The shape (OE. sceap Eng. created thing) of an object located in some space refers to the part of space occupied by the object as determined by its external boundary — abstracting from other aspects the object may have such as its colour, content, or the substance of which it is composed, as well as from the object's position and orientation in space, and its size. English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States
Simple two-dimensional shapes can be described by basic geometry such as points, line, curves, plane, and so on. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object Shapes that occur in the physical world are often quite complex; they may be arbitrarily curved as studied by differential geometry, or fractal, as for plants or coastlines). Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole"
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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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Euclidean geometry, a translation is moving every point a constant distance in a specified direction A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Euclidean geometry, uniform scaling or Isotropic scaling is a Linear transformation that enlarges or diminishes objects the Scale factor 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. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X
Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on the size of the object nor on changes in orientation/direction. However, a mirror image could be called a different shape. "Mirror Image" is an episode of the Television series The Twilight Zone. Shape may change if the object is scaled non uniformly. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal direction. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an Ellipse. In other words, preserving axes of symmetry (if they exist) is important for preserving shapes. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or Also, shape is not necessary 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) do not necessarily have the same shape, even though the outer boundary is identical.
Objects that can be transformed into each other only by rigid transformations and mirroring are congruent. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i An object is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. "Mirror Image" is an episode of the Television series The Twilight Zone. 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. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry This definition uses the fact, that geodesics (curves measured along the surface of the object) stay the same, independent of the isometric embedding. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces For the Mechanical engineering and Architecture usage see Isometric projection. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group 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. An ant climbing a bendable plant will not notice how the wind moves it around, as only bending and no stretching is involved. It is true that when a body is bent, the wind moves around it, not through it.
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. This article describes shape analysis to analyze and process geometric shapes. Statistical shape analysis is a geometrical analysis from a set of Shapes in which Statistics are measured to describe geometrical properties from similar In particular Procrustes analysis, which is a technique for analysing the statistical distributions of shapes. In Statistics, Procrustes analysis is a form of Statistical shape analysis used to analyse the distribution of a set of Shapes The name Procrustes These techniques have been used to examine the alignments of random points. Statistics shows that if you put a large number of random points on a bounded flat surface you can find many alignments of random points.
See also
External links
- Answers for many questions related to shapes and sizes of common objects
- American artist Allan McCollum's project to create a unique "shape" for every individual on the planet
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