# An Algebraic Approach to Geometry (Geometric Trilogy, Volume 2)

## Francis Borceux

Language: English

Pages: 440

ISBN: 3319347527

Format: PDF / Kindle (mobi) / ePub

This is a unified treatment of the various algebraic approaches to geometric spaces. The study of algebraic curves in the complex projective plane is the natural link between linear geometry at an undergraduate level and algebraic geometry at a graduate level, and it is also an important topic in geometric applications, such as cryptography.

380 years ago, the work of Fermat and Descartes led us to study geometric problems using coordinates and equations. Today, this is the most popular way of handling geometrical problems. Linear algebra provides an efficient tool for studying all the first degree (lines, planes) and second degree (ellipses, hyperboloids) geometric figures, in the affine, the Euclidean, the Hermitian and the projective contexts. But recent applications of mathematics, like cryptography, need these notions not only in real or complex cases, but also in more general settings, like in spaces constructed on finite fields. And of course, why not also turn our attention to geometric figures of higher degrees? Besides all the linear aspects of geometry in their most general setting, this book also describes useful algebraic tools for studying curves of arbitrary degree and investigates results as advanced as the Bezout theorem, the Cramer paradox, topological group of a cubic, rational curves etc.

Hence the book is of interest for all those who have to teach or study linear geometry: affine, Euclidean, Hermitian, projective; it is also of great interest to those who do not want to restrict themselves to the undergraduate level of geometric figures of degree one or two.

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+ x, y + y). When the second point is “infinitely close” to the first one, the ratio yx is the slope of the tangent (see Fig. 1.20). 26 1 The Birth of Analytic Geometry Let us make Fermat’s argument explicit in the case of the quartic (i.e. the curve of degree 4) with equation x 3 + 2x 2 y 2 = 3. The assertion that the point (x + x, y + y) is on the curve means (x + x)3 + 2(x + x)2 (y + y)2 = 3. Subtracting the two equations one obtains 3x 2 ( x) + 3x( x)2 + ( x)3 + 4x 2 y( y) + 2x 2 ( y)2 +

. . 1.1 Fermat’s Analytic Geometry . . . . . . . . 1.2 Descartes’ Analytic Geometry . . . . . . . 1.3 More on Cartesian Systems of Coordinates 1.4 Non-Cartesian Systems of Coordinates . . 1.5 Computing Distances and Angles . . . . . 1.6 Planes and Lines in Solid Geometry . . . . 1.7 The Cross Product . . . . . . . . . . . . . 1.8 Forgetting the Origin . . . . . . . . . . . . 1.9 The Tangent to a Curve . . . . . . . . . . 1.10 The Conics . . . . . . . . . . . . . . . . . 1.11 The Ellipse . . . . .

. . 6.8 Desargues’ Theorem . . . . . . . . . . . 6.9 Pappus’ Theorem . . . . . . . . . . . . 6.10 Fano’s Theorem . . . . . . . . . . . . . 6.11 Harmonic Quadruples . . . . . . . . . . 6.12 The Axioms of Projective Geometry . . 6.13 Projective Quadrics . . . . . . . . . . . 6.14 Duality with Respect to a Quadric . . . . 6.15 Poles and Polar Hyperplanes . . . . . . 6.16 Tangent Space to a Quadric . . . . . . . 6.17 Projective Conics . . . . . . . . . . . . 6.18 The Anharmonic Ratio Along a Conic .

their corresponding projections on the base half-line (see Fig. 1.2). Calling R the intersection of the lines OP and Q′ Q, the 1.1 Fermat’s Analytic Geometry 3 Fig. 1.1 Fig. 1.2 similarity of the triangles OP ′ P and OQ′ R yields P ′P Q′ R = . ′ OP OQ′ On the other hand, since P and Q are on the curve ax = by, we obtain P ′P a Q′ Q = = . ′ OP b OQ′ It follows at once that Q′ R = Q′ Q and thus R = Q. This proves that O, P , Q are on the same line. Next Fermat considers the case of the

improved in the expected way: Theorem 4.14.1 Let Q ⊆ E be a quadric in a finite dimensional Euclidean space (E, V ). There exists an orthonormal basis (0; e1 , . . . , en ) with respect to which the equation of the quadric takes one of the reduced forms: Type 1 Type 2 Type 3 n 2 i=1 ai Xi = 1; n 2 i=1 ai Xi = 0; n−1 2 i=1 ai Xi = Xn . Proof The proof is an easy adaptation of that of Theorem 2.24.2. We focus only on the necessary changes. Applying Theorem G.4.1 instead of Corollary G.2.8 in the