The three null-chains are all non-equivalent.

We call ω ∈ I (p) a principal point of p if there is a null-chain (C _n ) representing p such that C _n → {ω} as n → ∞. The set Π(p) of all principal points is a compact connected subset of I(p).

The main result is the Prime End Theorem of Carathéodory [11]. See, e.g., [19, p. 172] or [105, p. 31] for a proof.

Theorem 2.4

Let f map $\mathbb{D}$ conformally onto F. Then there is a bijective correspondence between $\mathbb{T}$ and the set of prime ends of F with the following property:

If $\zeta \in \mathbb{T}$ and if (C _n ) is a null-chain representing the prime end p corresponding to ζ, then

(2.2.5) $f^{-1}(C_n)\subset \left\{z\in \mathbb{D}:{\delta }_n<|z-\zeta |<{\delta }_n^{\prime }\right\}$

with $0<{\delta }_n<{\delta }_n^{\prime }\to 0$ as n → ∞.

The cluster set C(f, ζ) of f at the point $\zeta \in \mathbb{T}$ consists of all $w\in {\displaystyle \widehat{\mathbb{C}}}$ such that there exist sequences (z _n ) in $\mathbb{D}$ with

(2.2.6) $\begin{array}{lll}{z}_n\to \zeta ,\hfill & f(z_n)\to \omega \hfill & \text{as}n\to \infty \hfill \end{array}.$

If $E\subset \mathbb{D}$ then the cluster set C_E (f, ζ) along E is the set of all ω for which there are (z _n ) in E satisfying (2.2.6). Thus

(2.2.7) $C_E(f,\zeta )={\displaystyle {\displaystyle \underset{r>0}{\cap }}\text{clos}\left\{f(z):z\in E,|z-\zeta |<r\right\}}.$

The following consequence of the Prime End Theorem is due to Carathéodory [11] and Lindelöf [72]. See, e.g., [105, p. 34] for the proof. A Stolz angle at $\zeta \in \mathbb{T}=\partial \mathbb{D}$ is the interior of a triangle Δ with a vertex at ζ and ${\displaystyle \overline{\Delta }}\subset \mathbb{D}\cup \{\zeta \}$ .

Theorem 2.5

Let f map $\mathbb{D}$ conformally onto F and let p be the prime end corresponding to $\zeta \in \mathbb{T}$ . The impression satisfies

(2.2.8) $I(p)=C(f,\zeta )$

Fig. 5. The inclusion relation (2.2.5) in the Prime End Theorem.

Fig. 6. Two unsymmetric prime ends; the left prime end has only one principal point, the other a whole segment of them.

while the set of principal points satisfies

(2.2.9) $\Pi (p)={\displaystyle {\displaystyle \underset{\Gamma }{\cap }}{C}_{\Gamma }(f,\zeta )=C_{[0,\zeta )}(f,\zeta )=C_{\Delta }(f,\zeta )},$

where Γ runs through all curves in $\mathbb{D}$ ending at ζ and where Δ is any Stolz angle at ζ.

The Collingwood Category Theorem [17], [19, p. 76] implies that I(p) = Π(p) holds generically in topological terms:

Theorem 2.6

If $f:\mathbb{D}\to {\displaystyle \widehat{\mathbb{C}}}$ is continuous then

$\begin{array}{ll}C(f,\zeta )=C_{[0,\zeta )}(f,\zeta )\hfill & for\zeta \in \mathbb{T}\backslash B,\hfill \end{array}$

where B is of first Baire category.

The one-sided cluster sets C ^±(f, ζ) consist of all $\omega \in {\displaystyle \widehat{\mathbb{C}}}$ for which there exist (z _n ) with

(2.2.10) $\begin{array}{llll}\arg z_n\gtrless \arg \zeta ,\hfill & z_n\to \zeta ,\hfill & f(z_n)\to \omega \hfill & \text{as}n\to \infty \hfill \end{array}.$

The Collingwood Symmetry Theorem [18], [19, p. 82], [105, p. 38] states:

Theorem 2.7

If $f:\mathbb{D}\to {\displaystyle \widehat{\mathbb{C}}}$ is any function then

(2.2.11) $C(f,\zeta )=C^{+}(f,\zeta )=C^{-}(f,\zeta )$

except possibly for countably many $\zeta \in \mathbb{T}$ .

