Shape

Encyclopedia

The

in space, size

, and other properties such as colour, content, and material composition.

Mathematician and statistician David George Kendall

writes:

Simple shapes can be described by basic geometry

objects such as a set of two or more points

, a line

, a curve

, a plane, a plane figure (e.g. square

or circle

), or a solid figure (e.g. cube

or sphere

). 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 analyzed by differential geometry, or as fractal

s.

, two subsets of a Euclidean space

have the same shape if one can be transformed to the other by a combination of translations

, rotation

s (together also called rigid transformation

s), and uniform scaling

s. In other words, the

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 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 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 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 can 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.

s (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.

. In particular Procrustes analysis

, which is a technique for analyzing the statistical distributions of shapes. These techniques have been used to examine the alignments of random points

.

**shape**of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientationOrientation (geometry)

In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it is in....

in space, size

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

, and other properties such as colour, content, and material composition.

Mathematician and statistician David George Kendall

David George Kendall

David George Kendall FRS was an English statistician, who spent much of his academic life in the University of Oxford and the University of Cambridge. He worked with M. S...

writes:

In this paper ‘shape’ is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale and rotational effects are filtered out from an object.’

Simple shapes can be described by basic geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

objects such as a set of two or more points

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

, a line

Line (geometry)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

, a curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

, a plane, a plane figure (e.g. square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

or circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

), or a solid figure (e.g. cube

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

or sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

). 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 analyzed by differential geometry, or as fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

s.

## Rigid shape definition

In geometryGeometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, two subsets of a Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

have the same shape if one can be transformed to the other by a combination of translations

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

, rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s (together also called rigid transformation

Rigid transformation

In mathematics, a rigid transformation or a Euclidean transformation is a transformation from a Euclidean space to itself that preserves distances between every pair of points...

s), and uniform scaling

Scaling (geometry)

In Euclidean geometry, uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...

s. In other words, the

*shape*of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes.Having the same shape is an equivalence relation

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

, 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 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 and on changes in orientation/direction. However, a mirror image

Mirror image

A mirror image is a reflected duplication of an object that appears identical but reversed. As an optical effect it results from reflection off of substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures...

could be called a different shape. Shapes may change if the object is scaled non uniformly. For example, a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

becomes an ellipsoid when scaled differently in the vertical and horizontal directions. In other words, preserving axes of symmetry

Symmetry

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 perfection...

(if they exist) is 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 can be transformed into each other by rigid transformations and mirroring are congruent

Congruence (geometry)

In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...

. An object is therefore congruent to its mirror image

Mirror image

A mirror image is a reflected duplication of an object that appears identical but reversed. As an optical effect it results from reflection off of substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures...

(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

Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

. 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 sub-parts might be located at very different positions in space. This definition uses the fact that, geodesicGeodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

s (curves measured along the surface of the object) stay the same, independent of the isometric

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

. 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.## Philosophical skepticism of definitions

In Plato's Meno, Socrates questions Meno as to the most accurate definition of a figure/shape. By showing the possibility of there being more than one definition, Socrates shows that a definition cannot wholly and accurately describe something and that there is no one absolute definition for anything including shape.## Shape analysis

The modern definition of shape has arisen in the field of statistical shape analysisStatistical shape analysis

Statistical shape analysis is a geometrical analysis from a set of shapes in which statistics are measured to describe geometrical properties from similar shapes or different groups, for instance, the difference between male and female Gorilla skull shapes, normal and pathological bone shapes, etc...

. In particular Procrustes analysis

Procrustes analysis

In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off.To compare the shape of...

, which is a technique for analyzing the statistical distributions of shapes. These techniques have been used to examine the alignments of random points

Alignments of random points

Alignments of random points, as shown by statistics, can be found when a large number of random points are marked on a bounded flat surface. This might be used to show that ley lines exist due to chance alone .One precise definition which expresses the generally accepted meaning of "alignment"...

.