Cartesian coordinate system

The Cartesian coordinate system is a way to describe the position of points in space using numbers. It was named after René Descartes, who created it in the 1600s. This system helps us solve geometry problems using algebra and calculus.

In a flat plane, each point has two numbers, called coordinates, that show how far it is from two crossed lines called axes. The place where the axes cross is called the origin, and its coordinates are (0, 0). For points in three dimensions, like in our world, we use three coordinates.

With Cartesian coordinates, we can write equations to describe shapes. For example, a circle can be described by an equation using its coordinates. This system is important in many areas, such as astronomy, physics, engineering, and computer graphics. It is the most common way to work with points and shapes in mathematics and science.

History

The term Cartesian comes from the French mathematician and philosopher René Descartes. He shared this idea in 1637 while living in the Netherlands. Another mathematician, Pierre de Fermat, found the same idea but worked with three dimensions and did not share it publicly. Even earlier, a French teacher named Nicole Oresme used similar ideas long before Descartes and Fermat.

Descartes and Fermat both used one line to show positions. The idea of using two lines was added later when Descartes' book La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. They helped explain Descartes' work better.

This coordinate system was very important for the creation of calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The idea of two coordinates was later expanded to create vector spaces. Since then, many other coordinate systems have been made, like polar coordinates for flat space, and spherical and cylindrical coordinates for three-dimensional space.

Description

One dimension

Main article: Number line

A line with a chosen Cartesian coordinate system is called a number line. Every point on the line has a number that tells its position. We can pick two points on the line and give them numbers, like zero and one, to help find the positions of other points.

Two dimensions

Further information: Two-dimensional space

A Cartesian coordinate system in two dimensions uses two lines that cross at a point called the origin. Each point in the plane can be described by two numbers, called coordinates, that show how far the point is from each line. The first number is called the abscissa, and the second is called the ordinate.

Three dimensions

Further information: Three-dimensional space

A Cartesian coordinate system in three dimensions uses three lines that all meet at one point, the origin. Each point in space can be described by three numbers, called coordinates, that show how far the point is from each line. These numbers help us find the exact location of any point in space.

Higher dimensions

Cartesian coordinates can also be used in spaces with more than three dimensions. In these spaces, each point is described by a list of numbers, one for each dimension.

Generalizations

Cartesian coordinates can be changed so that the lines are not perpendicular or have different lengths. In these cases, we need new ways to find distances and angles.

Notations and conventions

The Cartesian coordinates of a point are usually written in parentheses and separated by commas, like (10, 5) or (3, 5, 7). The point where the axes meet is often called the origin and has coordinates (0, 0). In math, unknown coordinates are often called x and y for flat shapes, and x, y, and z for 3D shapes. This comes from a habit in algebra where letters at the end of the alphabet are used for unknown values.

We often name the axes after what they show. For example, in a graph showing how pressure changes over time, we might call the axes p and t. In computer programs, coordinates are sometimes written with subscripts like (x₁, x₂, ..., x_n), which helps organize the numbers in a list.

In school, children usually learn to read flat graphs by moving along the horizontal axis (called the x-axis) first and then up the vertical axis (called the y-axis). On computer screens, however, the vertical axis often points downward. For 3D shapes, the z-axis shows height, usually pointing up.

Quadrants and octants

Main articles: Octant (solid geometry) and Quadrant (plane geometry)

In a flat Cartesian system, the axes split the plane into four areas called quadrants. They are numbered I, II, III, and IV, going counter-clockwise from the top right. In 3D, the space is split into eight parts called octants, based on the signs (+ or −) of the x, y, and z values.

Cartesian formulae for the plane

Distance between two points

The Euclidean distance between two points in a plane with Cartesian coordinates ( x₁, y₁) and ( x₂, y₂) is

d = √[( x₂ − x₁ )² + ( y₂ − y₁ )²].

