Non-affinely

In mathematics and related fields, 'non-affinely' describes a characteristic or behavior that deviates from or does not adhere to affine transformations. An affine transformation, fundamentally, preserves collinearity (points lying on a line remain on a line after transformation) and ratios of distances along a line. Therefore, 'non-affinely' signifies transformations, mappings, or operations that do *not* maintain these properties. This can involve distortions, curves, or changes in relative distances that an affine transformation, such as scaling, rotation, translation, or shear, would not produce. It applies to various disciplines, including computer graphics, image processing, and data analysis, where the fidelity of the spatial relationship is not preserved or deliberately altered. Consequently, understanding 'non-affinely' is vital for understanding the intricacies and constraints that emerge in the realm of geometric transformations.

Non-affinely meaning with examples

• The image distortion, caused by the lens, demonstrated a non-affinely transformation, as the straight lines of the building facades appeared curved on the photograph. The preservation of straight lines wasn't maintained in the process, thus it was non-affinely represented.
• When a 3D object is projected onto a 2D plane using a perspective projection, the mapping is non-affinely due to the compression of space and the alteration of parallel lines that appear to converge towards a vanishing point. This effect adds depth and realism, yet destroys parallelism.
• In this specific algorithm, the data underwent a non-affinely deformation. The initial grid of points changed; the ratios between segments were altered during the process, showing its non-affinely nature.
• Consider a mapping that warps a square into a circular shape. Because parallel lines are not preserved, and distances are significantly altered; this transformation could accurately be described as non-affinely, showing the warping characteristics.
• The computer graphics program employed a non-affinely transformation to simulate the effect of a liquid's surface under distortion. Straight lines in the initial simulation were converted to curves and distortions of the geometry was achieved.