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46-644: Form is the shape , visual appearance , or configuration of an object. In a wider sense, the form is the way something happens. Form may also refer to: Shape A plane shape or plane figure is constrained to lie on a plane , in contrast to solid 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure ) may lie on a more general curved surface (a two-dimensional space ). Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles , quadrilaterals , pentagons , etc. Each of these

92-450: 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

138-410: 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

184-450: A member of an equivalence class formed by taking the set of all sets of k points in n dimensions, that is R and factoring out the set of all translations, rotations and scalings. A particular representation of shape is found by choosing a particular representation of the equivalence class. This will give a manifold of dimension kn -4. Procrustes is one method of doing this with particular statistical justification. Bookstein obtains

230-405: 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 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

276-599: 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 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 ). All similar triangles have

322-449: A reference orientation for the objects, which in the former technique is optimally determined, and in the latter one is arbitrarily selected. Scaling and translation are performed the same way by both techniques. When only two shapes are compared, GPA is equivalent to ordinary Procrustes analysis. The algorithm outline is the following: There are many ways of representing the shape of an object. The shape of an object can be considered as

368-455: A reference orientation. Fix the reference object and rotate the other around the origin, until you find an optimum angle of rotation θ {\displaystyle \theta \,\!} such that the sum of the squared distances ( SSD ) between the corresponding points is minimised (an example of least squares technique). A rotation by angle θ {\displaystyle \theta \,\!} gives where (u,v) are

414-421: 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

460-524: A representation of shape by fixing the position of two points called the bases line. One point will be fixed at the origin and the other at (1,0) the remaining points form the Bookstein coordinates. It is also common to consider shape and scale that is with translational and rotational components removed. Shape analysis is used in biological data to identify the variations of anatomical features characterised by landmark data, for example in considering

506-446: A right hand to a left hand. Thus, partial PS with reflection enabled preserves size but allows translation, rotation and reflection, while full PS with reflection enabled allows translation, rotation, scaling and reflection. Optimal translation and scaling are determined with much simpler operations (see below). Here we just consider objects made up from a finite number k of points in n dimensions. Often, these points are selected on

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552-428: 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: In this paper ‘shape’

598-435: A shape defined by n − 2 complex numbers S ( z j , z j + 1 , z j + 2 ) ,   j = 1 , . . . , n − 2. {\displaystyle S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.} The polygon bounds a convex set when all these shape components have imaginary components of the same sign. Human vision relies on

644-455: A shape is compared to another, or a set of shapes is compared to an arbitrarily selected reference shape, Procrustes analysis is sometimes further qualified as classical or ordinary , as opposed to generalized Procrustes analysis (GPA), which compares three or more shapes to an optimally determined "mean shape". To compare the shapes of two or more objects, the objects must be first optimally "superimposed". Procrustes superimposition (PS)

690-440: A triangle. 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: A polygon ( z 1 , z 2 , . . . z n ) {\displaystyle (z_{1},z_{2},...z_{n})} has

736-476: A wide range of shape representations. Some psychologists have theorized that humans mentally break down images into simple geometric shapes (e.g., cones and spheres) called geons . Meanwhile, others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their segmentability , compactness and spikiness . When comparing shape similarity, however, at least 22 independent dimensions are needed to account for

782-403: Is by homeomorphisms . Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated mathematical joke is that topologists cannot tell their coffee cup from their donut, since a sufficiently pliable donut could be reshaped to the form of

828-611: Is divided into smaller categories; triangles can be equilateral , isosceles , obtuse , acute , scalene , etc. while quadrilaterals can be rectangles , rhombi , trapezoids , squares , etc. Other common shapes are points , lines , planes , and conic sections such as ellipses , circles , and parabolas . Among the most common 3-dimensional shapes are polyhedra , which are shapes with flat faces; ellipsoids , which are egg-shaped or sphere-shaped objects; cylinders ; and cones . If an object falls into one of these categories exactly or even approximately, we can use it to describe

