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Rotation - Definition, Reflection, Translation, Resizing, Formula and Rotation in 3D

 

The earth spinning around its axis or a top spinning on its tip, physics considers these as prime examples of rotation. But what does rotation mean in mathematics? The answer is simple and similar to the one mentioned in physics. The circular motion of an item around a center or axis is what rotation means in math. There are four fundamental types of transformations in geometry. They are discussed below:

Rotation: A rotation is a transformation in which an object is "spun" around a fixed point called the center of rotation. The mathematical procedure is the most sophisticated of the transformations presented, even though the notion is basic.

Reflection: When a function is reflected, the graph seems to be a duplicate copy (like a mirror image) of the original function. This is accomplished by changing the sign of the input to the function. For example, let’s take a function x = 30y, the reflection of this function will be -x or x = -30y.

Translation: The addition or removal of a constant from a function causes the shift known as translation. Every point in a translation is moved in the same direction by a set distance. For instance, let us make a graph of a = 14b; now, if we add 3 to this function, it becomes a = 14b + 3, now the graph is 3 units above the original graph.

Resizing: Scaling is a transformation that alters the size or shapes of a function's graph. Resizing can modify the size or shape of a function; they simply have been able to shift the graphical output from one group of points to another. For example, let a function f(x) = xy, if we multiply 3 to this function it resizes to 3f(x) = 3xy.

In this article, we shall learn in detail about rotation, related formulas, and the complete mathematics behind it.

Definition and Formula:

The cyclic oscillation of an item around a center is known as rotation. Different forms can be rotated by an angle around the central point. The rotation may be done in both clockwise and counterclockwise directions. 90°, 180°, and 270° are the most frequent rotation angles. Rotation in the coordinate plane follows a set of laws. For instance, an object having coordinates (x, y) originally is rotated in a clockwise direction, then the image formed has the coordinates (y, -x) and in the anti- clockwise direction is (-y, x). For an object rotated in 180-degrees, the coordinates changes to (-x, -y). A 360-degree rotation brings back the object to its original position.

Rotation in 3D: Objects in three-dimensional forms can be rotated about an unlimited number of imaginary lines called the rotational axis. The main rotations are the rotations around the X, Y, and Z axes. Rotations around any axis can be accomplished by rotating around the X-axis, then the Y-axis, and finally the Z-axis.

Rotational Symmetry: Many geometric forms, such as circles, squares, and rectangles, exhibit rotational symmetry. Rotational symmetry exists in all regular polygons. When an object is rotated about its center, it appears precisely as before it was rotated. The item is thus said to be rotationally symmetric. The figure tally itself as it rotates through a complete 360 degrees. The order of symmetry is the number of times rotations of 360 degrees are performed by any object.

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