Analytical geometry

It is known him as analytical geometry the study of certain geometric objects by means of basic techniques of the mathematical analysis and algebra. It would be possible to be said that it is the historical development that begins with cartesian geometry and later concludes with the appearance of geometry differential with Gaussian and with the development of algebraic geometry.

The novel thing of Analytical Geometry is that x allows to represent geometric figures by means of formulas of type f (, y) = 0, where f represents a function. In particular, the straight lines can express like polinómicas equations of degree 1 (v.g.: 2x + 6y = 0) and the conical circumferences and the rest of like polinómicas equations of degree 2 (v.g.: circumference x + and = 4, the hyperbola xy = 1).

Fundamental constructions.

In a cartesian coordinate system, a point of the plane is determined by two numbers, that they are the abscissa and the ordinate of the point, so that, to all point of the plane two ordered real numbers always correspond (abscissa and ordinate), and reciprocally, to a ordered pair of numbers corresponds a single point of the plane.

Consequently the cartesian system establishes an biunivocal correspondence between a geometric concept as he is the one of the points of the plane and an algebraic concept as they are the ordered pairs of numbers. This correspondence constitutes the foundation of Analytical Geometry.

With Analytical Geometry it is possible to be determined flat geometric figures by means of equations and inequations with two unknown quantities. This one is an alternative method of resolution of problems, or to at least it provides a new point of view us with which to be able to attack the problem.

Location of a point in the cartesian plane.

In a flat plan two perpendicular straight lines (axes) - that by agreement draw up so that one of them is horizontal and the other vertical, and each point of the plane it is univocally determined by the distances of this point to each one of the axes, as long as a criterion also occurs to determine envelope what semiflat determined by each one of the straight lines it is necessary to take that distance, criterion that comes dice by a sign. That pair of numbers, the coordinates, will be represented by a ordered pair (x, y), being x the one of the axes (by agreement it will be the distance to the vertical axis) distance and distance to the other axis (to the horizontal).

In coordinate x, the positive sign (that usually is omitted) means that the distance is taken towards the right of the horizontal axis (axis of the abscissas), and the negative signal (never it is omitted) indicates that the distance is taken towards the left. For the coordinate and, the positive sign (also usually it is omitted) indicates downwards that the distance is taken upwards from the vertical axis (axis of ordinates), taking itself if the sign is negative (it is omitted either never in this case). To coordinate x usually denominates it abscissa of the point, whereas to and ordinate of the point is denominated.

The points of the X-axis have therefore equal ordinate to 0, so they will be of the form (x, 0), whereas those of the axis of ordinates have equal abscissa to 0, reason why will be of form (0, y).

The point where both axes are crossed will have therefore distance 0 to each one of the axes, soon its abscissa will be 0 and its ordinate also will be 0. To this point - (0,0) - origin of coordinates is denominated to him.

Equations of the straight line in the plane

A straight line in the plane imagines with the linear function of the form:

like general expression, although we can distinguish two particular cases. If a straight line does not cut to one of the axes, will be because she is parallel to him. As both axes are perpendicular, if it does not cut to one of them necessarily has to cut the other. We have then two cases:

Straight lines that do not cut to the axis of ordinates

These straight lines are parallel to this axis and vertical straight lines are denominated. The point of cuts with the X-axis is the point (x0,0). The equation of straight happiness is:

Straight lines that do not cut to the axis of the abscissas.

These straight lines are parallel to this axis and horizontal straight lines are denominated. The point of cuts with the axis of ordinates is the point (0, y0). The equation of straight happiness is:

Oblique straight lines

Any other type of straight line receives the name of oblique straight line. In them there is a point of cuts with the X-axis (to, 0) and another point of cuts with the axis of ordinates (0, b). The value to receives the name of abscissa in the origin, whereas the b denominates ordinate in the origin.

Conical

The result of the intersection of the surface of a cone, with a plane, gives rise to which they denominate Conic section, that is: the Parabola, the Ellipse and the Circumference as particular case and the Hyperbola

The Parabola (Figure A) in the plane has by formulates:

The Ellipse (Figure B) centered of axis to and b has by expression:

If both axes are equal and we called c:

the result is a circumference:

The Hyperbola (Figure C) has by expression:

Constructions in the three-dimensional space.

The reasonings on the construction of the Y-axes are equally valid for a point in the space and a short list ordinate of numbers, immediately than to introduce one third perpendicular straight line to x-axis and: z-axis. Nevertheless is no analog to the most important slope concept of a straight line.

Classification of Analytical Geometry within Geometry.

From the point of view of the classification of Klein of geometries (the Program of Erlangen), analytical geometry is not a geometry proper.

From the didactic point of view, Analytical Geometry is an indispensable deck between euclidiana geometry and other branches from the mathematical one and own geometry, as they are the own mathematical analysis, linear algebra, compatible geometry, euclidiana geometry, geometry differential or algebraic geometry.

History of analytical geometry.

A certain controversy exists on the true paternity of this method. Unique the certain thing is that it is published for the first time like “Analytical geometry”, appendix to the “Speech of the method”, of Discardings, although it is known that Pierre de Fermat knew and used the method before his publication by Discardings. Although Omar Khayyam already in century XI would use a very similar method to determine certain intersections between curves, is impossible that some of mentioned the mathematical French had access to their work.

The name of Analytical Geometry ran even to the one of Cartesian Geometry, being both indistinguishable ones. Nowadays, paradoxicalally, it is preferred to denominate Cartesian Geometry to the appendix of the Speech of the method, whereas it is understood that Analytical Geometry includes/understands not only to Cartesian Geometry (in the sense which we finished mentioning, that is to say, to the text appendix of the Speech of the method), but also all the later development of the Geometry that is based on the construction of Y-axes and the description of the figures by means of functions - algebraic or no, until the appearance of Geometry Differential of Gaussian (tenth paradoxicalally because the term is used indeed Cartesian Geometry for what own the Discardings baptized like Analytical Geometry). The problem is that during that period a clear difference between Analytical Geometry and Mathematical analysis does not exist - this lack of difference indeed must to the identification done at the time between the concepts of function and curve, reason why is sometimes very difficult to try to determine if the study that is being realized corresponds to one or the other branch.

Geometry Differential of curves yes that allows a study by means of a coordinate system, or in the plane or the three-dimensional space. But in the study of the surfaces, generally, they appear serious obstacles. Gaussian saves these obstacles creating Geometry Differential, and marking in this way the aim of Analytical Geometry like discipline. He is with the development of algebraic geometry when the overcoming of Analytical Geometry can be certified totally.

He is to emphasize that the denomination of analytical given to this form to study geometry caused that the previous way to study it (that is to say, the axiomatic-deductive way without the operation of coordinates) would be ended up denominating, by opposition, synthetic geometry, due to duality analysis-synthesis.

At the moment the term is only used in average lessons or technical careers in which a detailed study of Geometry is not realized.

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• Vestibule: Mathematical Content related to Mathematical.

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