Xavier Saint-Raymond

The Space of Spheres

We fix an n-dimensional Euclidean space E, and denote by Q ( E ) the space of affine quadratic forms q over E; \(\overrightarrow q \), an element of Q (\(\overrightarrow E \)), will be the symbol of q. A sphere is given by a form q, written as

Affine-projective Relationship: Applications

q = k{\left\| {\left. \cdot \right\|} \right.^2} + (\alpha \left| \cdot \right.) + h, where \alpha \in \overrightarrow E and k, h \in R;

The Sphere for Its Own Sake

it is an actual sphere if k ≠ 0 and ‖α‖2 > 4kh; if k ≠ 0 and ‖α‖2 = 4kh the image is a single point (sphere of zero radius), and if ‖α‖2 < 4kh the image is empty, and we say that q represents a sphere “of imaginary radius”. For k = 0, h ≠ 0, we obtain a hyperplane (if α ≠ 0); the case k = 0 and α = 0, h ≠ 0 represents the point at infinity of E.

Groups Operating on a Set: Nomenclature, Examples, Applications

We’ll be working in a d-dimensional real affine space X, for d finite. A polytope is a convex compact set with non-empty interior, which can be realized as the intersection of a finite number of closed half-spaces of X (cf. 2.G). We shall assume there are no superfluous half-spaces in the intersection. For d = 2 we use the word polygon.

Elliptic and Hyperbolic Geometry

We have shown, at the beginning of chapter 4 why it is necessary to go beyond the framework of affine spaces, adjoining to them points at infinity This is now possible in the following way: we associate to the affine space X its universal space \( \hat X\) (cf. 3.D), which is a vector space in which X is embedded as an affine hyperplane whose direction is a vector hyperplane of \(\hat X\). Considering now the projectivization \(\tilde X = P\left( {\hat X} \right)\) , we see that \(\tilde X\) is the disjoint union of two sets: P( X), which is canonically identified with X, and P (\(\vec X\)), which, being the space of lines in \(\vec X\), is also the space of directions of lines in X. We denote it by \({\infty _X} = P\left( {\vec X} \right)\) We write \( \tilde X = X \cup {\infty _X}\), and say that ∞x is the hyperplane at infinity in X.

Triangles, spheres and circles

To any four distinct points (a i ) i = 1,2,3,4 on a projective line (considered by itself or inside a projective space), we associate a scalar, denoted by [a i ] = [a l, a 2, a 3, a 4], and called the cross-ratio of these four points. For points on an affine line D, the cross-ratio is defined to be the same as on the completion \(\tilde D = D \cup {\infty _D}\) (cf. 5.A).

More about Euclidean Vector Spaces

The sphere is a compact connected topological space; for d ≥ 2, it is also simply connected, which means that every closed curve in S can be deformed into a point (such is not the case for either S 1 or P n (R)).

Problem Books in Mathematics

Let X be a set; we denote by S X the set of all bijections (permutations) f: X → X from X into itself, and we endow S X with the law of composition given by (f, g) ↦ f ° g. Thus S X becomes a group, called the permutation group or symmetric group of X.