Radians and Coordinate Systems

Definition of Angle

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

However, this definition restricts angles to the range \([0, 360^\circ]\), which poses difficulties for in-depth study. Other issues also cannot be explained clearly, such as: what does rotating \(720^\circ\) mean?

In high school mathematics, the dynamic definition of an angle is introduced: an angle is a figure formed by rotating a ray in a plane about its endpoint from one position to another.

The starting position is called the initial side, and the ending position is called the terminal side. It is stipulated that:

An angle formed by rotating counterclockwise is called a positive angle, and its measure is positive;
An angle formed by rotating clockwise is called a negative angle, and its measure is negative;
An angle where the terminal side does not rotate relative to the initial side is called a zero angle, and its measure is \(0^\circ\).

This extends the concept of angles to arbitrary angles.

Note

A zero angle has coinciding initial and terminal sides, but not all angles with coinciding initial and terminal sides are zero angles, such as angles that are multiples of \(360^\circ\).

Radian Measure

In practical applications, conversions from angles to various parameters are often required, and using radian measure to describe angles can reduce the use of coefficients. Therefore, radian measure is introduced next:

An angle whose arc length equals the radius length is defined as an angle of \(1\) radian, denoted by the symbol \(\text{rad}\), read as: radian.

According to the previous stipulations, positive angles have positive radian measures, negative angles have negative radian measures, and zero angles have a radian measure of \(0\). If the central angle \(\alpha\) of a circle with radius \(r\) subtends an arc of length \(l\), then:

\[ |\alpha|=\dfrac{l}{r} \]

Using this formula, the arc length and sector area formulas can also be derived, which are omitted here.

Thus, a \(360^\circ\) angle corresponds to \(2\pi\) radians. With this correspondence, conversions between degree measure and radian measure can be performed:

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

Consider an angle; if its terminal side is rotated another full revolution, or even multiple revolutions, while the initial side remains fixed, the terminal side position remains the same. Such angles are called angles with the same terminal side position.

The set of angles with the same terminal side position as angle \(\alpha\) is easily obtained as \(\{\varphi \mid \varphi = \alpha + 2k\pi, k \in \mathbf{Z}\}\).

This can be understood as: continuously adding full revolutions to this angle, the terminal side position remains unchanged.

The Two Mathematical Constants \(\pi\) and \(\tau\)

Currently, some perspectives in Western mathematics suggest that the "true circle constant" should be \(2\pi\), denoted by the Greek letter \(\tau\). Supporters of the new circle constant choose to celebrate the "true" Pi Day on June 28.

For example, in radian measure, a full rotation is \(2\pi\), and directly dividing \(2\pi\) yields equal divisions of a full rotation. Also, in complex analysis, combinations of \(2\pi\) frequently appear, etc.

To align with the conventions in various regions of China, OI Wiki uses the parameter \(\pi\) to represent the circle constant.

Common Practices for Writing Pi in Programming

In C/C++ languages, \(\pi\) is generally taken as acos(-1), as this value is the floating-point number closest to \(\pi\). Using acos(-1) or 4 * atan(1) yields \(\pi\) as \(3.14159265358979310000\).

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

If you have it memorized, you can also directly write \(3.1415926535897932\).

Cartesian Coordinate System

Two mutually perpendicular number lines with a common origin on the same plane constitute a Cartesian coordinate system (Rectangular Coordinates).

Typically, the two number lines are placed horizontally and vertically, with the rightward and upward directions taken as the positive directions of the two axes, respectively. The horizontal number line is called the \(x\)-axis (x-axis) or horizontal axis, and the vertical number line is called the \(y\)-axis (y-axis) or vertical axis. The \(x\)-axis and \(y\)-axis are collectively called the coordinate axes, and their common origin \(O\) is called the origin of the Cartesian coordinate system. The Cartesian coordinate system with point \(O\) as the origin is denoted as \(xOy\).

