Short answers, pulled from the story.
Analytic geometry was independently invented by René Descartes and Pierre de Fermat during the seventeenth century. The decisive step toward modern analytic geometry came later with René Descartes himself who published his work La Géométrie in 1637.
Apollonius of Perga used reference lines called diameters and tangents to establish a frame for measurement in his book Conics. He established relations between distances measured along the diameter from the point of tangency known as abscissas and segments parallel to the tangent known as ordinates.
Omar Khayyam helped close the gap between numerical and geometric algebra through his geometric solution of general cubic equations in his book Treatise on Demonstrations of Problems of Algebra published in 1070. His methods showed that geometry could solve problems previously thought to belong only to arithmetic.
Cartesian coordinates use x representing horizontal position and y representing vertical position typically appearing as ordered pairs like (x, y). Polar coordinates represent points by distance r from origin and angle θ measured counterclockwise from positive x-axis written as ordered pairs (r, θ).
The equation y equals x corresponds to all points whose x-coordinate matches their y-coordinate forming a line making y equals x the equation for that line. Linear equations specify lines while quadratic equations specify conic sections including ellipses parabolas hyperbolas and circles.