Geometry Introduction-b
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The document discusses different geometries including Euclidean, similarity, affine, projective, and topology. It lists key aspects of each like invariants, parameters, degrees of freedom, examples of transformations, and visual characteristics. The geometries range from 3 to 8 parameters and 2 to 8 degrees of freedom with different invariant properties like continuity, straightness, parallelism, angles, and distances.
Geometry Introduction-b
- 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 4 Congruence Distance 3 Geometry Euclidean Geometry
- 2. Congruence Geometry INVARIANTS Para. Transformation(Isometry) Continuity (Reflection, Rotation, Translation) Straightedge Parallelism Angle Distance 3
- 3. Congruence Geometry INVARIANTS Para. Transformation(Isometry) Continuity (Reflection, Rotation, Translation) Straightedge Parallelism Angle Distance 3
- 4. DOF 3 Congruence Geometry INV 5 Parameters Visuals Invariants Examples Transformation Continuity 3 Isometry Iso+metry Same+Measure Straightness 1 rotation Length Reflection 2 translation Parallelism Area Rotation 𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃 𝑡𝑥 Angle Angle Translation 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑡𝑦 0 0 1 Distance Glide Reflection Euclidean geometry describes shapes “as they are”
- 5. Similarity Geometry INVARIANTS Para. Continuity Transformation Straightedge Enlargement, Reduction Parallelism Angle 4 ?
- 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 Reduction 2 translation areas parallel Angle lines Enlargement ? It is the most general transformation that preserves shape..
- 7. Affine Geometry INVARIANTS Para. Continuity Transformation Straightedge Shear, Stretch Parallelism 6 ? ?
- 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 Parallelism 2 translation Line at infinity Affine ? parallel lines Transforma tion ratios of parallel ? lengths
- 9. Projective Geometry Transformation INVARIANTS Continuity Para. Perspective, Projective Transformation Straightedge 8 ? ? ?
- 11. 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, ? tangency 2 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
- 12. Topology INVARIANTS Continuity Transformation ? Topological Transformation ? A ? A B ? C B C