The prime end corresponding to ζ is called symmetric if (2.2.11) holds. Thus all but countably many prime ends are symmetric. The prime end in Figure 5 is symmetric.

The continuun ∂F is indecomposable [87, p. 58] if and only if C ⁺(f, ζ) or C ⁻(f, ζ) = ∂F for some ζ [51]. There exists F such that C(f, ζ) = ∂F for all ζ [51, Theorem 4], [71]. See, e.g., [81,100,19,94,112] for more information about prime ends and their classification.

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Handbook of Dynamical Systems

Vitaly Bergelson , ... Máté Wierdl , in Handbook of Dynamical Systems, 2006

Definition 4.11

The system X = (X, B, μ, T) is an ergodic extension of Y = (Y, $\mathcal{D}$ , ν, S) if the only T-invariant sets in B are preimages of the invariant sets in $\mathcal{D}$ . The system X is a weakly mixing extension of Y if X × _Y X is an ergodic extension of Y .

One can show that most properties of the "absolute" weak mixing (and in particular, items (i) through (iv) in Theorem 4.4) extend, with obvious modifications, to statements about relative weak mixing. For example, one has the following fact.

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Fiber Bundles

Henri Bourlès , in Fundamentals of Advanced Mathematics V3, 2019

3.4.6 Preimage of a fiber bundle

Let M be a vector bundle with base B and λ = (M, B, π) the associated fibration. Let B′ be a manifold and f ⁰ : B′ → B a morphism of manifolds. The preimage fibration was defined earlier by f ^0⁎ (λ) = (M′, B′, π′), where M′ := B′ × _B M and the canonical morphism is (f ⁰, f′) : f ^0⁎ (λ) → λ (Lemma-Definition 3.11).

Lemma-Definition 3.48

1)

Suppose that the restriction f b ′ ′  : M b ′ ′  → M f 0(b ′) (which is a diffeomorphism) is linear for every b′ ∈ B′. Then, (f 0, f′) is a morphism of vector bundles ( Definition 3.44 (A)) and f ′ b ′ −   1 transports the Banach space structure of M f 0(b ′) to M′ b′ ; thus, M′ can be equipped with a vector bundle structure with base B′, called the preimage of M under f 0 and written as f 0⁎ (M). The projection π′ is M b′ ↦ b′. The associated fibration is f 0⁎ (λ) = (M′, B′, π′) (the vector bundle f 0⁎ (M) is also called the vector bundle induced by f 0).

We say that rk _b′ (f′ _b ) ≤ ∞ is the vector rank of $f_{b^{\prime }}^{\prime }:f^{0\ast }{\left(M\right)}_{b^{\prime }}\stackrel{\sim }{\to }{M}_{f^0\left(b^{\prime }\right)}$ .

2)

Let N′ be a vector bundle with base B′ and suppose that g′ : N′ → M is an f ⁰ -morphism ( Lemma-Definition 3.11 (iii)). There exists a uniquely determined f ⁰-morphism u from N′ into f ^0⁎ (M) such that g′ = f′ ° u (see Lemma-Definition 3.11 (iii)).

Consider the situation of Lemma-Definition 3.48(2) with M = T (B), N′ = T (B′) = f ^0⁎ (T (B)). Then, T (f ⁰) : N′ → M (this is the mapping denoted g′ earlier), so there exists a unique f ⁰-morphism δ f ⁰ : T (B′) → T (B) (denoted u earlier) such that T (f ⁰) = f′ ° δ f ⁰. With this notation, we have the following result:

Corollary 3.49

i)

f ⁰ is an immersion (respectively a submersion) if and only if the sequence

$0\to T\left(B^{\prime }\right)\stackrel{\delta f^0}{\to }T\left(B\right)\to 0$

is exact locally direct at T (B′) (respectively T (B)). If so, the cokernel vector bundle coker (δ f ⁰) is said to be normal (or transversal) of f ⁰ (respectively the kernel vector bundle ker (f ⁰) is said to be tangent to the fibers of f ⁰ and is written as T (B′/B)).

