How to Plot Points and Convert Between Polar & Cartesian Coordinates
When you ask a child where a particular shop is, the child will answer roughly "just, over there" and point, describing the distance combined with the direction respectively. Or when you ask someone where their village is? They will answer "30 miles north of London" describing the distance and the direction rather than giving the latitude and longitude of their village.
In Mathematics, we have always been taught to represent the position of an object using cartesian coordinates which is not very natural or convenient. For a start, we should consider both the negative and the positive numbers to describe the points on the plane because using the direction and the distance as a means to describe the position is far more natural and convenient.
Therefore, we can describe a polar coordinate system as a method in which a point is described by its distance from a fixed point at the center of the coordinate space known as a pole and by the measurement of the angle formed by a fixed-line and a line from the pole through the given point. In the polar coordinate system, the coordinates of a point are represented as (r, θ), where r is the distance of the point from the pole, and θ is the measure of the angle.
The polar coordinate system is just like an alternative to the Cartesian coordinate system. On one hand where the Cartesian system determines the position east and north of a fixed point. On the other hand, the polar coordinate system determines the location using direction and distance from a fixed point.
Polar Coordinates Formula
With the help of the formula, we can drive an infinite number of polar coordinates for just one coordinate point. The formula can be represented as:
Where n is represented as an integer.
The value of θ will be positive if measured counterclockwise whereas it will be negative if measured clockwise. In the same way, the value of r will be positive if laid off the terminal side of θ whereas the value of r will be negative if laid off at the prolongation through the origin from the terminal side of θ. The side where the angle starts is known as the initial side whereas the ray where the measurement of the angle stops is known as the terminal side.
Plotting Points in Polar Coordinate System
(image will be uploaded soon)
The two points are 3,60 and 4,210.
In a two-dimensional Polar Coordinate system, there are two polar coordinates: r and θ i.e, the radial coordinate which represents the radial distance from the pole and the angular coordinate which represents the anticlockwise angle from the 0° ray, respectively. It is also known as the positive x-axis on the Cartesian coordinate plane.
We can look at some polar coordinates examples for a better grasp.
Consider that the polar coordinates (3,60°) are plotted as a point 3 units from the pole on the 60° ray. The coordinates (−3,240°) will also be plotted exactly at this point because the negative radial distance is measured as a positive distance on the opposite ray (240° − 180° = 60°).
Another important aspect of the Polar Coordinate System that is not present in the Cartesian coordinate system is the expressibility of a single point with an infinite number of different coordinates. Usually, the point (r, θ) can also be represented as (r, θ ± n × 360°) or (−r, θ ± (2n + 1)180°),
where n is the integer. If the r coordinate of a point is 0, then the point will be located at the pole regardless of the θ coordinate.
Converting Cartesian Coordinate System to Polar Coordinate System
If we know a point in Cartesian Coordinates (x,y) and want to convert it into Polar Coordinates (r,θ) we have to solve a right triangle with two known sides.
Example 1) What will be (12,5) in the Polar Coordinates system?
(image will be uploaded soon)
Solution 1) We can use Pythagoras theorem to find the hypotenuse
\[r^{2} = 12^{2} + 5^{2}\]
\[r = \sqrt{(12^{2} + 5^{2})}\]
\[r = \sqrt{(144 + 25)}\]
\[r = \sqrt{(169)}\]
r = 13.
Now, to find the angle, we will use the tangent function.
Tan ( θ ) = 5 / 12
θ = tan-1 ( 5 / 12 ) = 22.6°
Therefore, point (12,5) in the cartesian coordinate system will be (13, 22.6°) in the Polar Coordinate System.
Converting Polar Coordinate System to Cartesian coordinate System
Converting the polar coordinate system to Cartesian coordinate systems is relatively simple. We just have to take the cosine of θ in order to find the corresponding Cartesian x coordinate and sine of θ in order to find y.
Example 2) conversion from a polar coordinate system to the cartesian coordinate system.
(image will be uploaded soon)
Solution 2) With the help of basic trigonometry, it becomes easy to determine polar coordinates from a given pair of Cartesian coordinates.
\[r = \sqrt{x^{2} + y^{2}}\]
θ = tan-1(y/x)
FAQs on Polar Coordinates Explained: Concepts, Formulas & Uses
1. What are polar coordinates and how do they work?
Polar coordinates are a two-dimensional system used to locate points in a plane. Unlike the Cartesian system that uses (x, y) coordinates, the polar system uses a distance and an angle. Every point is defined by a pair (r, θ), where:
- r (the radial coordinate) is the direct distance from a central point called the pole (equivalent to the origin).
