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**Shapes Classification and Geometry**:
– Simple shapes like polygons (triangles, quadrilaterals, pentagons) and common shapes (points, lines, circles).
– 3D shapes including polyhedra, ellipsoids, and cones.
– Definition of 2D shapes by points, vertices, and closed chains of lines.
– Definition of 3D shapes by vertices, lines, and faces enclosed by lines.
– Examples of polygons (triangles, squares) and curved shapes (circles, spheres).

**Properties and Equivalence of Shapes**:
– Shapes remain invariant under translations, rotations, and scalings.
Shape defined as invariant geometrical information independent of size and placement.
– Mirror images may be considered different shapes.
– Shape changes with non-uniform scaling.
– Congruent and similar objects, their transformations, and size comparisons.

**Shape Analysis and Similarity Classes**:
– Statistical shape analysis methods like Procrustes analysis.
– Spectral shape analysis for non-rigid objects and homeomorphisms for non-rigid movements.
– Comparison of shapes by measuring deformations and distinguishing between rigid and non-rigid shapes.
– Similarity classes like similar triangles, complex numbers in classifying shapes, and affine transformations preserving shapes.

**Human Perception of Shapes**:
– Human vision processes shapes using different representations.
– Psychologists suggest breaking shapes into simple geometric forms for perception.
– Shape similarity requires multiple dimensions to account for natural variations.
– Shapes guide attention and influence perception, decomposable into features like segmentability and compactness.

**Additional Topics and References**:
– Glossary of shapes with metaphorical names and lists of shapes for examples and classifications.
– Solid geometry, shape factor, and the role of area and size in shape analysis.
– Region (mathematics) as a relevant field connected to shapes.
– Notable references like Kendall’s work on shape manifolds, Lester and Artzy’s classification methods, and other studies on object recognition and visual shape similarity.

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