Radians and Coordinate Systems

Definition of an angle

In primary or middle school, we learned the static definition of an angle: a figure formed by two rays with a common endpoint is called an angle.

However, this definition limits angles to $[0, 360^\circ]$, which creates difficulties for deeper study and cannot explain some questions clearly, such as: what does a rotation of $720^\circ$ mean?

High-school mathematics introduces the dynamic definition of an angle: a figure formed when a ray in a plane rotates around its endpoint from one position to another is called an angle.

The starting position is called the initial side, and the ending position is called the terminal side. We define:

• An angle formed by rotation in the counterclockwise direction is a positive angle, and its measure is positive;
• An angle formed by rotation in the clockwise direction is a negative angle, and its measure is negative;
• An angle whose terminal side has not rotated relative to its initial side is a zero angle, and its measure is $0^\circ$.

This extends the concept of an angle to an arbitrary angle.

Note

The initial and terminal sides of a zero angle coincide, but not every angle whose initial and terminal sides coincide is a zero angle, such as an angle that is a multiple of $360^\circ$.

Radian measure

In practical applications, angles are often converted to various parameters, and describing angles using radians reduces the use of coefficients. We therefore introduce radian measure:

A central angle subtending an arc whose length equals the radius is called a $1$-radian angle. It is denoted by $\text{rad}$ and read as radian.

According to the definitions above, positive angles have positive radian measures, negative angles have negative radian measures, and the zero angle has radian measure $0$. If a circle of radius $r$ has a central angle $\alpha$ subtending an arc of length $l$, then:

$$|\alpha|=\dfrac{l}{r}$$

This formula can also be used to derive the formulas for arc length and sector area, which we omit here.

Thus, a $360^\circ$ angle has radian measure $2\pi$. With this correspondence, degree values and radian measures can be converted:

$$k \operatorname{rad} = \frac{\pi}{180^\circ} n^\circ$$

Consider an angle. If its terminal side is rotated one full turn, or even multiple turns, while the initial side remains fixed, the terminal side always ends at the same position. Such angles are called coterminal angles.

The set of angles coterminal with angle $\alpha$ is easy to obtain: $\{\varphi \mid \varphi = \alpha + 2k\pi, k \in \mathbf{Z}\}$.

This can be understood as continuously adding full turns to the side of the angle without changing the terminal-side position.

The mathematical constants $\pi$ and $\tau$

Some views in the Western mathematical community hold that the “true circle constant” should be $2\pi$, denoted by the Greek letter $\tau$. Supporters of the new circle constant celebrate the “true” Pi Day on June 28.

For example, in radian measure, a full angle is $2\pi$, and directly dividing $2\pi$ gives equal divisions of a full angle. Also, combinations of $2\pi$ occur frequently in complex analysis, and so on.

To follow customary conventions in different regions of China, OI Wiki uses the parameter $\pi$ to denote the circle constant.

Conventional notation for $\pi$ in programming

In C/C++, $\pi$ is generally taken to be acos(-1), since this value is the floating-point number closest to $\pi$. Writing $\pi$ using acos(-1) or 4 * atan(1) gives $3.14159265358979310000$.

Using other values, such as acos(-1.0/2.0), acos(1.0/2.0), or asin(1.0/2.0), gives $3.14159265358979360000$, which is not the closest floating-point number to $\pi$.

If you can remember it, you can also write $3.1415926535897932$ directly.

Cartesian coordinate system

Two mutually perpendicular number axes with a common origin in the same plane form a Cartesian coordinate system (Rectangular Coordinates).

Usually, the two number axes are placed horizontally and vertically, with rightward and upward directions as their positive directions. The horizontal axis is called the $x$-axis (x-axis) or horizontal axis, and the vertical axis is called the $y$-axis (y-axis) or vertical axis. The $x$-axis and $y$-axis together are called the coordinate axes. Their common origin $O$ is called the origin of the Cartesian coordinate system (origin). A Cartesian coordinate system with $O$ as its origin is denoted by $xOy$.

The $x$-axis and $y$-axis divide the coordinate plane into four quadrants (quadrant). The upper-right part is the first quadrant; the other three parts, in counterclockwise order, are the second, third, and fourth quadrants. The axes form the boundaries of the quadrants, so points on either axis and the origin are not in any quadrant. Usually, the $x$- and $y$-axes use the same unit length, but different unit lengths may be used in special cases.

Describing positions in a Cartesian coordinate system

In a Cartesian coordinate system, every point in the plane corresponds to a unique ordered pair (that is, the point's coordinates (coordinates)); conversely, every ordered pair corresponds to a unique point in the plane.

