körparametrizálása

Körparametrizálása, often translated as "parameterization of a circle," refers to the mathematical process of defining the coordinates of points on a circle using a single variable, typically an angle. This allows for a systematic way to describe and work with all the points that lie on the circumference of a circle.

The most common parameterization of a circle centered at the origin (0,0) with radius 'r' uses the trigonometric functions sine and cosine. A point (x,y) on the circle can be represented by the parametric equations: x = r * cos(t) and y = r * sin(t). Here, 't' is the parameter, representing the angle in radians measured counterclockwise from the positive x-axis. As 't' varies from 0 to 2π (or 0 to 360 degrees), the point (x,y) traces out the entire circle.

If the circle is not centered at the origin but at a point (h,k), the parameterization is adjusted to: x = h + r * cos(t) and y = k + r * sin(t). This method is fundamental in various fields, including geometry, calculus, physics (e.g., describing circular motion), and computer graphics, where it enables the efficient generation and manipulation of circular shapes. The parameterization simplifies calculations involving arc length, area of sectors, and motion along a circular path.