Cartesian coordinate planes (Polar to cartesian)

Cartesian coordinate planes are a set of values that determine a point’s position in space. These values are often represented numerically and are based on a reference system.

Types of Coordinate Systems

1. System of Cartesian Coordinates

• consists of perpendicular axes, which in three dimensions are typically denoted by the letters x, y, and z.
• The distances that a point is from these axes characterize it.
• 2D Cartesian Coordinates: A point is represented as  (x,y).
• 3D Cartesian Coordinates: A point is represented as $(x,y,z)$

2. Polar Coordinate System (2D).

• Uses a radius (𝑟) and an angle (𝜃) measured from a reference direction.
• Points are expressed as (𝑟, 𝜃) .

3. Cylindrical Coordinate System (3D)

• Combines polar coordinates in the x-y plane (𝑥−𝑦) with a height (𝑧).
• Points are expressed as (𝑟, 𝜃, 𝑧) .

4. Spherical Coordinate System (3-D)

• Uses a radius (r), polar angle (𝜙), and azimuthal angle (𝜃).
• A point is represented as (𝑟, 𝜃, 𝜙) .

What are Cartesian coordinates?

Axes

•  x-axis  horizontal line in 2D or a principle axis in 3D.
•  y-axis: might be a vertical line in 2D or another principle axis in 3D.
•  z-axis (3D): Adds depth, perpendicular to both x and y axes.

Quadrants (2D):

• The plane is divided into four regions:
  1. (+x,+y)
  2. (−x,+y)
  3. (−x,−y)
  4. (+x,−y)

Converting Between Coordinate Systems

1. From Polar to Cartesian:

                                                                                         x=rcos⁡(θ), y=rsin⁡(θ)

2. From Cartesian to Polar:

3. Between Cartesian and Spherical:

Polar to Cartesian

Equation: r=2cos⁡θ 

Step 1: Convert to Cartesian coordinates. We apply the following polar-to-Cartesian relationships:

Substitute r=2cosθ into these

Thus, the Cartesian form is:

Or alternatively, using the trigonometric identity:

Applications of Cartesian coordinate planes 

• Maps and Navigation: GPS employs a coordinate system to find locations.
• Graphics and Animation: Coordinates describe object placements in 2D and 3D spaces.
• Physics and engineering: Explain motion, forces, and structures.
• Robotics focuses on positioning and movement planning.