Rectangular Polar Conversion Calculator

Rectangular to polar conversion

With the rectangular-to-polar conversion calculator, find the counterpart of any point of the Cartesian plane in the polar coordinate system. 

Rectangular Coordinate System

In the rectangular coordinate system, also known as Cartesian coordinates, any point in the 2D plane is described by an ordered pair of numbers (x, y).

Here

•  'x' refers to the horizontal distance of the point from the origin (along the x-axis) 
• 'y' stands for the vertical distance from the origin (along the y-axis).

For example, a point with coordinates (2, 3) indicates a place that is 2 units to the right and 3 units up from the origin. This system is known as "rectangular" because any point can be reached by moving along a rectangle's sides.

Polar Coordinate System

Contrarily, the polar coordinate system is a two-dimensional system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

For a point (r,θ) in polar coordinates 

• 'R' is the distance of a point from the origin 
• 'θ' is the angle that the line connecting the point and the origin makes with the positive x-axis. This angle θ is measured counterclockwise from the positive x-axis.

For example, a point with polar coordinates (2, 60°) is located 2 units away from the origin, and the line connecting this point to the origin forms an angle of 60 degrees with the positive x-axis.

Conversion from Rectangular to Polar Coordinates

The relationships between rectangular and polar coordinates can be derived from the Pythagorean theorem and the definition of the inverse trigonometric function of arctan.

The formulas for conversion from rectangular to polar coordinates are:

r = √(x² + y²)

θ = tan⁻¹(y/x)

Please note, the second equation for θ requires some careful consideration. As the range of arctan (or tan⁻¹) is between -90° and 90°, this formula only works when x is positive. We need to adjust θ based on the quadrant of (x, y) as follows:

Quadrant I: θ = tan⁻¹(y/x)

Quadrant II: θ = tan⁻¹(y/x) + 180°

Quadrant III: θ = tan⁻¹(y/x) + 180°

Quadrant IV: θ = tan⁻¹(y/x) + 360°

Worked-Out Examples:

Example 1:

Let's convert the rectangular coordinates (3, 4) into polar coordinates.

Using our formulas:

r = √(3² + 4²) = 5

θ = tan⁻¹(4/3) ≈ 53.13° (since we're in the first quadrant)

So, the polar coordinates are (5, 53.13°).

Example 2:

Now let's convert the rectangular coordinates (-4, -5) into polar coordinates.

r = √((-4)² + (-5)²) = 6.40

θ = tan⁻¹((-5)/(-4)) ≈ 51.34°.

As the point is in the third quadrant, add 180 to the angle. 51.34° + 180° = 231.34°

So, the polar coordinates are (5, 231.34°).

Applications of Rectangular to Polar Conversion

Conversion from rectangular coordinates to polar coordinates is used extensively in various fields of study. Here are a few examples:

Physics: Polar coordinates are particularly useful when dealing with problems involving circular symmetry, like the motion of objects under the influence of central forces.

Engineering: In electrical engineering, particularly in signal processing and control systems, the polar form of complex numbers simplifies mathematical calculations and interpretation of data.

Computer Graphics: The conversion is often used in image processing, such as when applying filters or transformations that are more naturally described in polar coordinates.

Mathematics: In calculus, the conversion aids in integrating functions over circular regions or calculating areas and volumes of circular or cylindrical shapes.