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affine bundle A bundle (A,M,π; A) which affine geometry The geometry of affine
has an affine space A as a standard fiber and tran- spaces.
sition functions acting on A by means of affine
affine map Let X and Y be vector spaces
transformations.
and C a convex subset of X. A map T : C →
If the base manifold is paracompact then any
Y is called an affine map if T((1 − t)x + ty)
affine bundle allows global sections. Examples
= (1 − t)T x + tT y for all x, y ∈ C, and all
of affine bundles are the bundles of connections
0 ≤ t ≤ 1.
(transformation laws of connections are affine)
k+1 k+1 k
and the jet bundles π : J C → J C. affine mapping (1) Let A be an affine space
k
with difference space E. Let P → P be a map-

affine connection A connection on the ping from A into itself subject to the following
frame bundle F(M) of a manifold M. conditions:

(i.) P Q = P Q implies P Q = P Q ;

1
2
1
2
affinecoordinates Anaffinecoordinatesys- 1 1 2 2
tem (0; x ,x , ..., x ) in an affine space A con- (ii.) The mapping φ : E → E defined by
n
1
2

φ(PQ) = P Q is linear.

sists of a fixed pont 0 ∈ A, and a basis {x } (i =
i
1, ..., n) of the difference space E. Then every Then P → P is called an affine mapping.

pont P ∈ A determines a system of n numbers If a fixed origin 0 is used in A, every affine
n
i i
{ξ } (i = 1, ..., n) by 0P = i=1 ξ x . The mapping x → x can be written in the form
i
i
numbers {ξ } (i = 1, ..., n) are called the affine x = φx + b, where φ is the induced linear map-

coordinates of P relative to the given system. ping and b = 00 .

The origin 0 has coordinates ξ = 0.
i (2) Let M, M be Riemannian manifolds.A
map f : M → M is called an affine map if the

affine equivalence A special case of para- tangent map Tf : TM → TM maps every hori-

metric equivalence, where the mapping ) is an zontal curve into a horizontal curve. An affine
affine linear mapping, that is, )(x) := Ax + t map f maps every geodesic of M into a geodesic
n
with a regular matrix A ∈ R n,n and t ∈ R (n the of M .

dimension of the ambient space).
For affine equivalent families of finite ele- affine representation A representation of a
ment spaces on simplicial meshes in dimension Lie group G which operates on a vector space V
n the usual reference element is the unit simplex such that all φ : V → V are affine maps.
g
n
spanned by the canonical basis vectors of R . affine space Let E be a real n-dimensional
In the case of a shape regular family {; }
h h∈H vector space and A a set of elements P, Q, ...
of meshes and affine equivalence, there exist which will be called points. Assume that a rela-
constants C > 0, i = 1, ..., 4, such that tion between points and vectors is defined in the
i
following way:
n
n
C diam(K) ≤| det A |≤ C diam(K) ,
2
1
K
(i.) To every ordered pair (P,Q) of A there
A ≤ C diam(K), is an assigned vector of E, called the difference
K 3

−1
A ≤ C diam(K) −1 ∀K ∈ ; ,h ∈ H. vector and denoted by PQ;
K 4 h
(ii.) To every point P ∈ A and every vector
Here . denotes the Euclidean matrix norm, and
x ∈ E there exists exactly one point Q ∈ A such
the matrix A belongs to that unique affine map-

that PQ = x;
ping taking a suitable reference element on K.
(iii.) If P, Q, R are arbitrary points inA, then
These relationships pave the way for assessing
PQ + QR = PR.

the behavior of norms under pullback.
Then A is called an n-dimensional affine space
affine frame An affine frame on a manifold with difference space E.
n
M at x ∈ M consists of a point p ∈ A (M) The affine n-dimensional space A is distin-
x
n
(where A (M) is the affine space with differ- guished from R in that there is no fixed origin;
x
n
ence space E = T M) and an ordered basis thus the sum of two points of A is not defined,
x
(X , ..., X ) of T M (called a linear frame at but their difference is defined and is a vector
1 n x
n
x). It is denoted by (p; X , ..., X ). in R .
1 n
© 2003 by CRC Press LLC