KnowraAffine geometryLinked fromLinked fromThe 17 pages that link to Affine geometry, each with the reason it gives.All 17Broader topic 1Related 7Narrower topic 5Compared with 4Projective geometryCompared with: Affine geometry preserves parallelism, which projective transformations need not preserve.Affine spaceNarrower topic: It studies geometric structure using the operations available in affine spaces.GeometryRelated: It preserves alignment and parallelism while allowing lengths and angles to change.Line segmentRelated: Segments remain straight and their internal ratios are preserved under affine maps.Distance geometryCompared with: Affine transformations can change distances, so affine structure is not the focus here.Complete quadrilateralCompared with: In an affine plane, parallel defining lines lack finite intersections, unlike in projective completion.Projective dualityCompared with: Projective duality relies on projective incidence and does not preserve affine notions such as parallelism.Vector geometryRelated: It retains vector-like affine combinations while setting aside distances and angles.Affine subspaceNarrower topic: Affine subspaces are its basic flat figures, independent of distance and angle.Menelaus's theoremRelated: Collinearity and ratios along a line are preserved by affine transformations.Erlangen programBroader topic: Its affine transformations sit between Euclidean and projective geometry in what they preserve.BetweennessRelated: Affine structure supports a natural notion of a point lying between two others on a line.Line–line intersectionNarrower topic: It preserves the incidence and parallelism rules governing planar line intersections.Butterfly theoremRelated: Midpoints are affine data, making affine transformations useful in reformulating the conclusion.Midpoint theoremNarrower topic: The theorem depends on parallelism and ratios, not on lengths or angles alone.Droz-Farny line theoremNarrower topic: The midpoint conclusion is affine, although perpendicularity supplies the Euclidean hypothesis.Pasch's theoremRelated: The theorem's crossing structure can be expressed using affine and order concepts.
KnowraAffine geometryLinked fromLinked fromThe 17 pages that link to Affine geometry, each with the reason it gives.All 17Broader topic 1Related 7Narrower topic 5Compared with 4Projective geometryCompared with: Affine geometry preserves parallelism, which projective transformations need not preserve.Affine spaceNarrower topic: It studies geometric structure using the operations available in affine spaces.GeometryRelated: It preserves alignment and parallelism while allowing lengths and angles to change.Line segmentRelated: Segments remain straight and their internal ratios are preserved under affine maps.Distance geometryCompared with: Affine transformations can change distances, so affine structure is not the focus here.Complete quadrilateralCompared with: In an affine plane, parallel defining lines lack finite intersections, unlike in projective completion.Projective dualityCompared with: Projective duality relies on projective incidence and does not preserve affine notions such as parallelism.Vector geometryRelated: It retains vector-like affine combinations while setting aside distances and angles.Affine subspaceNarrower topic: Affine subspaces are its basic flat figures, independent of distance and angle.Menelaus's theoremRelated: Collinearity and ratios along a line are preserved by affine transformations.Erlangen programBroader topic: Its affine transformations sit between Euclidean and projective geometry in what they preserve.BetweennessRelated: Affine structure supports a natural notion of a point lying between two others on a line.Line–line intersectionNarrower topic: It preserves the incidence and parallelism rules governing planar line intersections.Butterfly theoremRelated: Midpoints are affine data, making affine transformations useful in reformulating the conclusion.Midpoint theoremNarrower topic: The theorem depends on parallelism and ratios, not on lengths or angles alone.Droz-Farny line theoremNarrower topic: The midpoint conclusion is affine, although perpendicularity supplies the Euclidean hypothesis.Pasch's theoremRelated: The theorem's crossing structure can be expressed using affine and order concepts.