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2016 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

The Universe of Conics

From the ancient Greeks to 21st century developments

verfasst von: Georg Glaeser, Hellmuth Stachel, Boris Odehnal

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This text presents the classical theory of conics in a modern form. It includes many novel results that are not easily accessible elsewhere. The approach combines synthetic and analytic methods to derive projective, affine and metrical properties, covering both Euclidean and non-Euclidean geometries.

With more than two thousand years of history, conic sections play a fundamental role in numerous fields of mathematics and physics, with applications to mechanical engineering, architecture, astronomy, design and computer graphics.

This text will be invaluable to undergraduate mathematics students, those in adjacent fields of study, and anyone with an interest in classical geometry.

Augmented with more than three hundred fifty figures and photographs, this innovative text will enhance your understanding of projective geometry, linear algebra, mechanics, and differential geometry, with careful exposition and many illustrative exercises.

Inhaltsverzeichnis

Frontmatter
1. Introduction
Abstract
Our Universe is full of conics, even if we cannot always see them – like the orbits of the planets. It needed very accurate observations to detect (JOHANNES KEPLER), and a great physicist (ISAAC NEWTON) to prove that.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
2. Euclidean plane
Abstract
A pencil of planes meets a cone of revolution in a family of conics which maps to a pencil of conics in the top view. These conics in the top view share a focal point and the associated directrix.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
3. Differential Geometry
Abstract
The picture shows a triaxial ellipsoid with its curvature lines. The top and front view of the curvature lines of the ellipsoid are affine images of confocal conics. The four umbilical points U 1, … ,U 4 (singularities of the curvature line parametrization) map to the common focal points in the top and front view.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
4. Euclidean 3-space
Abstract
The central projection of circles with different radii may result in conics of any type. Depending on whether the projection cone C, i.e., the connection of the circles and the center C of the projection, avoids, touches, or intersects the vanishing plane, the image of the circle is an ellipse, a parabola, or a hyperbola.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
5. Projective Geometry
Abstract
Projective Geometry is the proper frame work for understanding the geometry of conics. It differs from Euclidean Geometry and allows us to treat points and lines in a unifying way: There is no difference between points at infinity and proper points. The lines in a projective plane are closed like any conic, and the line at infinity is a line like any other. The above surface was discovered in 1901 by WERNER BOY and is an immersion of the real projective plane into threespace.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
6. Projective conics
Abstract
Studying conics in the framework of Projective Geometry leads to a much deeper understanding of their properties. The results are independent on the choice of the model as is the case for example with PASCAL’s theorem stating that any six points on a conic define a Pascal axis. The distinction between the three affine types (ellipse, parabola, hyperbola) is no longer necessary.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
7. Polarities and pencils
Abstract
A hyperbolic pencil of circles gives rise to a hyperbolic pencil of spheres by simply revolving it about its axis. Pencils of circles are special families of conics which are classified by means of a projectivity induced on the circles’ common diameter line.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
8. Affine Geometry
Abstract
An ellipse can be the image of a circle under a parallel projection. The terms ellipse, parabola, hyperbola are typical notions of affine geometry.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
9. Special problems
Abstract
A one-sheeted hyperboloid is a ruled quadric and carries two one-parameter families of lines. A top view shows the pattern of rulings as a part of a Poncelet grid. According to theoretical kinematics, the framework of crossing rods is flexible if all crossings of generators are realized as hinges.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
10. Other geometries
Abstract
At the ‘spherical Wankel engine’ the common path t of the three vertices of the rotor is a trochoid. It can be generated when the circle m which is attached to the rotor, is rolling along the fixed circle o. The angular velocities of the rotor’s rotations about M and additionally about the fixed center O build the constant ratio −3 ∶ 2.
Georg Glaeser, Hellmuth Stachel, Boris Odehnal
Backmatter
Metadaten
Titel
The Universe of Conics
verfasst von
Georg Glaeser
Hellmuth Stachel
Boris Odehnal
Copyright-Jahr
2016
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-45450-3
Print ISBN
978-3-662-45449-7
DOI
https://doi.org/10.1007/978-3-662-45450-3