For any point $C$ in the plane, draw perpendiculars through $C$ to the $x$- and $y$-axes. The corresponding feet on the $x$- and $y$-axes, denoted by $a, b$, are respectively called the abscissa and ordinate of $C$. The ordered pair (ordered pair) $(a, b)$ is called the Cartesian coordinate of $C$. A point has different coordinates in different quadrants or on different axes.

Polar coordinate system

Consider a practical situation such as navigation: one would say “point $B$ is $30^\circ$ north of east from point $A$, at a distance of $100$ meters”, rather than “construct a Cartesian coordinate system with $A$ as the origin, giving $B(50,50\sqrt 3)$”.

Thus:

1. Choose a fixed point $O$ in the plane, called the pole;
2. Draw a ray $Ox$ from the pole, called the polar axis;
3. Choose a unit length (usually $1$ in mathematical problems), an angular unit (usually radians), and its positive direction (usually counterclockwise);

This establishes a polar coordinate system.

Describing positions in a polar coordinate system

Let $A$ be a point in the plane.

• The distance $|OA|$ between the pole $O$ and $A$ is called the radius vector, denoted by $\rho$;
• The angle $\angle xOA$ whose initial side is the polar axis and terminal side is $OA$ is called the polar angle, denoted by $\varphi$;

Then the ordered pair $(\rho,\varphi)$ is the polar coordinate of $A$.

By the definition of coterminal angles, $(\rho,\varphi)$ and $(\rho,\varphi + 2k\pi)\ (k\in \mathbf{Z})$ represent the same point. In particular, the pole has polar coordinates $(0,\varphi)\ (\varphi \in \mathbf{R})$, so points in the plane have infinitely many polar-coordinate representations.

If we require $\rho \ge 0,0 \le \varphi < 2\pi$, every point in the plane other than the pole can be represented by a unique ordered pair $(\rho,\varphi)$, and the point represented by $(\rho,\varphi)$ is uniquely determined.

Conversion between Cartesian and polar coordinate systems

Sometimes it is inconvenient to study a figure in polar coordinates. To study it in Cartesian coordinates, use the conversion formulas. The Cartesian coordinates $(x,y)$ of point $A(\rho,\varphi)$ are:

$$\begin{aligned} x &= \rho \cos \varphi \\ y &= \rho \sin \varphi \end{aligned}$$

Consequently:

$$\begin{aligned} \rho^2 &= x^2 + y^2\\ \tan \varphi &= \frac{y}{x}\ \ \ \ (x\not =0) \end{aligned}$$

Thus $\rho = \sqrt{x^2+y^2}$.

However, $\tan\varphi$ values with the same $\dfrac{y}{x}$ may correspond to two possible values of $\varphi$, so the direction must also be determined from the values of $x, y$. Specifically, define:

$$\operatorname{atan2}(y, x) = \begin{cases} \arctan(\frac{y}{x}) & \text{if } x > 0 \\ \arctan(\frac{y}{x}) + \pi & \text{if } y \ge 0, x < 0 \\ \arctan(\frac{y}{x}) - \pi & \text{if } y < 0, x < 0 \\ \pi/2 & \text{if } y > 0, x = 0 \\ -\pi/2 & \text{if } y < 0, x = 0 \\ \text{any} & \text{if } y = 0, x = 0 \end{cases}$$

Then $\varphi = \operatorname{atan2}(y, x)$. Note that the range of this function is $(-\pi, \pi]$.

The function is defined in the <math.h> or <cmath> library in C/C++; call it with atan2(y, x).

Cartesian coordinate system in space

Establish a Cartesian coordinate system in space as follows:

1. Choose a point $O$ in space;
2. Through $O$, draw three mutually perpendicular number axes $\overrightarrow{Ox}, \overrightarrow{Oy}, \overrightarrow{Oz}$, called the $x$-axis (horizontal axis), $y$-axis (vertical axis), and $z$-axis (upward axis), collectively called the coordinate axes. Their positive directions follow the right-hand rule: hold the $z$-axis with the right hand; when the four fingers turn from the positive direction of the $x$-axis toward the positive direction of the $y$-axis, the thumb points in the positive direction of the $z$-axis;
3. Set the length unit on each axis, usually to $1$.

This forms the spatial Cartesian coordinate system $O-xyz$. The fixed point $O$ is the origin of this coordinate system.

Any two coordinate axes determine a plane, giving three mutually perpendicular planes, collectively called coordinate planes. The coordinate plane determined by the $x$- and $y$-axes is the $xOy$ plane; similarly, there are the $yOz$ and $zOx$ planes. The three coordinate planes divide space into eight parts, each called an octant.

Describing positions in a spatial Cartesian coordinate system

After fixing the spatial Cartesian coordinate system $O-xyz$, a one-to-one correspondence can be established between points in space and triples.

