What Is Analytic Geometry?
Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. Analytic geometry is a great invention of Descartes and Fermat.
In plane analytic geometry, points are defined as ordered pairs of numbers, say,
ax_{1} + by_{1} + c = 0 ax_{2} + by_{2} + 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 of set remains undefined.
Geometry of the three-dimensional space is modeled with triples of numbers
Within the framework of analytic geometry one may (and does) model non-Euclidean geometries as well. For example, in plane projective geometry a point is a triple of homogenous coordinates
(tx, ty, tz) = (x, y, z), |
for all
ax + bx + cz = 0. |
In analytic geometry, conic sections are defined by second degree equations:
ax² + bxy + cy² + dx + ey + f = 0. |
That part of analytic geometry that deals mostly with linear equations is called Linear Algebra.
Cartesian analytic geometry is geometry in which the axes
The components of n-tuple x = (x_{1}, ..., x_{n}) are known as its coordinates. When
References
- D. A. Brannan, M. F. Esplen, J. J. Gray, Geometry, Cambridge University Press, 2002
- D. Pedoe, Geometry: A Comprehensive Course, Dover, 1988
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