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Encyclopedia

From Wikipedia, the free encyclopedia

The shape (from Old English ȝesceap, shap, etc., originally meaning created thing) of an object located in some space is the part of that space occupied by the object, as determined by its external boundary – abstracting from other properties such as colour, content, and material composition, as well as from the object's other spatial properties (position and orientation in space; size).

Mathematician and statistician David George Kendall defined shape this way:[1]

Shape is all the geometrical information that remains when location, scale and rotational effects are filtered out from an object.

Simple two-dimensional shapes can be described by basic geometry such as points, line, curves, plane, and so on. (A shape whose points belong all the same plane is called a plane figure.) Most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so arbitrary as to defy traditional mathematical description – in which case they may be analysed by differential geometry, or as fractals.

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Rigid shape definition

In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings. In other words, the shape of a set is all the geometrical information that is invariant to position (including rotation) and scale.

Having the same shape is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape.

Shapes of physical objects are not equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape depends on the size of the object and on changes in orientation/direction. However, a mirror image could be called a different shape. Shapes may change if the object is scaled non uniformly. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal directions. In other words, preserving axes of symmetry (if they exist) is not important for preserving shapes. Also, shape is determined by only the outer boundary of an object. For example, a solid ice cube and a second ice cube containing an inner cavity (air bubble) have the same shape.

Objects that cannot be transformed into each other by rigid transformations and mirroring are congruent. An object is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Objects that have the same shape or one has the same shape as the other's mirror image (or both if they are themselves symmetric) are called geometrically similar. Thus congruent objects are always geometrically similar, but geometrical similarity additionally allows uniform scaling.

Non-rigid shape definition

A more flexible definition of shape takes into consideration the fact that we often deal with deformable shapes in reality, (e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions). By allowing also isometric (or near-isometric) deformations like bending, the intrinsic geometry of the object will stay the same, while subparts might be located at very different positions in space. This definition uses the fact that, geodesics (curves measured along the surface of the object) stay the same, independent of the isometric embedding. This means that the distance from a finger to a toe of a person measured along the body is always the same, no matter how the body is posed.

Colloquial shape definition

Shape can also be more loosely defined as "the appearance of something, especially its outline". This definition is consistent with the above, in that the shape of a set does not depend on its position, size or orientation. However, it does not always imply an exact mathematical transformation. For example it is common to talk of star-shaped objects even though the number of points of the star is not defined.

Shape analysis

The modern definition of shape has arisen in the field of statistical shape analysis. In particular Procrustes analysis, which is a technique for analysing the statistical distributions of shapes. These techniques have been used to examine the alignments of random points.

See also

References

  1. ^ Kendall, D.G. (1984). "Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces". Bulletin of the London Mathematical Society 16: 81 -- 121. 

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Simple English

Simple English Wiktionary has the word meaning for:

A Shape is a geometric figure that can be mathematically defined. Shapes are defined in terms of the mathematical objects they are embedded in: Two-dimensional shapes are defined in terms of the plane, three-dimensional objects in terms of the Space.

Two shapes are said to be equal, if one can be transformed into the other using either rotation or movement.

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Ordinary shapes

2D shapes

These are two-dimensional shapes or flat plane geometry shapes. Their sides are made of straight or curved lines. They can have any number of sides. Plane figures made of lines are called polygons. Triangles and squares are examples of polygons.

3D shapes

These are three-dimensional shapes. Their sides are made of flat or curved surfaces.

Special shapes

Some shapes are special. These special shapes either cannot be made in real space, or they look unusual.


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