2Drotatsioonid

2Drotatsioonid, or two-dimensional rotations, are transformations that move points in a plane around a fixed point, known as the center of rotation. These transformations are fundamental in geometry and have various applications in fields such as computer graphics, physics, and engineering. A rotation in a 2D plane is typically defined by the angle of rotation and the center of rotation.

The angle of rotation determines the amount of clockwise or counterclockwise movement. Positive angles correspond to counterclockwise rotations, while negative angles correspond to clockwise rotations. The center of rotation is the point around which all other points in the plane are rotated.

Rotations in a 2D plane can be represented using matrix multiplication. For a rotation by an angle θ around the origin, the transformation matrix is given by:

[ cos(θ) -sin(θ) ]

[ sin(θ) cos(θ) ]

This matrix can be applied to any point (x, y) in the plane to find its new position (x', y') after the rotation. The resulting coordinates are calculated as follows:

x' = x * cos(θ) - y * sin(θ)

y' = x * sin(θ) + y * cos(θ)

Rotations are isometric transformations, meaning they preserve the distances between points. This property makes them useful in various applications, such as image processing and computer graphics, where preserving the shape and size of objects is crucial. Additionally, rotations are reversible, as rotating an object by an angle θ and then by -θ will return it to its original position.

In summary, 2Drotatsioonid are essential transformations in geometry and have wide-ranging applications. They involve moving points in a plane around a fixed center by a specified angle, preserving distances and shapes. Represented using matrix multiplication, rotations are fundamental in fields requiring precise spatial transformations.