Analytic geometry

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

This article is about coordinate geometry. For the geometry of analytic varieties, see Algebraic geometry § Analytic geometry.

Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.

History[edit]

Ancient Greece[edit]

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.^[1]

Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others.^[2] Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations (expressed in words) of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.^[3]

Persia[edit]

The 11th-century Persian mathematician Omar Khayyam saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra^[4] with his geometric solution of the general cubic equations,^[5] but the decisive step came later with Descartes.^[4] Omar Khayyam is credited with identifying the foundations of algebraic geometry, and his book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe.^[6] Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.^[7]^: 248

m is the or gradient of the line.

slope

b is the of the line.

y-intercept

x is the of the function y = f(x).

independent variable

Changing $x$ to $x-h$ moves the graph to the right $h$ units.

Changing $y$ to $y-k$ moves the graph up $k$ units.

Changing $x$ to $x/b$ stretches the graph horizontally by a factor of $b$ . (think of the $x$ as being dilated)

Changing $y$ to $y/a$ stretches the graph vertically.

Changing $x$ to $x\cos A+y\sin A$ and changing $y$ to $-x\sin A+y\cos A$ rotates the graph by an angle $A$ .

Transformations are applied to a parent function to turn it into a new function with similar characteristics.

The graph of $R(x,y)$ is changed by standard transformations as follows:

There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations.

For example, the parent function $y=1/x$ has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if $y=f(x)$ , then it can be transformed into $y=af(b(x-k))+h$ . In the new transformed function, $a$ is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative $a$ values, the function is reflected in the $x$ -axis. The $b$ value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like $a$ , reflects the function in the $y$ -axis when it is negative. The $k$ and $h$ values introduce translations, $h$ , vertical, and $k$ horizontal. Positive $h$ and $k$ values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.

Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.

Suppose that $R(x,y)$ is a relation in the $xy$ plane. For example,

Geometric axis[edit]

Axis in geometry is the perpendicular line to any line, object or a surface.

Also for this may be used the common language use as a: normal (perpendicular) line, otherwise in engineering as axial line.

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the three-dimensional case a surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.

Applied geometry

Cross product

Rotation of axes

Translation of axes

Vector space

Boyer, Carl B. (2004) [1956], History of Analytic Geometry, Dover Publications, 978-0486438320

ISBN

Cajori, Florian (1999), A History of Mathematics, AMS, 978-0821821022

ISBN

(1885) Analytic Geometry of the Point, Line, Circle, and Conic Sections, link from Internet Archive.

John Casey

Katz, Victor J. (1998), , Reading: Addison Wesley Longman, ISBN 0-321-01618-1

A History of Mathematics: An Introduction (2nd Ed.)

(1982) Lectures in Geometry Semester I Analytic Geometry via Internet Archive

Mikhail Postnikov

Struik, D. J. (1969), A Source Book in Mathematics, 1200-1800, Harvard University Press, 978-0674823556

ISBN

with interactive animations

Analytic geometry

History[edit]

Ancient Greece[edit]

Persia[edit]

slope

y-intercept

independent variable

Geometric axis[edit]

Applied geometry

Cross product

Rotation of axes

Translation of axes

Vector space

ISBN

ISBN

John Casey

A History of Mathematics: An Introduction (2nd Ed.)

Mikhail Postnikov

ISBN

Coordinate Geometry topics