The \(x\)-axis and \(y\)-axis divide the coordinate plane into four quadrants. The upper-right part is called the first quadrant, and the other three parts are called the second, third, and fourth quadrants in counterclockwise order. Quadrants are bounded by the axes; points on the axes and the origin are not in any quadrant. Generally, the \(x\)-axis and \(y\)-axis have the same unit length, but in special cases, they may have different unit lengths.

Describing Positions in the Cartesian Coordinate System

In the Cartesian coordinate system, for any point on the plane, there is a unique ordered pair (i.e., the coordinates of the point) corresponding to it; conversely, for any ordered pair, there is a unique point on the plane corresponding to it.

For any point \(C\) in the plane, drop perpendiculars from \(C\) to the \(x\)-axis and \(y\)-axis. The feet of the perpendiculars on the \(x\)-axis and \(y\)-axis, \(a\) and \(b\), are called the abscissa and ordinate of point \(C\), respectively. The ordered pair \((a, b)\) is called the Cartesian coordinates of point \(C\). A point in different quadrants or on the axes has different coordinates.

Polar Coordinate System

Consider practical situations, such as navigation, where one might say "point \(B\) is \(100\) meters away from point \(A\) in the direction of \(30^\circ\) east of north," rather than "establish a Cartesian coordinate system with \(A\) as the origin, \(B(50,50\sqrt 3)\)."

Thus:

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

This establishes the polar coordinate system.

Describing Positions in the Polar Coordinate System

Let \(A\) be a point on the plane.

The distance \(|OA|\) from the pole \(O\) to \(A\) is called the radial distance, denoted by \(\rho\);
The angle \(\angle xOA\) from the polar axis to \(OA\) is called the polar angle, denoted by \(\varphi\);

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

From the definition of angles with the same terminal side, \((\rho,\varphi)\) and \((\rho,\varphi + 2k\pi)\ (k\in \mathbf{Z})\) actually represent the same point. In particular, the polar coordinates of the pole are \((0,\varphi)\ (\varphi \in \mathbf{R})\), so the polar coordinate representation of points in the plane has infinitely many forms.

If it is stipulated that \(\rho \ge 0,0 \le \varphi < 2\pi\), then except for the pole, every point in the plane can be uniquely represented by an ordered pair \((\rho,\varphi)\), and the point represented by polar coordinates \((\rho,\varphi)\) is uniquely determined.

Conversion Between Cartesian and Polar Coordinate Systems

Of course, sometimes studying graphs in the polar coordinate system is inconvenient. To switch to the Cartesian coordinate system for study, there are conversion formulas. The Cartesian coordinates \((x,y)\) of point \(A(\rho,\varphi)\) can be expressed as:

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

Furthermore:

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

Thus, \(\rho = \sqrt{x^2+y^2}\).

However, for a given \(\dfrac{y}{x}\), \(\tan\varphi\) corresponds to two possible values of \(\varphi\). At this point, the signs of \(x\) and \(y\) are needed to determine the direction. Specifically, define the function:

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

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

In C/C++ language, the <math.h> or <cmath> library defines this function. Call atan2(y, x) to use it.

Three-Dimensional Cartesian Coordinate System

Establish a three-dimensional Cartesian coordinate system using the following method:

Select a point \(O\) in space;
Through point \(O\), draw three mutually perpendicular number lines \(\overrightarrow{Ox}, \overrightarrow{Oy}, \overrightarrow{Oz}\), called the \(x\)-axis (abscissa axis), \(y\)-axis (ordinate axis), and \(z\)-axis (vertical axis), collectively called the coordinate axes; their positive directions follow the right-hand rule, i.e., grasp the \(z\)-axis with the right hand, and when the four fingers of the right hand rotate from the positive direction of the \(x\)-axis to the positive direction of the \(y\)-axis by an angle, the direction of the thumb is the positive direction of the \(z\)-axis;
Set the length units on each axis, usually all set to \(1\).

This constitutes a three-dimensional Cartesian coordinate system, denoted as \(O-xyz\). The fixed point \(O\) is called the origin of this coordinate system.

