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, (x, y), while the straight lines are in turn defined as the sets of points that satisfy linear equations, see the excellent expositions by D. Pedoe or D. 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 line ax + by + c = 0 that passes through these points. Its coefficients a, b, c can be found (up to a constant factor) from the linear system of two equations

ax1 + by1 + c = 0
ax2 + by2 + c = 0,

  1. Conic Section
  2. Vector in space