Polar and Cartesian Coordinates Converter
Understanding Coordinate Systems
Cartesian Coordinates $(x, y)$: This system, also known as the rectangular coordinate system, uses two perpendicular axes (the horizontal x-axis and the vertical y-axis) to specify the position of a point. The coordinates $(x, y)$ represent the horizontal distance from the origin and the vertical distance from the origin, respectively. It's like navigating a city grid where you move a certain number of blocks east/west and then north/south.
Polar Coordinates $(r, \theta)$: This system describes a point's position using a distance ($r$) from a fixed point called the pole (usually the origin in a Cartesian system) and an angle ($\theta$) from a fixed direction, called the initial line (usually the positive x-axis). Imagine yourself standing at the origin; $r$ tells you how far away the point is, and $\theta$ tells you in which direction to look.
The angle $\theta$ is typically measured counter-clockwise from the positive x-axis. It can be expressed in either **radians** (where $2\pi$ radians equals $360^\circ$) or **degrees**. The radius $r$ is generally considered to be non-negative. When converting from Cartesian to Polar, the angle $\theta$ is often chosen to be in the range $(-\pi, \pi]$ (from $-180^\circ$ to $180^\circ$) or $[0, 2\pi)$ (from $0^\circ$ to $360^\circ$).
Cartesian $(x,y)$ to Polar $(r,\theta)$
Input Cartesian Coordinates
Polar $(r,\theta)$ to Cartesian $(x,y)$
Input Polar Coordinates
Students Also Ask
What are the key differences between Cartesian and Polar coordinates?
The primary difference lies in how a point's location is defined:
- Cartesian Coordinates $(x, y)$: A point is located by its horizontal distance ($x$) from the y-axis and its vertical distance ($y$) from the x-axis. Think of it like a city grid system where you move east/west, then north/south. The grid lines are perpendicular.
- Polar Coordinates $(r, \theta)$: A point is located by its distance ($r$) from the origin (called the "pole") and the angle ($\theta$) measured from a fixed reference direction (usually the positive x-axis). Imagine a radar screen; $r$ is how far the blip is from the center, and $\theta$ is its direction. The grid lines are concentric circles and radial lines.
Cartesian: $(x, y)$
Polar: $(r, \theta)$
Why do we use two different coordinate systems?
Different problems and shapes are more naturally and simply described in one system over the other. Choosing the right coordinate system can significantly simplify calculations and understanding:
- Cartesian Coordinates are ideal for:
- Linear relationships and straight lines (e.g., $y = mx + b$).
- Rectangular shapes (squares, rectangles).
- Problems involving horizontal and vertical movements.
- Polar Coordinates are ideal for:
- Circular motion, rotations, and phenomena with radial symmetry (e.g., orbits, sound waves spreading from a source).
- Spirals and curves that revolve around a central point.
- Simplifying equations for circles (e.g., a circle centered at the origin, $x^2 + y^2 = R^2$, becomes simply $r = R$ in polar coordinates).
For example, to describe a circle of radius 5 centered at the origin:
In Cartesian: $x^2 + y^2 = 5^2$ or $x^2 + y^2 = 25$
In Polar: $r = 5$
Clearly, the polar equation is much simpler for describing a circle.
Can the radius ($r$) be negative in polar coordinates?
By the most common and standard definition of polar coordinates, the radius $r$ is always considered to be non-negative ($r \ge 0$). This ensures a unique representation for most points.
However, in some advanced mathematical contexts or for certain applications, an extended definition allows $r$ to be negative. If $r$ is negative, a point $(-r, \theta)$ is interpreted as going a distance $|r|$ in the direction *opposite* to the angle $\theta$. This means the point is equivalent to $(|r|, \theta + \pi)$ radians (or $\theta + 180^\circ$).
Example: The point $(-3, 0)$ in polar coordinates is the same as $(3, \pi)$ or $(3, 180^\circ)$.
For the purpose of this converter, we adhere to the standard definition where $r \ge 0$. If you input a negative value for $r$ in the "Polar to Cartesian" conversion, the converter will flag it as an error, as this is the standard expectation for basic coordinate conversions.