Shape
A shape is a
A plane shape or plane figure is constrained to lie on a
Classification of simple shapes
Some simple shapes can be put into broad categories. For instance,
Other common shapes are
.Among the most common 3-dimensional shapes are
If an object falls into one of these categories exactly or even approximately, we can use it to describe the shape of the object. Thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk.
In geometry
A geometric shape consists of the
Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles, squares, and pentagons. Other shapes may be bounded by curves such as the circle or the ellipse. Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional faces enclosed by those lines, as well as the resulting interior points. Such shapes are called polyhedrons and include cubes as well as pyramids such as tetrahedrons. Other three-dimensional shapes may be bounded by curved surfaces, such as the ellipsoid and the sphere.
A shape is said to be convex if all of the points on a line segment between any two of its points are also part of the shape.
Properties
There are several ways to compare the shapes of two objects:
- Congruence: Two objects are congruent if one can be transformed into the other by a sequence of rotations, translations, and/or reflections.
- Similarity: Two objects are similar if one can be transformed into the other by a uniform scaling, together with a sequence of rotations, translations, and/or reflections.
- deformationsthat do not tear the object or put holes in it.
Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the letters "b" and "d" are a reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, a hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have the same shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same.
Simple shapes can often be classified into basic
Some common shapes include:
Equivalence of shapes
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 of points is all the geometrical information that is invariant to translations, rotations, and size changes. 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.
Mathematician and statistician David George Kendall writes:[2]
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[3] and rotational effects are filtered out from an object.’
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 and placement in space of the object. For instance, a "d" and a "p" have the same shape, as they can be perfectly superimposed if the "d" is translated to the right by a given distance, rotated upside down and magnified by a given factor (see
Congruence and similarity
Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are congruent. An object is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes, and have the same size.
Objects that have the same shape or mirror image shapes are called
Homeomorphism
A more flexible definition of shape takes into consideration the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions.
One way of modeling non-rigid movements is by
A described shape has external lines that you can see and make up the shape. If you were putting you coordinates on and coordinate graph you could draw lines to show where you can see a shape, however not every time you put coordinates in a graph as such you can make a shape. This shape has a outline and boundary so you can see it and is not just regular dots on a regular paper.
Shape analysis
The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of statistical shape analysis. In particular, Procrustes analysis is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example Spectral shape analysis).
Similarity classes
All
The shape of a quadrilateral is associated with two complex numbers p, q. If the quadrilateral has vertices u, v, w, x, then p = S(u,v,w) and q = S(v,w,x). Artzy proves these propositions about quadrilateral shapes:
- If then the quadrilateral is a parallelogram.
- If a parallelogram has | arg p | = | arg q |, then it is a rhombus.
- When p = 1 + i and q = (1 + i)/2, then the quadrilateral is square.
- If and sgn r = sgn(Im p), then the quadrilateral is a trapezoid.
A polygon has a shape defined by n − 2 complex numbers The polygon bounds a convex set when all these shape components have imaginary components of the same sign.[6]
Human perception of shapes
Human vision relies on a wide range of shape representations.[7][8] Some psychologists have theorized that humans mentally break down images into simple geometric shapes (e.g., cones and spheres) called geons.[9] Others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their segmentability, compactness and spikiness.[10] When comparing shape similarity, however, at least 22 independent dimensions are needed to account for the way natural shapes vary. [7]
There is also clear evidence that shapes guide human attention.[11]
See also
- Area
- Glossary of shapes with metaphorical names
- Lists of shapes
- Shape factor
- Size
- Solid geometry
- Region (mathematics)
References
- ^ .
- .
- uniform scaling, as non-uniform scaling would change the shape of the object (e.g., it would turn a square into a rectangle).
- ISBN 978-0-387-94377-0.
- ^ J.A. Lester (1996) "Triangles I: Shapes", Aequationes Mathematicae 52:30–54
- ^ Rafael Artzy (1994) "Shapes of Polygons", Journal of Geometry 50(1–2):11–15
- ^ PMID 34061825.
- .
- ^ Marr, D., & Nishihara, H. (1978). Representation and recognition of the spatial organization of three-dimensional shapes. Proceedings of the Royal Society of London, 200, 269–294.
- PMID 32315405.
- PMID 26180505.