Basic Analytic Geometry: Coordinates, Lines and Distances

The Cartesian plane, created by René Descartes in the 17th century, is a reference system formed by two perpendicular axes: the x-axis (horizontal, called the abscissa axis) and the y-axis (vertical, called the ordinate axis). The intersection of the two axes is the origin O = (0, 0).

Each point on the plane is identified by an ordered pair (x, y). The x value indicates the horizontal displacement from the origin (positive to the right, negative to the left) and the y value indicates the vertical displacement (positive upward, negative downward). For example, the point (3, −2) is 3 units to the right and 2 units below the origin.

The two axes divide the plane into four quadrants. In the 1st quadrant, x > 0 and y > 0. In the 2nd, x < 0 and y > 0. In the 3rd, x < 0 and y < 0. In the 4th, x > 0 and y < 0. Identifying the quadrant of a point is a quick way to check whether a result is correct.

The distance between two points P1 = (x1, y1) and P2 = (x2, y2) is derived directly from the Pythagorean theorem: d = √((x2 − x1)² + (y2 − y1)²). The segment joining the two points is the hypotenuse of a right triangle whose legs have lengths |x2 − x1| and |y2 − y1|.

The midpoint M between P1 and P2 has coordinates equal to the arithmetic mean of the coordinates of the endpoints: M = ((x1 + x2)/2, (y1 + y2)/2). For example, the midpoint of (2, 4) and (8, 10) is ((2+8)/2, (4+10)/2) = (5, 7). This result is intuitive: the midpoint is 'halfway' in each direction.

The slope m of a line passing through two points measures how much y changes for each unit change in x: m = (y2 − y1) / (x2 − x1). A horizontal line has m = 0; an increasing line has m > 0; a decreasing line has m < 0. Vertical lines have no defined slope.

The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept (the value of y when x = 0, i.e., where the line crosses the y-axis). Given the slope m and a point (x0, y0) on the line, find b by substitution: b = y0 − m·x0.

Two lines are parallel if they have the same slope: m1 = m2. Two lines are perpendicular if the product of their slopes is −1: m1 × m2 = −1, or equivalently, m2 = −1/m1. These relationships are widely used in analytic geometry to solve problems involving distances and angles between lines.

GPS uses a spherical coordinate system (latitude and longitude), but local navigation calculations — trajectories in small areas — are performed using Cartesian plane approximations. Map applications calculate distances between points of interest, shortest routes and midpoints using exactly the formulas of analytic geometry.

In computer graphics, each pixel on a screen is identified by coordinates (x, y). Rotations, translations and scalings of 2D objects are geometric transformations applied to coordinates. 2D games, animations and graphical interfaces constantly compute midpoint transformations and distances to detect collisions and render scenes.

In data analysis, scatter plots represent points on the Cartesian plane. Linear regression — a fundamental technique in statistics and machine learning — finds the line y = mx + b that best fits a set of points, minimising the distances between the points and the line. Mastering basic analytic geometry is therefore a prerequisite for understanding machine learning.