Example: In Euclidean geometry, Cartesian coordinates are affine coordinates with respect to an orthonormal framework, that is, an affine framework (o, v1, …, vn), so that (v1, …, vn) is an orthonormal basis. Specifically, an affine space E with the associated vector space E → {displaystyle {overrightarrow {E}}} , F is an affine subspace of the direction F → {displaystyle {overrightarrow {F}}} and D is an additional subspace of F → {displaystyle {overrightarrow {F}}} in E → {displaystyle {overrightarrow {E}}} (this means that each vector of E → {displaystyle {overrightarrow {E}}} can be uniquely decomposed as the sum of an element of F → {displaystyle { overrightarrow {F}}} and an element of D). For each point x of E, its projection at F parallel to D is the unique point p(x) in F, so, for example, V itself is an affine space connected to V, where f is given by f (v, w) = w-v. When the rest of the sentence is added, “I`m trying to learn my geometry lesson,” the whole thing needs to be reconstructed. In addition, we define A∨B as the smallest apartment in A(V), which contains both A and B. According to Zorn`s lemma, A∨B. As A∨B is also unique, ∨ is well defined. This makes A(V) an upper half-grid. If S1 is the corresponding subspace of A and S2 is the corresponding subspace of B, then span(S1∪S2) is the corresponding subspace of A∨B. The definition of ∨ can be extended to any number of apartments, so that ⋁S is the smallest apartment containing all the apartments in S⊆A. In fact, it`s not hard to see that A(V) is a complete half-grid. if the affine coordinates of a point are above the affine frame, then its barycentric coordinates are above the barycentric frame Suppose further that V is a left vector space on a division ring D and (A,f) an affine space connected to V. An affine subspace of A is the collection B of the points of A, which is mapped by the induced function f(P,-) for a point P∈A to a vector subspace S of V.
In other words, B is the inverse image of S under the function f(P,-): A walk through the city center near Dunne Park offers attentive observers a hidden fair with distorted geometry. There is a natural injective function from an affine space in the set of prime ideals (i.e. the spectrum) of its ring of polynomial functions. If affine coordinates are selected, this function forms the point of the coordinates ( a 1 , . , a n ) {displaystyle left(a_{1},dots ,a_{n}right)} to the maximum ideal ⟨ X 1 − a 1 , . , X n − a n ⟩ {displaystyle leftlangle X_{1}-a_{1},dots ,X_{n}-a_{n}rightrangle }. This function is a homeomorphism (for the Zariski topology of affine space and the spectrum of the ring of polynomial functions) of the affine space on the image of the function. Barycentric coordinates define an affine isomorphism between the affine space A and the affine subspace of kn + 1, defined by the equation λ 0 + ⋯ + λ n = 1 {displaystyle lambda _{0}+dots +lambda _{n}=1}.
We therefore identify as affine theorems all the geometric results that can be given in invariant terms under the affine group. An example of the plane geometry of triangles is the theorem on the correspondence of lines that connect each vertex to the center of the opposite side (center or barycenter). The idea of the center is an affine invariant. There are other classical examples (Ceva`s theorems, Menelaus). His thesis became the book Geometric Perturbation Theory in Physics on new developments in differential geometry. This implies the following generalization of Playfair`s axiom: At a given direction V, for each point a of A there is one and only one affine subspace of direction V, which traverses a, namely the subspace a + V. Each vector space V can be considered an affine space above itself. This means that any element of V can be considered a point or vector.
This affine space is sometimes called (V, V) to emphasize the dual role of the elements of V. When the null vector is considered a period, it is usually called o (or O if capital letters are used for the points) and called the origin. The affine subspaces of A are the subsets of A of the form Known formulas such as half of the base multiplied by height for the area of a triangle or a third of the base multiplied by height for the volume of a pyramid, are also affine invariants. While the latter is less obvious for the general case than the former, it is easy to see for the sixth of the unit cube, which consists of an area (zone 1) and the center of the cube (height 1/2). Therefore, it applies to all pyramids, even oblique, whose vertex is not directly above the center of the base, and for those with base, a parallelogram instead of a square. The formula is further generalized to pyramids, the basis of which can be broken down into parallelograms, including cones, allowing an infinite number of parallelograms (taking due account of convergence). The same approach shows that a four-dimensional pyramid has a 4D hypervolume, a quarter of the 3D volume of its parallelepiped base multiplied by height, and so on for higher dimensions. Explicitly, the above definition means that the action is a mapping commonly referred to as addition, the linear subspace associated with an affine subspace is often called :target~.vanchor-text{background-color:#b1d2ff}]]>direction, and two subspaces that share the same direction are called parallels. An affine transformation or endomorphic space A {displaystyle A} is an affine representation of that space to itself. An important family of examples are translations: Given is a vector v → {displaystyle {overrightarrow {v}}}, the translation map T v →: A → A {displaystyle T_{overrightarrow {v}}:Arightarrow A}, which sends a ↦ a + v → {displaystyle amapsto a+{overrightarrow {v}}} for each a {displaystyle a} in A {displaystyle A}, is an affine map. Another important family of examples are linear images centered at an origin: Given is a point b {displaystyle b} and a linear map M {displaystyle M} , so that one can define an affine map L M, b: A: A → A {displaystyle L_{M,b}:Arightarrow A} from In Euclidean geometry, the common term “affine property” refers to a property, which can be proven in affine parts.
That is, it can be proven without using the square shape and the internal product associated with it. In other words, an affine property is a property that does not include lengths and angles.