Analytic geometry

The analytic geometry (including vector geometry ) is a branch of geometry that provides algebraic tools (mainly linear algebra ) to solve geometric problems. It makes it possible in many cases, geometric problems mathematically to solve, without taking your intuition to help.

Geometry, which bases its rates without reference to a number system on an axiomatic basis is, in contrast, called synthetic geometry.

The method of analytic geometry are applied in all the natural sciences, but especially in physics, such as in the description of planetary orbits. Originally analytic geometry dealt only with issues of plane and spatial ( Euclidean ) geometry. In a general sense, however, describes the Analytic geometry affine spaces of arbitrary dimension over arbitrary fields.

• 5.1 points in space
• 5.2 straight line in space
• 5.3 planes in space
• 5.4 surfaces of the second order in space

• 7.1 Incidence review 7.1.1 In the two-dimensional space
• 7.1.2 In the three-dimensional space

• 7.2.1 In the two-dimensional space
• 7.2.2 In the three-dimensional space

The coordinate system

Ultimate resource of analytical geometry, a coordinate system. In practice, one usually uses a system of Cartesian coordinates. For some simple questions, such as the determination of line intersections, the study of straight lines parallelism or the calculation of partial ratios, but an oblique coordinate system would be enough. Indispensable is a Cartesian coordinate system when the distance and angle to be calculated.

Vectors

Many bills of analytic geometry are unified and simplified by the methods of vector analysis. Although the entire analytic geometry was invented without vectors and of course still may be practiced without vectors and vice versa, the vector space can be defined as an abstract algebraic construct without geometric reference, the use of vectors in Cartesian coordinate systems appear so natural that " Linear Algebra and Analytic Geometry " in the upper secondary and mathematical-physical- technical basic studies are generally taught as a course.

Coordinates and parametric equations

More complex geometric structures such as lines, planes, circles, spheres, etc., are regarded as point sets and described by equations. It may be coordinate equations or parametric equations:

• Implicit coordinate equation: One of the coordinates ( x, y, ...) -dependent arithmetic expression is set to 0.

• Explicit coordinate equation: One of the coordinates is expressed by the other.

• Parametric equation: the radius vector of an arbitrary point of the structure is given by a vector calculation expression that includes one or more parameters.

Analytical geometry of the plane

Points in the plane

Each point P of the plane is described by two coordinates, eg, P ( 2 | -1.5 ). The coordinates are called typically ( in order) the x-coordinate (also: abscissa) and the y coordinate (also: ordinate). Common are also the names and.

The combined coordinates of points form ordered pairs in the plane case.

Lines in the plane

• Coordinates equation (implicitly):

This is also called the normal (s) form the linear equation, since the vector perpendicular (normal) is the straight line.

• Parameter equation:

Here, the position vector of an arbitrary but fixed chosen point of the line is ( base ); is a so-called direction vector, that is a vector whose direction is parallel to the straight line.

Quadratic curves in the plane

Through an (implicit coordinate ) second degree equation

Is generally given a conic. Depending on the values ​​of the coefficients, this may be an ellipse (special case: circle) act, parabola, or hyperbola.

Analytical geometry of Euclidean space

Points in space

Each point P of the space is determined by three coordinates, eg, P = P ( 4 | -0.5 | -3). Assigns to each point P on its position vector, which is the connection vector of the origin of the coordinate system with the given point. Its coordinates correspond to those of the point P, but they are written as a column vector:

The coordinates are ( in that order) as the x-, y -and z- coordinate, or as - designates coordinates and -.

The combined coordinates of points in the spatial form Case 3 - tuple.

Straight line in space

• Coordinate equations: straight line in space can not be described by a single coordinate equation. It is a straight line but always as an average ( intersection) conceive of two planes and coordinate equations of these two levels ( see below) to the line clearly defined.

• Parameter equation:

Thus, the equation has the same form as in the two-dimensional case.

