# Categories

- A
- Algebraic Curves and Surfaces
- Ia
- Ellipsoids
- Ib
- Hyperbolic paraboloids
- Ic
- One-sheet hyperboloids
- Id
- Two-sheet hyperboloids
- Ie
- Elliptic paraboloids
- If
- Cones and cylinders
- Ig
- Confocal systems, conics
- II
- Generation of conics and surfaces of second order
- III
- Curves of third and fourth order
- IV
- Surfaces of third order
- V
- Cyclides
- VI
- Other surfaces of fourth and higher order
- VII
- Line geometry
- VIII
- Singularities
- IX
- Descriptive geometry
- XX
- Others

- B
- Combinatorical Geometry
- C
- Topology
- D
- Kinematics and Mechanics
- E
- Differential Geometry
- F
- Algebra
- G
- Analysis, Probability Theory
- H
- Theory of Functions of Complex Variables
- J
- Differential Equations, Wave Theory
- K
- Geometrical Optics
- L
- Instruments and Devices.
- M
- History of Mathematis and Astronomy
- Z
- Other

Algebraic curves and surfaces are determined by polynomial equations. The order respectively the degree of a curve or surface is given by the maximal sum of exponents. The order has an geometric interpretation in \(\mathbb{R}^3\): The intersection of a surface of order \(n\) and a line has a maximal number of /(n/) points.

## Ellipsoids

Ellipsoids in \(\mathbb{R}^3\) are the solutions of \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\) with \(a,b,c\) positive real numbers. This numbers rerpresent the length of the semiaxis.

## Hyperbolic paraboloids

Hyperbolic paraboloids are solutions of the equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\) with \(a,b,c\) positive reals. Many sections of paraboloids are parabola for example all sections paralell to the \(xz\) - or \(yz\) -plane.

## One-sheet hyperboloids

Hyperbolic paraboloids are solutions of the equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\) with \(a,b,c\) positive reals. For \(a=b\) the surface can be generated as a surface of revolution of a hyperbola or of two lines (model 24).>>

## Two-sheet hyperboloids

Hyperboloids of two sheets are determined by the equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1\) with \(a,b,c\) positive real numbers. The sections with planes normal to the \(z\)-axis are ellipses. For \(a=b\) the surface is an surface of revolation of an hyperbola.

## Elliptic paraboloids

Elliptic paraboloids are often called paraboloids. They are solutions of the equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=z\) for positve real numbers \(a\) and \(b\). The sections with planes normal to the \)z\)-axis are ellipses.

## Cones and cylinders

A cone can be constructed by connecting every point of a circle with one point outside the plane the surface lies in. For a cylinder one slides a planar curve along a line non parralel to the curve. If the line is normal to the planar curve, then the cylinder is calles a right cylinder.

## Confocal systems, conics

A confocal system describes conic sections with the same foci respectively their body of rotation.

## Generation of conics and surfaces of second order

There are exactly three types of conic sections if one allows only the intersection of planes not through the apex: Ellipse (circle), hyperbola and parabola

## Curves of third and fourth order

Algebraic curves are determined by polynomial equations. The order of a curve or surface is given by the maximal sum of exponents. The order has an geometric interpretation in \(\mathbb{R}^3\): The intersection of a curve of order \(n\) and a line has a maximal number of /(n/) points.

## Surfaces of third order

For the classification of surfaces of order 3 it's reasonnable to look on the different types of singularities. A singularity of the surface is a point with vanishing derivation. A surface without singularities is calles a smooth surface.

## Cyclides

Cyclids are algebraic surfaces of order four. In the manner of differential geometry they are the surfaces with circular lines of curvature.

## Other surfaces of fourth and higher order

Algebraic surfaces are determined by polynomial equations. The order of a surface is given by the maximal sum of exponents. The order has an geometric interpretation in \(\mathbb{R}^3\): The intersection of a surface of order \(n\) and a line has a maximal number of /(n/) points.

## Line geometry

In line-geometry the geometry of the space of all lines in the three dimensional space are concidered. For a better understanding one intruduces some special coordinates.

## Singularities

A singularity of the surface is a point with vanishing derivation. A surface without singularities is calles a smooth surface.

## Descriptive geometry

In descriptive geometry one studies the behaviour of projections of three dimensional objects onto the plane.