ii)

If f ⁰ is a submersion, then the fiber T (B′/B) _b′ is the tangent space of the submanifold (f ⁰)^{−   1}(f ⁰({b ^′})) at b′ ∈ B′ and the sequence

$0\to T{\left(B^{\prime }/B\right)}_{b^{\prime }}\stackrel{\iota }{\to }{T}_{b^{\prime }}\left(B^{\prime }\right)\stackrel{T_{b^{\prime }}\left(f^0\right)}{\to }{T}_{f^0\left(b^{\prime }\right)}\left(B\right)\to 0$

(where ι is the canonical injection) is exact.

iii)

f ⁰ is étale (respectively a subimmersion) if and only if f ⁰ is an isomorphism (respectively is locally direct).

If M is locally of finite rank and the sections s ₁, …,s _n form a frame of M over U  ⊂ B, then the preimage sections (Lemma-Definition 3.19(5)) s′₁, …,s′ _n , form a frame of M′ over (f ⁰)^–1 (U) and the diagram [3.8] (section 3.3.4) commutes.

Lemma 3.50

Let f ⁰ : B′ → B be a morphism of manifolds and T ^∨ (B) the cotangent bundle of B. Consider its preimage f ^0⁎ (T ^∨ (B)), with base B′. There exists a unique B′-morphism

$w:f^{0\ast }\left(T^{\vee }\left(B\right)\right)\to T^{\vee }\left(B^{\prime }\right)$

such that, for b′ ∈ B′, w _b′ is the continuous linear mapping formed by the composition

$f^{0\ast }{\left(T^{\vee }\left(B\right)\right)}_{b^{\prime }}\stackrel{f_{b^{\prime }}}{\to }{T}_{f^0\left(b^{\prime }\right)}^{\vee }\left(B\right)\stackrel{{}^t{T}_{b^{\prime }}\left(f^0\right)}{\to }{T}_{b^{\prime }}^{\vee }\left(B^{\prime }\right)$

where f _b′ is the canonical morphism ( Proposition 3.45 ). If Φ: ξ (U) → ξ′ (U′) is the local expression of f ⁰, then the local expression of w is

$\left(b^{\prime },{\mathbf{h}}^{\vee }\right)\mapsto \left(b^{\prime }{,}^{t}D\mathrm{\Phi }\left(b^{\prime }\right).{\mathbf{h}}^{\vee }\right).$

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Geometries for CAGD

Helmut Pottmann , Stefan Leopoldseder , in Handbook of Computer Aided Geometric Design, 2002

3.3.4 Other applications of line geometry in geometric computing

Line geometry is a basic entity in the formulation of the so-called generalized stereographic projection σ, also known as Hopf mapping. It maps points in projective 3-space P ³ onto points of the Euclidean sphere S ². The preimage of a point on S ² under this mapping σ is a straight line in P ³. All fibers of σ form a so-called elliptic linear line congruence in P ³. It may be seen as intersection of two appropriate linear complexes, and the Klein image of the line congruence is an oval quadric in $M_2^4$ . Dietz, Hoschek, and Jüttler [22] have shown that the mapping σ is well-suited to construct rational curve and surface patches on the sphere. Applying a projective mapping, one can work on other oval quadrics as well. It can also be used for the definition of a B-spline like intrinsic control structure for NURBS curves on the sphere [80]. There are similar mappings for ruled quadrics and singular quadrics [21], whose fibers are line congruences (intersections of two linear complexes). Such mappings are useful for the design of curves and surface patches on quadrics (see also chapter 31), and they can also be used to construct rational blending surfaces between quadrics [109].

A generalization of the mapping σ to the construction of rational curves and surface patches on Dupin cyclides has been studied by C. Mäurer [62].

Line geometry also appears in manufacturing, such as sculptured surface machining [91],[111] and wire cut EDM [96]. For further applications and detailed discussions, we refer the reader to Pottmann and Wallner [94].

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