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

Describing Positions in the Three-Dimensional Cartesian Coordinate System

After establishing the three-dimensional Cartesian coordinate system \(O-xyz\), a one-to-one correspondence between points in space and ordered triples can be established.

Let point \(M\) be a point in space. Through point \(M\), draw planes perpendicular to the \(x\)-axis, \(y\)-axis, and \(z\)-axis, respectively. Let the intersections of these three planes with the \(x\)-axis, \(y\)-axis, and \(z\)-axis be \(P, Q, R\), respectively. Points \(P, Q, R\) are called the projections of point \(M\) on the \(x\)-axis, \(y\)-axis, and \(z\)-axis, respectively. Let the coordinates of points \(P, Q, R\) on the \(x\)-axis, \(y\)-axis, and \(z\)-axis be \(x, y, z\), respectively. Then point \(M\) determines an ordered triple \((x, y, z)\).

Conversely, given an ordered triple \((x, y, z)\), take point \(P\) with coordinate \(x\) on the \(x\)-axis, point \(Q\) with coordinate \(y\) on the \(y\)-axis, and point \(R\) with coordinate \(z\) on the \(z\)-axis. Then, through points \(P, Q, R\), draw three planes perpendicular to the \(x\)-axis, \(y\)-axis, and \(z\)-axis, respectively. These three planes intersect at a point \(M\) in space, which is the point determined by the ordered triple \((x, y, z)\).

Thus, a one-to-one correspondence is established between points \(M\) in space and ordered triples \((x, y, z)\). The ordered triple \((x, y, z)\) is called the coordinates of point \(M\), denoted as \(M(x, y, z)\), where \(x\) is called the abscissa, \(y\) the ordinate, and \(z\) the applicate.

Cylindrical Coordinate System

The cylindrical coordinate system extends polar coordinates to three dimensions: starting from the polar coordinate system applied to planar work, then adding a \(z\)-axis perpendicular to that 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 "upward" or "downward" along the \(z\)-axis according to the \(z\) coordinate.

Conversion Between Cylindrical and Cartesian Coordinate Systems

The \(z\) values are the same in both coordinate systems.

The conversion between \((x,y)\) and \((\rho, \varphi)\) is described in the previous section Conversion Between Cartesian and Polar Coordinate Systems.

Spherical Coordinate System

Spherical coordinates can be determined as follows:

Stand at the origin, facing the direction of the horizontal polar axis; the vertical axis points from feet to head;
Raise your arm, pointing in the direction of the vertical polar axis;
Rotate counterclockwise by angle \(\varphi\);
Lower your arm by angle \(\vartheta\); your arm now points in the direction specified by \(\varphi\) and \(\vartheta\);
Move a distance \(r\) from the origin along that direction.

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

Warning

Due to various reasons, some sources use \(\phi\) for the zenith angle and \(\theta\) for the azimuth angle. When reading articles involving spherical coordinates, be sure to pay attention to this.

Also, when writing articles, if spherical coordinates are used, it is advisable to declare clearly which symbols represent the zenith and azimuth angles.

Conversion Between Cylindrical and Spherical Coordinate Systems

The \(\varphi\) values are the same in both coordinate systems.

From cylindrical to spherical coordinates:

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

Note that for the point \((0,0,0)\) in cylindrical coordinates, its zenith angle \(\vartheta\) in spherical coordinates is undefined.

From spherical to cylindrical coordinates:

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

Conversion Between Cartesian and Spherical Coordinate Systems

You can combine the previous section Conversion Between Cartesian and Polar Coordinate Systems and the previous section Conversion Between Cylindrical and Spherical Coordinate Systems, or directly use the following formulas:

From Cartesian to spherical coordinates:

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

where \(\operatorname{atan2}\) is defined in Conversion Between Cartesian and Polar Coordinate Systems.

Note that for the point \((0,0,0)\) in Cartesian coordinates, its zenith angle \(\vartheta\) and azimuth angle \(\varphi\) in spherical coordinates are undefined.

From spherical to Cartesian coordinates:

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

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