ROTATE AN IMAGE WITH A MATHS OPERATOR

In Cartesian coordinates, we marked two points P and P1. The coordinates of P is x, y and that of P1 is x1, y1. Both are connected to the origin by straight lines. They make angles A and B with the x-axis respectively. The length of the lines connecting both the points to the origin is equal and it is ' r'. That means, P1 is rotation of the point P by an angle B.

From the figure we can write

x = r*cosA
y = r*sinA

similarly,
x1 = r*cos (A+B)
y1 = r*sin (A+B)

using trigonometry formula, we can expand the equations.
x1 = r*(cosAcosB - sinAsinB)
y1 = r*(sinAcosB+cosAsinB)

If we substitute x=r*cosA and y= r*sinA in the above equations we get,
x1 = xcosB -ysinB
y1 = xsinB+ycosB

writing in matrix form,

So R(B) is a matrix rotation operator. If we multiply coordinates of any point by the matrix operator R(B), the point will be rotated by an angle B and we will get the new coordinates

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The image shown in the figure is letter T. Assume each square is a pixel (picture element). Each pixel carries colour information of red, green and blue (RGB intensities)
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If you want to rotate the image, then coordinates of each point (pixel) of the image should be multiplied by matrix operator R to get the new coordinates. Next, RGB info. should be written in the new locations. Finally we will get tilted image as shown.

This is the mathematics of a "rotating tool" in an image editing software.

Science updates: Reaching absolute zero - 273.15 degree c temperature is proved to be impossible mathematically.

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