Coordinate Geometry - History and Fundamentals

This set covers the historical and mathematical foundations of coordinate geometry, including key figures, terminology of the Cartesian plane, and the geometric distance formula.

System of coordinates

A structured framework, like a grid on a map, that enables the use of numbers to describe the exact physical locations of points or objects.

Sindhu-Sarasvatī Civilisation

Ancient civilisation that practiced the first systematic use of grids on an urban scale, with streets constructed in North–South and East–West directions about $10\,metres$ apart.

Baudhāyana

Scholar who developed the Baudhāyana–Pythagoras Theorem and used geometric constructions based on East–West and North–South lines.

Ujjayinī

The ancient world's central longitude meridian city from which all other locations were measured, later referred to as ‘Ozine’ by Ptolemy.

Āryabhaṭa

Mathematician who replaced Greek ‘chords’ with ‘sines’ and mapped the sky using Celestial Coordinates measured from the ecliptic.

Ecliptic

The path of the sun used as a reference point for celestial coordinate measurements.

Brahmagupta

Formalised the concept of zero and negative numbers as algebraic entities, enabling the development of the four-quadrant coordinate plane.

Arin

The Arabic name for the Ujjayinī meridian, which served as the zero-longitude reference for early Arabic maps.

Al-Bĩrũnĩ

An influential Arab scholar who used Indian trigonometric methods to calculate city coordinates and perfected the astrolabe.

Astrolabe

A handheld device that allowed sailors to find their coordinates by observing the positions of stars.

Ömar Khayyām

The first mathematician to solve algebraic problems using geometry by interpreting them in terms of coordinates in the plane.

René Descartes

Formalised the definition of a point in a two-dimensional plane using two numbers representing distances from two perpendicular axes.

x-axis

The horizontal line in a two-dimensional coordinate system.

y-axis

The vertical line in a two-dimensional coordinate system.

Origin

The point of intersection of the $x$-axis and $y$-axis, denoted as $O$ with coordinates $(0, 0)$.

Coordinate axes

The plural term for the $x$-axis and $y$-axis used together to locate points in space.

Cartesian plane

The plane in which the coordinate axes are situated, also called the coordinate plane or $xy$-plane.

Quadrants

The four parts into which the coordinate axes divide the Cartesian plane.

x-coordinate

The perpendicular distance of a point from the $y$-axis, measured along the $x$-axis.

y-coordinate

The perpendicular distance of a point from the $x$-axis, measured along the $y$-axis.

Quadrant I

The region where both the $x$-coordinate and $y$-coordinate are positive $(+, +)$.

Quadrant II

The region where the $x$-coordinate is negative and the $y$-coordinate is positive $(-, +)$.

Quadrant III

The region where both the $x$-coordinate and $y$-coordinate are negative $(-, -)$.

Quadrant IV

The region where the $x$-coordinate is positive and the $y$-coordinate is negative $(+, -)$.

Distance Formula

Based on the Baudhāyana–Pythagoras Theorem, the distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.