Image rotation is a type of affine transformation, a mathematical operation from linear algebra that preserves straight lines and parallelism while allowing translation, scaling, rotation, and shearing. Affine transformations are represented by matrices, and rotation specifically uses a 2x2 rotation matrix combined with translation to rotate points around a center. For a rotation of angle theta, each pixel at position (x, y) maps to a new position: x' = x*cos(theta) - y*sin(theta), y' = x*sin(theta) + y*cos(theta). This elegant trigonometric relationship comes directly from the geometry of circles and is the foundation of all 2D rotation in computer graphics.

For 90-degree increments (90, 180, 270 degrees), the rotation matrix simplifies dramatically because cos and sin of these angles are exactly 0, 1, or -1. A 90-degree clockwise rotation maps (x, y) to (y, -x), which means every pixel moves to a new grid-aligned position with no fractional coordinates. This is why 90-degree rotations are perfectly lossless: each source pixel maps exactly to one destination pixel with no interpolation needed. The pixels are simply rearranged, and for 90 or 270 degrees, the image width and height are swapped.

Non-90-degree rotations present a fundamental challenge because the rotated pixel positions almost never land exactly on the output grid. When a source pixel at (100, 50) rotates by 15 degrees, it maps to approximately (89.26, 74.05), a position between four output pixels. The output pixel at (89, 74) must be assigned a color value, but no single source pixel maps directly to it. This is where interpolation becomes necessary. The algorithm performs inverse mapping, asking for each output pixel: which source position maps here? That source position typically falls between source pixels, and interpolation (nearest-neighbor, bilinear, or bicubic) computes the output value from surrounding source pixels. Nearest-neighbor simply picks the closest source pixel, producing blocky results. Bilinear interpolation blends the four surrounding pixels, producing smooth but slightly soft results. Bicubic interpolation considers sixteen surrounding pixels for the sharpest results.

Another consequence of arbitrary rotation is that the rotated image may not fit within the original rectangular bounds. The corners of the original rectangle extend beyond the original dimensions after rotation, so the output canvas must be enlarged to accommodate the full rotated image. The areas outside the original image but within the new bounding rectangle are typically filled with a transparent or user-specified background color. Some tools crop to the largest axis-aligned rectangle that fits within the rotated image to avoid these empty corners.