Planes in space

• Coordinates equation (implicitly):

This type of plane equation is called the normal (s) form, as the vector perpendicular (normal) to the plane is.

• Parameter equation:

Is the position vector of an arbitrary but fixed chosen point of the plane ( base ); and are linearly independent direction vectors (or tension vectors), ie vectors parallel to the plane, the " span " the plane.

Surfaces of the second order in space

The general Cartesian equation of the second degree

Describes a second-order surface. The most important special cases are:

Ellipsoid, elliptic paraboloid, hyperbolic paraboloid, hyperboloid single-shell, double-shell hyperboloid, cone, elliptic cylinder, parabolic cylinder, hyperbolic cylinder

Generalization: Analytical Geometry of an arbitrary affine space

The concepts of analytic geometry can be generalized by allowing permits coordinates from any body and any desired dimensions.

If V is a vector space over a field K and R is an affine space associated to V, so can a k- dimensional subspace of R described by the parametric equation

Here, the position vector of an arbitrary but fixed chosen point of the subspace is ( base ); the vectors are linearly independent vectors, in other words a base of the sub-vector space V, the observed part of the subspace of R.

For k = 1 is the equation of a straight line for k = 2 by the equation of a plane. If k is 1 less than the dimension of R and V, it is called a hyperplane.

In analogy to the quadratic curves ( conics ) of plane geometry and to surfaces of the second order of the spatial geometry is considered in n-dimensional affine space, so-called quadrics, which are hypersurfaces of second order ( with the dimension n-1), the defined by coordinates equations of the second degree are:

Typical tasks of analytic geometry

Incidence review

The aim here is to determine whether a given point at a given point set belongs ( approximately to a straight line ).

In the two-dimensional space

As an example, the straight line with the explicit coordinates equation

Be considered.

The point lies on this line, as obtained by inserting the coordinates x = 2 and y = 1 ( point sample) recognizes:

The point, however, is not on the line. For x = 4 and y = 2 namely

In three-dimensional space

It should be checked whether the point on the line is the following parametric form:

Used for the position vector of, this leads to the following three equations:

Since in all three cases has the same value (here), lies on the line.

Determining the intersection of two point sets

Determining the intersection of two sets of points (for example, the point of intersection of two straight lines ) amounts to solving a system of equations. Depending on the form in which the two sets of points will be described, the process varies a little:

In the two-dimensional space

It should be examined whether and where the graphs of the functions and cut. This corresponds to and:

To calculate the points of intersection, function terms in the equations of the two features will now be set equal. In this way we find the coordinate ( s), for which the two features have the same coordinate:

Solving this quadratic function leads to the solutions: and.

By substitution in one of the two initial equations gives the intersections of at: and.

In three-dimensional space

It should be examined whether and to what point the two lines and intersect. The two lines may be defined as follows:

As in the two-dimensional space, the two equations are equated here:

The vector equation can be decomposed into the following three equations:

Adding the first and the last equation yields respectively. From the first equation results in order to use through: ie. This solution also satisfies the second equation, because.

The position vector of the point of intersection of the straight line obtained by substituting one of the two parameters calculated () in the corresponding line ( )

History

The analytical geometry was founded by the French mathematician and philosopher René Descartes. Significant enhancements are due to Leonhard Euler, who dealt in particular with the curves and surfaces of the second order. The development of vector calculus ( including by Hermann Grassmann ) allowed the now common vector notation.

David Hilbert proved that the three-dimensional analytic geometry is completely equivalent to the ( synthetic ) Euclidean geometry in the requirements set by him form. In practical terms, it is far superior to this. In the first half of the 20th century, therefore, the opinion was that geometry in the way they have been taught since Euclid, is only of historical interest.

N. Bourbaki even went a step further: he decided against geometric conceptions such as point, line, etc., and held with treatment of linear algebra everything you need to say. This of course - as always with Bourbaki - completely apart from the needs of applied mathematics.