This is the Cartesian version of Pythagoras's theorem. In three-dimensional space, the distance between points ( x₁, y₁, z₁) and ( x₂, y₂, z₂) is

d = √[( x₂ − x₁ )² + ( y₂ − y₁ )² + ( z₂ − z₁ )²],

which can be obtained by using Pythagoras' theorem twice.

Euclidean transformations

The Euclidean transformations or Euclidean motions are mappings of points in the plane that keep distances the same. There are four types: translations, rotations, reflections, and glide reflections.

Translation

Translating points in the plane means moving them by the same fixed amount. If a point has coordinates ( x, y), after translation by amounts a and b, its new coordinates become

( x′, y′ ) = ( x + a, y + b ).

Rotation

To rotate a figure around the origin by an angle θ, each point's coordinates change to

x′ = x cos θ − y sin θ
y′ = x sin θ + y cos θ.

Reflection

Reflecting a point across the x-axis changes its y-coordinate to its opposite, while reflecting across the y-axis changes its x-coordinate to its opposite.

Glide reflection

A glide reflection combines a reflection across a line with a translation in the same direction.

General matrix form of the transformations

All these transformations can be described using matrices. The coordinates ( x, y) of a point can be written as a column matrix ( x y ). The new coordinates ( x′, y′ ) after a transformation are given by a formula involving a matrix A and a column matrix b.

Among these, the Euclidean transformations are those where the matrix A has special properties that keep distances the same.

Affine transformation

Affine transformations of the plane change lines to lines but may change distances and angles. They can also be described using matrices.

Some affine transformations that are not Euclidean have special names.

Scaling

Scaling makes a figure larger or smaller by multiplying all coordinates by the same number m. If ( x, y) are the coordinates of a point, the new coordinates become

( x′, y′ ) = ( mx, my ).

Shearing

A shearing transformation changes a square into a parallelogram. Horizontal shearing changes coordinates to

( x′, y′ ) = ( x + y s, y ).

Vertical shearing changes coordinates to

( x′, y′ ) = ( x, xs + y ).

Orientation and handedness

Main article: Orientability

See also: Right-hand rule and Axes conventions

In two dimensions

When we draw a flat space, we pick one line to be the x-axis and another line that crosses it at a right angle to be the y-axis. The point where they cross is called the origin. We can choose which side of each axis is positive (pointing forward) and which is negative (pointing backward). The usual way is to have the x-axis point to the right and the y-axis point upward. This is called the standard or right-handed orientation.

A helpful way to remember this is to use the right-hand rule. If you place your right hand on the flat space with your thumb pointing up, your fingers will point from the x-axis to the y-axis in a positive direction. The left-hand rule does the same but with the left hand.

In three dimensions

After choosing the x- and y-axes, the z-axis can point in two different directions. These two choices create what we call right-handed and left-handed coordinate systems. The standard, or right-handed, system has the xy-plane as a flat floor and the z-axis pointing upward. This matches the right-hand rule: point your index finger forward, bend your middle finger at a right angle, and your thumb points to the third direction. Your thumb shows the x-axis, your index finger the y-axis, and your middle finger the z-axis. Using the left hand gives the left-handed system.

Figure 7 shows both a left and a right-handed system. Because we see these in a flat picture, it can look confusing. The axis that seems to point downward and to the right is really pointing toward you, while the middle axis points away. The red circle shows how the x-axis turns into the y-axis.

Figure 8 tries to show a right-handed system in another way. It might look like it flips between a solid cube and a bent corner. To see it correctly, imagine the x-axis pointing toward you, which makes the corner look bent inward.

Representing a vector in the standard basis

In a Cartesian coordinate system, any point can be shown as a vector, which is like an arrow pointing from the origin (the point where the axes cross) to the point. In two dimensions, this vector can be written using special symbols i and j, which point along the x-axis and y-axis respectively.

For example, a point with coordinates (x, y) can be shown as a combination of these symbols. The same idea works in three dimensions, where we also use a symbol k to point along the z-axis.