polárkoordinátákra

Polárkoordinátákra refers to the concept of polar coordinates in mathematics. In a 2D plane, polar coordinates represent a point by its distance from a fixed point, the origin, and the angle from a fixed direction. The distance is typically denoted by 'r' and the angle by 'theta' (θ). The origin in polar coordinates is the same as the origin in Cartesian coordinates. The fixed direction is usually the positive x-axis.

To convert from polar coordinates (r, θ) to Cartesian coordinates (x, y), the following formulas are used: x = r * cos(θ) and y = r * sin(θ). Conversely, to convert from Cartesian coordinates (x, y) to polar coordinates, the distance r can be found using the Pythagorean theorem: r = sqrt(x² + y²). The angle θ can be found using the arctangent function: θ = atan2(y, x). The atan2 function is preferred over a simple arctan because it correctly determines the quadrant of the angle.

Polar coordinates are particularly useful for describing circular or spiral phenomena. They simplify the representation of curves that are symmetric around a point, such as circles, spirals, and cardioids. For example, a circle centered at the origin with radius 'a' is described simply as r = a in polar coordinates, whereas in Cartesian coordinates it is x² + y² = a². This simplification makes polar coordinates a valuable tool in various fields, including physics, engineering, and computer graphics.