# 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 Axis

### 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**.

### Spherical Coordinates

Definition:Spherical Coordinates/Polar Axis

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## 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.

(*The Elements*: Book $\text{XI}$: Definition $19$)

### 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.

(*The Elements*: Book $\text{XI}$: Definition $22$)

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.

(*The Elements*: Book $\text{XI}$: Definition $15$)

### 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.

## Sources

- 1971: Wilfred Kaplan and Donald J. Lewis:
*Calculus and Linear Algebra*... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**axis**(*plural***axes**) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**axis**(*plural***axes**) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**axis (axes)**