# Definition:Axis

## Definition

An axis is the name used for a general infinite straight line which is particularly significant in some particular way in the study of linear transformations of a real vector space.

### Coordinate Axes

Consider a coordinate system.

One of the reference lines of such a system is called an axis.

## Cartesian Coordinates

### $x$-Axis

In a cartesian coordinate system, the $x$-axis is the one usually depicted and visualised as going from left to right.

It consists of all the points in the real vector space in question (usually either $\R^2$ or $\R^3$) at which all the elements of its coordinates but $x$ are zero.

### $y$-Axis

In a cartesian coordinate system, the $y$-axis is the one usually depicted and visualised as going from "bottom" to "top" of the paper (or screen).

It consists of all the points in the real vector space in question (usually either $\R^2$ or $\R^3$) at which all the elements of its coordinates but $y$ are zero.

### $z$-Axis

In a cartesian coordinate system, the $z$-axis is the axis passing through $x = 0, y = 0$ which is perpendicular to both the $x$-axis and the $y$-axis.

It consists of all the points in the real vector space in question (usually $\R^3$) at which all the elements of its coordinates but $z$ are zero.

## Polar Coordinates

### Polar Coordinates

Let $O$ be the pole of the polar coordinate plane.

A ray is drawn from $O$, usually to the right, and referred to as the polar axis.

## Directions

### Positive Direction

Consider a coordinate system whose axes are each aligned with an instance of the real number line $\R$.

The direction along an axis in which the corresponding elements of $\R$ are increasing is called the positive direction.

### Negative Direction

Consider a coordinate system whose axes are each aligned with an instance of the real number line $\R$.

The direction along an axis in which the corresponding elements of $\R$ are decreasing is called the negative direction.

## Axis of Solid Figure

### Axis of Cone

Let $K$ be a cone whose base has a center of symmetry $C$.

Let $\LL$ be the straight line from the apex of $K$ to $C$.

Then $\LL$ is known as the axis of $K$.

### Axis of Right Circular Cone

Let $K$ be a right circular cone.

Let point $A$ be the apex of $K$.

Let point $O$ be the center of the base of $K$.

Then the line $AO$ is the axis of $K$.

In the words of Euclid:

The axis of the cone is the straight line which remains fixed and about which the triangle is turned.

### Axis of Right Circular Cylinder

In the words of Euclid:

The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned.

In the above diagram, the axis of the cylinder $ACBEFD$ is the straight line $GH$.

### Axis of Sphere

By definition, a sphere is made by turning a semicircle around a straight line.

That straight line is called the axis of the sphere.

In the words of Euclid:

The axis of the sphere is the straight line which remains fixed about which the semicircle is turned.

### Axis of Helix

The axis of a helix $\HH$ is the fixed line to which the tangent to $\HH$ makes a makes a constant angle.

## Also see

The term number axis is sometimes used to refer to the real number line.

## Linguistic Note

The plural of axis is axes, which is pronounced ax-eez not ax-iz.

Compare basis.