Aplane shape orplane figure is constrained to lie on aplane, in contrast tosolid 3D shapes.Atwo-dimensional shape ortwo-dimensional figure (also:2D shape or2D figure) may lie on a more general curvedsurface (atwo-dimensional space).
Among the most common three-dimensional shapes arepolyhedra, which are shapes with flat faces;ellipsoids, which are egg-shaped or sphere-shaped objects;cylinders; andcones.
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 amanhole cover is adisk, because it is approximately the same geometric object as an actual geometric disk.
Ageometric shape consists of thegeometric information which remains whenlocation,scale,orientation andreflection are removed from the description of ageometric object.[1] That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape.
Many two-dimensional geometric shapes can be defined by a set ofpoints orvertices andlines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are calledpolygons and includetriangles,squares, andpentagons. Other shapes may be bounded bycurves such as thecircle or theellipse.
Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensionalfaces enclosed by those lines, as well as the resulting interior points. Such shapes are calledpolyhedrons and includecubes as well aspyramids such astetrahedrons. Other three-dimensional shapes may be bounded by curved surfaces, such as theellipsoid and thesphere.
A shape is said to beconvex if all of the points on a line segment between any two of its points are also part of the shape.
There are multiple ways to compare the shapes of two objects:
Congruence: Two objects arecongruent if one can be transformed into the other by a sequence of rotations, translations, and/or reflections.
Similarity: Two objects aresimilar if one can be transformed into the other by a uniform scaling, together with a sequence of rotations, translations, and/or reflections.
Isotopy: Two objects areisotopic if one can be transformed into the other by a sequence ofdeformations that do not tear the object or put holes in it.
Figures shown in the same color have the same shape as each other and are said to be similar.
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 basicgeometric objects such as aline, acurve, aplane, aplane figure (e.g.square orcircle), or a solid figure (e.g.cube orsphere). However, most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed bydifferential geometry, or asfractals.
In geometry, two subsets of aEuclidean space have the same shape if one can be transformed to the other by a combination oftranslations,rotations (together also calledrigid transformations), anduniform scalings. In other words, theshape of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is anequivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being anequivalence class of subsets of a Euclidean space having the same shape.
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 (seeProcrustes superimposition for details). However, amirror image could be called a different shape. For instance, a "b" and a "p" have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written. Even though they have the same size, there's no way to perfectly superimpose them by translating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if the object is scaled non-uniformly. For example, asphere becomes anellipsoid when scaled differently in the vertical and horizontal directions. In other words, preserving axes ofsymmetry (if they exist) is important for preserving shapes. Also, shape is determined by only the outer boundary of an object.
Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) arecongruent. An object is therefore congruent to itsmirror 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 calledgeometrically similar, whether or not they have the same size. Thus, objects that can be transformed into each other by rigid transformations, mirroring, and uniform scaling are similar. Similarity is preserved when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have different size.
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 byhomeomorphisms. Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, asquare and acircle are homeomorphic to each other, but asphere and adonut are not. An often-repeatedmathematical joke is that topologists cannot tell their coffee cup from their donut,[4] since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle.
A described shape has external lines that you can see and make up the shape. If you were putting your coordinates on a 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.
The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field ofstatistical 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 exampleSpectral shape analysis).
Allsimilar triangles have the same shape. These shapes can be classified usingcomplex numbersu,v,w for the vertices, in a method advanced by J.A. Lester[5] andRafael Artzy. For example, anequilateral triangle can be expressed by the complex numbers 0, 1,(1 + i√3)/2 representing its vertices. Lester and Artzy call the ratiotheshape of triangle(u,v,w). Then the shape of the equilateral triangle isFor anyaffine transformation of thecomplex plane, a triangle is transformed but does not change its shape. Hence shape is aninvariant ofaffine geometry.The shapep = S(u,v,w) depends on the order of the arguments of function S, butpermutations lead to related values. For instance, AlsoCombining these permutations gives Furthermore, These relations are "conversion rules" for shape of a triangle.
The shape of aquadrilateral is associated with two complex numbersp,q. If the quadrilateral has verticesu,v,w,x, thenp = S(u,v,w) andq = S(v,w,x). Artzy proves these propositions about quadrilateral shapes:
If a parallelogram has| argp | = | argq |, then it is arhombus.
Whenp = 1 + i andq = (1 + i)/2, then the quadrilateral issquare.
If andsgnr = sgn(Imp), then the quadrilateral is atrapezoid.
Apolygon has a shape defined byn − 2 complex numbers The polygon bounds aconvex set when all these shape components have imaginary components of the same sign.[6]
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) calledgeons.[9] Meanwhile, others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like theirsegmentability,compactness andspikiness.[10] When comparing shape similarity, however, at least 22 independent dimensions are needed to account for the way natural shapes vary.[7]
Experimental work suggests that an object'scontour, whether sharp-angled or curved, has a critical influence on people's attitude toward that object.[11] Studies ofprinted designs on packaging report that consumers show "a preference for rounded shapes," which can shift overall evaluations of labeled products.[12] Other studies similarly show thatlogo and packaging can shift judgments of attributes (for example, circular cues "softness," angular cues "hardness").[13]
There is also clear evidence that shapes guide humanattention.[14]
^abKendall, D.G. (1984). "Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces".Bulletin of the London Mathematical Society.16 (2):81–121.doi:10.1112/blms/16.2.81.
^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.
^Bar, Moshe; Neta, Maital (2006). "Humans prefer curved visual objects".Psychological Science.17 (8):645–648.doi:10.1111/j.1467-9280.2006.01759.x.PMID16913943.The type of contour a visual object possesses—whether the contour is sharp angled or curved—has a critical influence on people's attitude toward that object.
^Westerman, Stephen J.; Sutherland, Elaine J.; Gardner, Paul H. (2013). "The design of consumer packaging: Effects of manipulations of shape, orientation, and alignment of graphical forms on consumers' assessments".Food Quality and Preference.27 (1):8–17.doi:10.1016/j.foodqual.2012.05.007.Participants' evaluations indicated a preference for rounded shapes...
^Jiang, Yuwei; Gorn, Gerald J.; Galli, Maria; Chattopadhyay, Amitava (2016). "Does Your Company Have the Right Logo? How and Why Circular- and Angular-Logo Shapes Influence Brand Attribute Judgments".Journal of Consumer Research.42 (5):709–726.doi:10.1093/jcr/ucv049.circular- versus angular-logo shapes activate "softness" and "hardness" associations, respectively
