Facts about the Cartesian Coordinate System for Kids
Gathered by: T.Price
- A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.
- Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0,0).
- The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
- One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes.
- The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.
- The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions, and did not publish the discovery.
- Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—means choosing a point O of the line (the origin), a unit of length, and an orientation for the line.
- The lines are commonly referred to as the x and y-axes where the x-axis is taken to be horizontal and the y-axis is taken to be vertical.
- In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the abscissa) is measured along a horizontal axis, oriented from left to right.
- However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise.
- The Euclidean transformations or Euclidean motions are the (bijective) mappings of points of the Euclidean plane to themselves which preserve distances between points.
- Translation Translating a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (a,b) to the Cartesian coordinates of every point in the set.
- That is, if the original coordinates of a point are (x,y), after the translation they will be  Rotation To rotate a figure counterclockwise around the origin by some angle is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x’,y’), where Thus: Reflection If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the Y axis), as if that line were a mirror.
- In more generality, reflection across a line through the origin making an angle with the x-axis, is equivalent to replacing every point with coordinates (x,y) by the point with coordinates (x’,y’), where Thus: Glide reflection A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line.
- If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero.
- The usual way of orienting the axes, with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the “first” and the y-axis the “second” axis) is considered the positive or standard orientation, also called the right-handed orientation.
- Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the x-axis to the y-axis, in a positively oriented coordinate system.