What Is Analytic Geometry?Analytic Geometryis a branch ofalgebrathat is used to modelgeometricobjects -points, (straight)lines, andcirclesbeing the most basic of these. Analytic geometry is a great invention of Descartes and Fermat.In plane analytic geometry, points aredefinedas ordered pairs of numbers, say,(x, y),while the straight lines are in turndefinedas the sets of points that satisfy linear equations, see the excellent expositions byD. PedoeorD. Brannan et al. From the view of analytic geometry, geometric axioms are derivable theorems. For example, for any two distinct points(x1, y1)and(x2, y2),there is a single lineax + by + c = 0that passes through these points. Its coefficients a, b, c can be found (up to a constant factor) from the linear system of two equationsax1+ by1+ c = 0ax2+ by2+ c = 0,

or directly from the determinant equation

However, no axiomatic theory may escape using undefined elements. In Set Theory that underlies much of mathematics and, in particular, analytic geometry, the most fundamental notion ofsetremains undefined.Geometry of the three-dimensional space is modeled with triples of numbers(x, y, z)and a 3D linear equationax + by + cz + d = 0defines a plane. In general, analytic geometry provides a convenient tool for working in higher dimensions.Within the framework of analytic geometry one may (and does) modelnon-Euclideangeometries as well. For example, inplane projective geometrya point is a triple ofhomogenous coordinates(x, y, z),not all 0, such that(tx, ty, tz) = (x, y, z),

for allt 0,while a line is described by a homogeneous equationax + bx + cz = 0.

In analytic geometry,conic sectionsare defined bysecond degree equations:ax + bxy + cy + dx + ey + f = 0.

That part of analytic geometry that deals mostly with linear equations is calledLinear Algebra.Cartesian analytic geometryis geometry in which the axesx = 0andy = 0are perpendicular.The components of n-tuplex= (x1, ..., xn) are known as itscoordinates. Whenn = 2orn = 3,the first coordinates is calledabscissaand the secondordinate.