874-423: Is not a pure shape analysis as it is not only sensitive to shape differences, but also to size differences. Both full and partial PS will never manage to perfectly match two objects with different shape, such as a cube and a sphere, or a right hand and a left hand. In some cases, both full and partial PS may also include reflection . Reflection allows, for instance, a successful (possibly perfect) superimposition of

920-504: Is not performed (i.e. the size of the objects is preserved). Notice that, after full PS, the objects will exactly coincide if their shape is identical. For instance, with full PS two spheres with different radii will always coincide, because they have exactly the same shape. Conversely, with partial PS they will never coincide. This implies that, by the strict definition of the term shape in geometry , shape analysis should be performed using full PS. A statistical analysis based on partial PS

966-467: Is performed by optimally translating , rotating and uniformly scaling the objects. In other words, both the placement in space and the size of the objects are freely adjusted. The aim is to obtain a similar placement and size, by minimizing a measure of shape difference called the Procrustes distance between the objects. This is sometimes called full , as opposed to partial PS, in which scaling

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1012-480: 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

1058-477: 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 geometrically 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

1104-406: 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.’ 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

1150-1596: The complex plane , z ↦ a z + b , a ≠ 0 , {\displaystyle z\mapsto az+b,\quad a\neq 0,}   a triangle is transformed but does not change its shape. Hence shape is an invariant of affine geometry . The shape p = S( u , v , w ) depends on the order of the arguments of function S, but permutations lead to related values. For instance, 1 − p = 1 − u − w u − v = w − v u − v = v − w v − u = S ( v , u , w ) . {\displaystyle 1-p=1-{\frac {u-w}{u-v}}={\frac {w-v}{u-v}}={\frac {v-w}{v-u}}=S(v,u,w).} Also p − 1 = S ( u , w , v ) . {\displaystyle p^{-1}=S(u,w,v).} Combining these permutations gives S ( v , w , u ) = ( 1 − p ) − 1 . {\displaystyle S(v,w,u)=(1-p)^{-1}.} Furthermore, p ( 1 − p ) − 1 = S ( u , v , w ) S ( v , w , u ) = u − w v − w = S ( w , v , u ) . {\displaystyle p(1-p)^{-1}=S(u,v,w)S(v,w,u)={\frac {u-w}{v-w}}=S(w,v,u).} These relations are "conversion rules" for shape of

1196-415: 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

1242-620: The shape of triangle ( u , v , w ) . Then the shape of the equilateral triangle is 0 − 1 + i 3 2 0 − 1 = 1 + i 3 2 = cos ⁡ ( 60 ∘ ) + i sin ⁡ ( 60 ∘ ) = e i π / 3 . {\displaystyle {\frac {0-{\frac {1+i{\sqrt {3}}}{2}}}{0-1}}={\frac {1+i{\sqrt {3}}}{2}}=\cos(60^{\circ })+i\sin(60^{\circ })=e^{i\pi /3}.} For any affine transformation of

1288-404: 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. There are multiple ways to compare the shapes of two objects: 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

1334-406: The continuous surface of complex objects, such as a human bone, and in this case they are called landmark points . The shape of an object can be considered as a member of an equivalence class formed by removing the translational , rotational and uniform scaling components. For example, translational components can be removed from an object by translating the object so that the mean of all

1380-467: The coordinates of a rotated point. Taking the derivative of ( u 1 − x 1 ) 2 + ( v 1 − y 1 ) 2 + ⋯ {\displaystyle (u_{1}-x_{1})^{2}+(v_{1}-y_{1})^{2}+\cdots } with respect to θ {\displaystyle \theta } and solving for θ {\displaystyle \theta } when

1426-454: The derivative is zero gives When the object is three-dimensional, the optimum rotation is represented by a 3-by-3 rotation matrix R , rather than a simple angle, and in this case singular value decomposition can be used to find the optimum value for R (see the solution for the constrained orthogonal Procrustes problem , subject to det ( R ) = 1). The difference between the shape of two objects can be evaluated only after "superimposing"