Let $M$ be a point in space. Through $M$, draw planes perpendicular to the $x$-, $y$-, and $z$-axes. Let their intersections with the $x$-, $y$-, and $z$-axes be $P, Q, R$, respectively. The points $P, Q, R$ are called the projections of $M$ onto the $x$-, $y$-, and $z$-axes. If the coordinates of $P, Q, R$ on the $x$-, $y$-, and $z$-axes are $x, y, z$, respectively, then $M$ determines a triple $(x, y, z)$.

Conversely, given a triple $(x, y, z)$, take on the $x$-axis the point $P$ with coordinate $x$, on the $y$-axis the point $Q$ with coordinate $y$, and on the $z$-axis the point $R$ with coordinate $z$. Through $P, Q, R$, draw planes perpendicular to the $x$-, $y$-, and $z$-axes, respectively. They intersect at a point $M$ in space, and $M$ is the point determined by the triple $(x, y, z)$.

Thus, a one-to-one correspondence is established between a spatial point $M$ and a triple $(x, y, z)$. The triple $(x, y, z)$ is called the coordinate of $M$, written as $M(x, y, z)$; $x$ is the abscissa, $y$ the ordinate, and $z$ the applicate.

Cylindrical coordinate system

A spatial cylindrical coordinate system extends polar coordinates to three dimensions: start with the polar coordinate system used in the plane, then add a $z$-axis perpendicular to the plane through the pole $O$, pointing upward.

To find the point described by cylindrical coordinates $(\rho, \varphi, z)$, first handle $\rho$ and $\varphi$ in the polar coordinate system, then move “up” or “down” along the $z$-axis according to the $z$ coordinate.

Conversion between cylindrical and spatial Cartesian coordinate systems

The values of $z$ are the same in both coordinate systems.

For the conversion between $(x,y)$ and $(\rho, \varphi)$, see Conversion between Cartesian and polar coordinate systems.

Spherical coordinate system

Spherical coordinates can be determined as follows:

1. Stand at the origin facing the direction of the horizontal polar axis; the vertical axis points from the feet toward the head;
2. Raise the arm toward the vertical polar-axis direction;
3. Rotate counterclockwise by angle $\varphi$;
4. Rotate the arm downward by angle $\vartheta$; the arm points in the direction specified by $\varphi$ and $\vartheta$;
5. Move a distance $r$ from the origin along that direction.

This reaches the point described by spherical coordinates $(r,\vartheta,\varphi)$. Here $\vartheta$ is called the zenith angle, and $\varphi$ the azimuth angle.

Warning

For various reasons, some sources use $\phi$ for the zenith angle and $\theta$ for the azimuth angle. Be sure to pay attention to this when reading an article about spherical coordinates.

When writing about spherical coordinates, it is recommended to state in advance which symbols represent the zenith and azimuth angles.

Conversion between cylindrical and spherical coordinate systems

The values of $\varphi$ are the same in both coordinate systems.

From cylindrical coordinates to spherical coordinates:

$$\begin{aligned} r &= \sqrt{\rho^2 + z^2} \\ \vartheta &= \begin{cases} \arctan\left(\frac{\rho}{z}\right) & \text{if }z > 0 \\ \pi/2 & \text{if }z = 0, \rho \not= 0 \\ \arctan\left(\frac{\rho}{z}\right) + \pi & \text{if }z < 0 \\ \end{cases} \end{aligned}$$

For the point $(0,0,0)$ in cylindrical coordinates, its spherical $\vartheta$ is undefined.

From spherical coordinates to cylindrical coordinates:

$$\begin{aligned} \rho &= r \sin \vartheta \\ z &= r \cos \vartheta \end{aligned}$$

Conversion between spatial Cartesian and spherical coordinate systems

This can be used together with Conversion between Cartesian and polar coordinate systems and Conversion between cylindrical and spherical coordinate systems, or the formulas below can be used directly:

From spatial Cartesian coordinates to spherical coordinates:

$$\begin{aligned} r &= \sqrt{x^2 + y^2 + z^2} \\ \vartheta &= \arccos\left(\frac{z}{\sqrt{x^2 + y^2 + z^2}}\right) \\ \varphi &= \operatorname{atan2}(y, x) \end{aligned}$$

The definition of $\operatorname{atan2}$ is given in Conversion between Cartesian and polar coordinate systems.

For the point $(0,0,0)$ in the spatial Cartesian coordinate system, the values of its spherical $\vartheta$ and $\varphi$ are undefined.

From spherical coordinates to spatial Cartesian coordinates:

$$\begin{aligned} x &= r \sin \vartheta \cos \varphi \\ y &= r \sin \vartheta \sin \varphi \\ z &= r \cos \vartheta \end{aligned}$$