Polar Coordinates

A Basic Overview of Polar Coordinates

Coordinate systems are essential for many real-world uses, including mapping, robotics, and navigation. The purpose of this article is to present and examine the operation of polar coordinates, an other coordinate system.Polar Coordinates

Why is a coordinate system necessary?

Providing a consistent way to define points in space in relation to a reference point is the main function of a coordinate system. Points in the Cartesian coordinate system are identified by their separations from the origin, both vertically and horizontally. For example, the coordinate (2, 5) on a Cartesian plane denotes a point that is 5 units vertically above and 2 units to the right of the origin (0, 0).

We may represent additional things like lines, circles, curves, vectors, and more using the same coordinate system once we have established this basic geometric element: the point.Polar Coordinates

A positive x-coordinate suggests movement to the right, whereas a negative x-coordinate suggests movement to the left. The Cartesian system uses the sign of the values to represent direction. Likewise, upward movement is indicated by a positive y-coordinate, and downward movement is indicated by a negative y-coordinate.

The Polar Coordinate System

An easy way to describe point locations within a grid is to use the polar coordinate system. Examine point P on the grid that follows. First, we may use its distance from the origin, which we’ll call “r,” to indicate its location. On the other hand, as the picture on the right illustrates, an endless number of points could be equally spaced from the origin, creating a circle around it.

The angle created with the x-axis is another characteristic included by the polar coordinate system to add more clarity and differentiate point P from others.

A polar coordinate system uses the values 6 and 45o to uniquely identify P, as the above picture illustrates.

The position of point P in this system is defined as (r, θ), where “r” is the distance from the origin and “θ” is the angle formed by the line segment (OP) with the x-axis. In accordance with standard practice, this angle is counterclockwise from the x-axis and is measured in radians.

Measurement of angle in radians

An other method of quantifying angles, especially in trigonometry and calculus, is the radian measure. Radians are based on the ratio of the arc length to the circle’s radius, as opposed to degrees, which split a circle into 360 equal pieces. In this system, the angle that an arc with a length equal to the radius subtends at the center of a circle is equal to one radian.

Since a circle’s diameter is 2π times its radius, a complete circle has 2π radians. The following formula can be used to convert degrees to radians:

90 degrees, for instance, would be:

Screenshot taken at 4.55.54 PM on April 23, 2024
90 degrees is equal to π/2 radians when simplified.

Example #1

What are the polar coordinates of the point that is equally spaced from the center but facing the opposite direction as P?

Step 1: We can begin by marking the grid’s P1 point.

Step 2: Calculate the angle it forms with the x-axis and its distance from the origin (r).

OP1 has a length of 6 units and a counterclockwise angle of 225 degrees between the line segment and the x-axis.

We need to convert 225o into radians.

Step 3: Write the point’s polar coordinates (r,θ).

P1 has the polar coordinates (6, 5π/4).