COLLINEATION Meaning and
Definition
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Collineation is a term used in projective geometry to describe a fundamental geometric transformation that preserves the alignment of points, lines, and planes. It is a mapping that associates each point in one space to a corresponding point in another space, while maintaining the collinearity (lying on the same line) between points.
A collineation can be viewed as a generalization of a linear transformation in a projective space, where the concept of parallelism between lines is introduced. In a collineation, parallel lines are mapped to parallel lines, and non-parallel lines intersect at a unique point that is also preserved by the transformation. This property highlights the importance of preserving the fundamental geometric relationships.
Furthermore, collineations are characterized by a set of properties and transformations that they uphold. These transformations include the preservation of incidence (points lying on lines or planes), cross-ratio (a measure of the arrangement of four points on a line), and harmonic ratio (a special case of cross-ratio that relates three points).
In simpler terms, collineation is a geometric transformation that preserves the concept of parallelism and intersects lines. It maintains the relative arrangement of points, lines, and planes, ensuring that they remain collinear. Collineation is a crucial concept in projective geometry, providing a foundational understanding of geometric transformations.
Common Misspellings for COLLINEATION
- xollineation
- vollineation
- follineation
- dollineation
- cillineation
- ckllineation
- clllineation
- cpllineation
- c0llineation
- c9llineation
- coklineation
- coplineation
- coolineation
- colkineation
- colpineation
- coloineation
- colluneation
- colljneation
- collkneation
- colloneation
Etymology of COLLINEATION
The word "collineation" derives from the Latin term "collineare", which is a combination of the prefix "col-" (meaning "with" or "together") and the verb "lineare" (meaning "to make straight"). The Latin root implies the notion of aligning or making straight lines together. This word was later anglicized into "collineation", which is commonly used in mathematics and geometry to describe a transformation that preserves collinearity (the property of points lying on the same line).
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