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# Geometry Introduction-b

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### Geometry Introduction-b

1. 1. Geometry INVARIANTS Parameter Topology Continuity Projective Geometry Spherical Geometry Straightedge 8 Affine Hyperbolic Elliptic Geometry Geometry Geometry Parallelism 6 Similarity Non-Euclidean Geometry Geometry Angle 4Congruence Distance 3Geometry Euclidean Geometry
2. 2. Congruence Geometry INVARIANTS Para.Transformation(Isometry) Continuity(Reflection, Rotation, Translation) Straightedge Parallelism Angle Distance 3
3. 3. Congruence Geometry INVARIANTS Para.Transformation(Isometry) Continuity(Reflection, Rotation, Translation) Straightedge Parallelism Angle Distance 3
4. 4. DOF 3 Congruence Geometry INV 5Parameters Visuals Invariants Examples Transformation Continuity 3 Isometry Iso+metry Same+Measure Straightness 1 rotation Length Reflection2 translation Parallelism Area Rotation𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃 𝑡𝑥 Angle Angle Translation𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑡𝑦 0 0 1 Distance Glide Reflection Euclidean geometry describes shapes “as they are”
5. 5. Similarity Geometry INVARIANTS Para. ContinuityTransformation StraightedgeEnlargement, Reduction Parallelism Angle 4 ?
6. 6. DOF 4 Similar Geometry INV 4 Parameters Visuals Invariants Examples Transformation Continuity 4 ratios of length 1 scale Straightness Isometry+Scale angle 1 rotation Parallelism ratios of Reduction2 translation areas parallel Angle lines Enlargement ? It is the most general transformation that preserves shape..
7. 7. Affine Geometry INVARIANTS Para. ContinuityTransformation StraightedgeShear, Stretch Parallelism 6 ? ?
8. 8. DOF 6 Affine Geometry INV 3 Parameters Visuals Invariants Examples Transformation Continuity 6 Ratio of area Straightness Linear 2 scale Midpoint Transforma 2 rotation Centroid tion Parallelism2 translation Line at infinity Affine ? parallel lines Transforma tion ratios of parallel ? lengths
9. 9. Projective GeometryTransformation INVARIANTS Continuity Para.Perspective, Projective Transformation Straightedge 8 ? ? ?
10. 10. DOF 8 Projective Geometry INV 2 Parameters Visuals Invariants Examples Transformation Continuity 8 Concurrency Projective Transformation 2 scale Collinearity Straightness Incidence Perspectivity Order of contact 2 rotation intersection, ? tangency2 translation inflection ? Cross ratio 2 line at conic section infinity ? Projective geometry describes objects “as they appear” Line at infinity becomes finite, allows to observe vanishing points
11. 11. Topology INVARIANTS ContinuityTransformation ?Topological Transformation ? A ? A B ? C B C