analytic geometry
analytic geometry, branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates, and in which the approach to geometric problems is primarily algebraic. Its most common application is in the representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. For example, the linear equation ax+by+c=0 represents a straight line in the xyplane, and the linear equation ax+by+cz+d=0 represents a plane in space, where a, b, c, and d are constant numbers (coefficients). In this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. Conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimated or sometimes given exactly by geometric means, e.g., plotting curves and surfaces and determining points of intersection.
In plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slope m of a straight line is the (trigonometric) tangent of the angle it makes with the xaxis. If the line is parallel to the xaxis, its slope is zero. Two or more lines with equal slopes are parallel to one another. In general, the slope of the line through the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by m= (y_{2}y_{1}) / (x_{2}x_{1}). The conic sections are treated in analytic geometry as the curves corresponding to the general quadratic equation ax^{2}+bxy+cy^{2}+dx+ey+f=0, where a, b, … , f are constants and a, b, and c are not all zero.
In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, λ, μ, and ν, the cosines of the angles the line makes with the x, y, and zaxes, respectively; these satisfy the relationship λ^{2}+μ^{2}+ν^{2}= 1. In the same way that the conic sections are studied in two dimensions, the 17 quadric surfaces, e.g., the ellipsoid, paraboloid, and elliptic paraboloid, are studied in solid analytic geometry in terms of the general equation ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+px+qy+rz+s=0.
The methods of analytic geometry have been generalized to four or more dimensions and have been combined with other branches of geometry. Analytic geometry was introduced by René Descartes in 1637 and was of fundamental importance in the development of the calculus by Sir Isaac Newton and G. W. Leibniz in the late 17th cent. More recently it has served as the basis for the modern development and exploitation of algebraic geometry.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"analytic geometry." The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. 24 Feb. 2018 <http://www.encyclopedia.com>.
"analytic geometry." The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. (February 24, 2018). http://www.encyclopedia.com/reference/encyclopediasalmanacstranscriptsandmaps/analyticgeometry
"analytic geometry." The Columbia Encyclopedia, 6th ed.. . Retrieved February 24, 2018 from Encyclopedia.com: http://www.encyclopedia.com/reference/encyclopediasalmanacstranscriptsandmaps/analyticgeometry
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.
quadrant
quadrant In plane geometry, a quarter of a circle, bounded by radii at right angles to each other and by the arc of the circle. In analytic geometry, it is one of the four sections of a plane divided by an x axis and a y axis. A quadrant is also a device for measuring angles, based on a 90° scale.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"quadrant." World Encyclopedia. . Encyclopedia.com. 24 Feb. 2018 <http://www.encyclopedia.com>.
"quadrant." World Encyclopedia. . Encyclopedia.com. (February 24, 2018). http://www.encyclopedia.com/environment/encyclopediasalmanacstranscriptsandmaps/quadrant
"quadrant." World Encyclopedia. . Retrieved February 24, 2018 from Encyclopedia.com: http://www.encyclopedia.com/environment/encyclopediasalmanacstranscriptsandmaps/quadrant
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.