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1472-581: The literature. Removing the rotational component is more complex, as a standard reference orientation is not always available. Consider two objects composed of the same number of points with scale and translation removed. Let the points of these be ( ( x 1 , y 1 ) , … ) {\displaystyle ((x_{1},y_{1}),\ldots )} , ( ( w 1 , z 1 ) , … ) {\displaystyle ((w_{1},z_{1}),\ldots )} . One of these objects can be used to provide

1518-537: The literature. We showed how to superimpose two shapes. The same method can be applied to superimpose a set of three or more shapes, as far as the above mentioned reference orientation is used for all of them. However, Generalized Procrustes analysis provides a better method to achieve this goal. GPA applies the Procrustes analysis method to optimally superimpose a set of objects, instead of superimposing them to an arbitrarily selected shape. Generalized and ordinary Procrustes analysis differ only in their determination of

1564-454: The naming convention of the Greek derived prefix with '-gon' suffix: Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon... See polygon 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

1610-448: 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. Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are congruent . An object

1656-399: The object's points (i.e. its centroid ) lies at the origin. Mathematically: take k {\displaystyle k} points in two dimensions, say The mean of these points is ( x ¯ , y ¯ ) {\displaystyle ({\bar {x}},{\bar {y}})} where Now translate these points so that their mean is translated to

1702-503: The origin ( x , y ) → ( x − x ¯ , y − y ¯ ) {\displaystyle (x,y)\to (x-{\bar {x}},y-{\bar {y}})} , giving the point ( x 1 − x ¯ , y 1 − y ¯ ) , … {\displaystyle (x_{1}-{\bar {x}},y_{1}-{\bar {y}}),\dots } . Likewise,

1748-407: 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 by differential geometry , or as fractals . Some common shapes include: Circle , Square , Triangle , Rectangle , Oval , Star (polygon) , Rhombus , Semicircle . Regular polygons starting at pentagon follow

1794-422: 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 geometric objects such as a line , a curve , a plane , a plane figure (e.g. square or circle ), or a solid figure (e.g. cube or sphere ). However, most shapes occurring in

1840-482: The same shape. These shapes can be classified using complex numbers u , v , w for the vertices, in a method advanced by J.A. Lester and Rafael Artzy . For example, an equilateral triangle can be expressed by the complex numbers 0, 1, (1 + i√3)/2 representing its vertices. Lester and Artzy call the ratio S ( u , v , w ) = u − w u − v {\displaystyle S(u,v,w)={\frac {u-w}{u-v}}}

1886-400: The scale component can be removed by scaling the object so that the root mean square distance ( RMSD ) from the points to the translated origin is 1. This RMSD is a statistical measure of the object's scale or size : The scale becomes 1 when the point coordinates are divided by the object's initial scale: Notice that other methods for defining and removing the scale are sometimes used in

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1932-434: The shape of jaw bones. One study by David George Kendall examined the triangles formed by standing stones to deduce if these were often arranged in straight lines. The shape of a triangle can be represented as a point on the sphere, and the distribution of all shapes can be thought of a distribution over the sphere. The sample distribution from the standing stones was compared with the theoretical distribution to show that

1978-454: 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. A geometric shape consists of the geometric information which remains when location , scale , orientation and reflection are removed from the description of a geometric object . That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in

2024-403: 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 Procrustes superimposition for details). However, a mirror image could be called a different shape. For instance, a " b " and

2070-409: The two objects by translating, scaling and optimally rotating them as explained above. The square root of the above mentioned SSD between corresponding points can be used as a statistical measure of this difference in shape: This measure is often called Procrustes distance . Notice that other more complex definitions of Procrustes distance, and other measures of "shape difference" are sometimes used in

2116-467: The way natural shapes vary. There is also clear evidence that shapes guide human attention . 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 ( Greek : Προκρούστης ) refers to a bandit from Greek mythology who made his victims fit his bed either by stretching their limbs or cutting them off. In mathematics: When

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