Math 331 - May 7, 1999

What is the Synthetic Euclidean Plane?

Intuitively, the n-dimensional synthetic Euclidean space is the object that results when one begins with R^{n} and discards coordinate axes and origin while keeping points, distance between points, lines, angles between intersecting lines, and congruences. Note that one keeps all that is necessary to do school geometry.

Somewhat more precisely, one may say that the study of synthetic Euclidean space is the study of those aspects of n-dimensional real Cartesian space that do not become altered under a rigid change of affine coordinates, i.e., a coordinate change in which two coordinate columns are related by an equation of the form x' = U x + b where U is an orthogonal matrix.

One still seeks a definition of synthetic n-dimensional Euclidean space that says exactly what it is to the extent that this is logically possible. This may be done based on the concept of vector space. First, it is necessary to observe that the notion of action of a group on a set may be extended to the notion of a vector space action on a set by observing that any vector space forms a group under addition.

Definition. An action of a vector space on a set is an action of the additive group of the vector space on the set.

Definition. An (abstract) affine space is a triple (X, V, alpha) where X is a set, V is a vector space, and alpha is an action of V on X with the property that for any points x, y of X there is one and only one element v in V for which alpha(v, x) = y.

The vector space V is called the vector space of translations of the given affine space. Although a vector space does not come with a given basis (coordinate system), a vector space does have an origin. An affine space has neither a given basis nor a given origin.

Example 1. X = V = R^{n} with alpha given by alpha(v, x) = x + v.

Example 2. X = V = any vector space with alpha given by alpha(v, x) = x + v.

Let (X, V, alpha) be an affine space. For given a in X let xi_{a} be the map from V to X that is defined by xi_{a}(v) = alpha(v, a). Then xi_{a} is a bijection. The set X may be made into a vector space V_{a} by declaring xi_{a} to be linear. That is, there is a unique vector addition and a unique multiplication by scalars on the set X for which xi_{a} is a linear isomorphism.

Proposition. If (X, V, alpha) is an affine space, a a given point of X, xi_{a} the map from V to X, x_{1}, ..., x_{r} given points of X, and c_{1}, ..., c_{r} given scalars, then the expression

xi_{a}(SUM_{i = 1}^{r}[ c_{i} xi_{a}^{-1}(x_{i}) ]) with SUM_{i = 1}^{r}[ c_{i} ] = 1 ,

which represents a point of X, is independent of the choice of the point a.

This is a consequence of the fact that if w in V is the unique vector for which alpha(w, a) = b, then xi_{b} = xi_{a} \circ tau_{w} where tau_{w} denotes the translation in V by w. The proposition ensures that the barycentric linear combinations in X that may be defined using xi_{a} do not depend on the choice of a.

An affine map from an affine space (X, V, alpha) to an affine space (Y, W, beta) is a map f: X --> Y that preserves barycentric combinations. An invertible affine map is an isomorphism of affine spaces.

The dimension of an affine space is the dimension of its vector space of translations. Just as any two finite-dimensional vector spaces are isomorphic when they have the same dimension, any two n-dimensional affine spaces are isomorphic. If an affine space is given, one uses xi_{a}, where a is chosen arbitarily from X, to establish an isomorphism of affine spaces from V (as an affine space under example 2 above) with X.

Note that for any affine map f from X to Y and any a in X the map Phi_{f} from the vector space V of translations of X to the vector space W of translations of Y defined by the formula Phi_{f} = xi_{f(a)}^{-1} \circ f \circ xi_{a} is a linear map that depends on f but not on the choice of a. One has the relation beta(Phi_{f}(v), f(x)) = f(alpha(v, x)).

Definition. A synthetic n-dimensional Euclidean space is an affine space for which the n-dimensional vector space of translations has a given inner product. In an affine space the distance from a to b is the length of the unique vector v for which alpha(v, a) = b relative to the given inner product in V. An isometry is a distance-preserving affine isomorphism.

If f is an affine map from a synthetic Euclidean X space to to another Y, then f is an isometry if and only if Phi_{f} is an orthogonal linear isomorphism from the vector space V of translations of X to the vector space W of translations of Y relative to the given inner products. Note that for each point a in a synthetic Euclidean space X the affine map xi_{a} from its vector space of translations to X is an isometry. This means that every synthetic n-dimensional Euclidean space just amounts to R^{n} without its coordinate system: any two isometric ways of identifying with R^{n} differ (composition-wise) by an isometry of the latter.


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