projectively

Projectively refers to a concept in geometry that deals with projections. In projective geometry, parallel lines are often considered to meet at a point at infinity. A projective transformation, also known as a collineation, is a transformation that preserves lines. This means that if three points are collinear in one figure, their images under a projective transformation will also be collinear. The concept of projective transformations is fundamental to understanding perspective in art and computer graphics, as well as in the study of algebraic curves and surfaces. It is a more general type of transformation than affine or Euclidean transformations, as it does not necessarily preserve distances or angles. A key aspect of projective geometry is the use of homogeneous coordinates, which allows for a unified treatment of points at infinity and finite points. This framework enables the representation of all projective transformations as matrix multiplications. The term "projectively" can also be used in a broader sense to describe anything related to or achieved through projection.