- θ (the angular coordinate) is the angle measured counter-clockwise from a fixed ray called the polar axis (equivalent to the positive x-axis).
This system is like giving directions by saying “5 km in the north-east direction” instead of using latitude and longitude.
2. How do you convert coordinates from the Cartesian system (x, y) to the polar system (r, θ)?
To convert a Cartesian point (x, y) to its polar equivalent (r, θ), you can use the Pythagorean theorem and basic trigonometry. The formulas are:
- To find the distance r: r = √(x² + y²)
- To find the angle θ: θ = tan⁻¹(y/x)
It is crucial to consider the quadrant in which the (x, y) point lies to find the correct angle θ, as the arctan function on a calculator typically only returns values between -90° and +90°.
3. What are the formulas to convert polar coordinates (r, θ) back to Cartesian coordinates (x, y)?
Converting from polar coordinates (r, θ) back to Cartesian coordinates (x, y) is a straightforward process based on right-angled triangle trigonometry. The conversion formulas are:
- For the x-coordinate: x = r cos(θ)
- For the y-coordinate: y = r sin(θ)
These formulas directly calculate the horizontal (x) and vertical (y) components of the point based on its distance 'r' from the pole and its angle 'θ'.
4. Why can a single point be represented by an infinite number of polar coordinates?
A single point has a unique (x, y) address in the Cartesian system, but it can have infinite addresses in the polar system. This is due to two main reasons:
- Co-terminal Angles: The angle θ can be increased or decreased by any multiple of 360° (or 2π radians) without changing the direction. For example, the coordinates (5, 30°), (5, 390°), and (5, -330°) all represent the exact same point.
- Negative Radius: The radius 'r' can be negative. A negative radius, -r, means you move 'r' units in the opposite direction (180° away) of the angle θ. Therefore, the point (r, θ) is identical to the point (-r, θ + 180°).
5. In which scenarios are polar coordinates more advantageous than Cartesian coordinates?
Polar coordinates are significantly more useful in situations involving circular, radial, or rotational symmetry. Key examples include:
- Physics: Describing the motion of planets in orbit, rotating bodies, or electromagnetic waves radiating from a source.
- Engineering: Mapping the pickup patterns of microphones, designing radar systems that scan in a circle, or analysing stresses in circular pipes.
- Mathematics: Graphing complex shapes like cardioids and spirals, which have very complicated equations in Cartesian form but simple ones in polar form (e.g., r = 1 + cos(θ)).
In contrast, Cartesian coordinates are better for describing linear or grid-based systems.
6. How does the 2D concept of polar coordinates extend into three dimensions?
Polar coordinates are inherently a 2D system. However, they form the basis for two common 3D coordinate systems:
- Cylindrical Coordinates (r, θ, z): This is the most direct extension. It takes the 2D polar coordinates (r, θ) for the x-y plane and simply adds a vertical z-axis, identical to the one in the Cartesian system. It's useful for describing objects with an axis of symmetry, like a cylinder or a helix.
- Spherical Coordinates (ρ, θ, φ): This system uses one distance and two angles to define a point in 3D space. Here, ρ (rho) is the direct distance from the origin to the point, θ (theta) is the same azimuthal angle as in polar coordinates, and φ (phi) is the polar angle measured down from the positive z-axis. It is ideal for describing spherical objects like planets.
7. What are some practical, real-world examples where polar coordinates are used?
Polar coordinates are fundamental in many real-world technologies and fields. For instance:
- Air Traffic Control: Radar systems display aircraft locations using a distance and an angle (bearing) from the radar station, which are polar coordinates.
- Robotics: The position of a robotic arm is often defined by the rotation of its joints (angles) and the extension of its segments (distances), a direct application of polar principles.
- Computer Graphics: Used to create rotating objects, circular patterns, or radial gradients in digital design and gaming.
- Audio Engineering: The directional sensitivity of a microphone is often plotted on a polar graph, showing how well it picks up sound from different angles.
8. What is a polar curve?
A polar curve is a graph of a polar equation, which is typically an equation of the form r = f(θ). Unlike a Cartesian curve (y = f(x)) that relates vertical and horizontal positions, a polar curve describes how the distance from the pole (r) changes as the angle (θ) sweeps around. This allows for the easy creation of beautiful and complex shapes that are difficult to describe in Cartesian coordinates, such as:
- Cardioids: Heart-shaped curves, e.g., r = a(1 - cos(θ)).
- Lemniscates: Figure-eight shaped curves.
- Roses: Flower-like shapes with multiple petals, e.g., r = a cos(nθ).
