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51-558: [REDACTED] Look up figure in Wiktionary, the free dictionary. Figure may refer to: General [ edit ] A shape , drawing , depiction , or geometric configuration Figure (wood) , wood appearance Figure (music) , distinguished from musical motif Noise figure , in telecommunication Dance figure , an elementary dance pattern A person's figure, human physical appearance Figure–ground (perception) ,

102-541: A plane is a flat two- dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below: The Schläfli symbol { n } {\displaystyle \{n\}} represents

153-419: A regular n -gon . The regular monogon (or henagon) {1} and regular digon {2} can be considered degenerate regular polygons and exist nondegenerately in non-Euclidean spaces like a 2-sphere , 2-torus , or right circular cylinder . There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share

204-450: 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

255-410: 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

306-405: 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

357-599: 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

408-421: 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

459-533: A region D in R of a function f ( x , y ) , {\displaystyle f(x,y),} and is usually written as: The fundamental theorem of line integrals says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let φ : U ⊆ R 2 → R {\displaystyle \varphi :U\subseteq \mathbb {R} ^{2}\to \mathbb {R} } . Then with p , q

510-428: 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’

561-435: 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

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612-440: 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

663-454: A type of floating block (text, table, or graphic separate from the main text) Figure of speech , also called a rhetorical figure Christ figure , a type of character in typesetting, text figures and lining figures Accounting [ edit ] Figure, a synonym for number Significant figures in a decimal number Science [ edit ] Figure of the Earth ,

714-473: A vector A by itself is which gives the formula for the Euclidean length of the vector. In a rectangular coordinate system, the gradient is given by For some scalar field f  : U ⊆ R → R , the line integral along a piecewise smooth curve C ⊂ U is defined as where r : [a, b] → C is an arbitrary bijective parametrization of the curve C such that r ( a ) and r ( b ) give

765-476: 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

816-408: Is a one-dimensional manifold . In a Euclidean plane, it has the length 2π r and the area of its interior is where r {\displaystyle r} is the radius. There are an infinitude of other curved shapes in two dimensions, notably including the conic sections : the ellipse , the parabola , and the hyperbola . Another mathematical way of viewing two-dimensional space

867-403: 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

918-437: Is characterized as being the unique contractible 2-manifold . Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not simply connected . In graph theory , a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such

969-430: Is defined as: A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by ‖ A ‖ {\displaystyle \|\mathbf {A} \|} . In this viewpoint, the dot product of two Euclidean vectors A and B is defined by where θ is the angle between A and B . The dot product of

1020-582: Is different from Wikidata All article disambiguation pages All disambiguation pages 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

1071-611: 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

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1122-440: Is found in linear algebra , where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors . The dot product of two vectors A = [ A 1 , A 2 ] and B = [ B 1 , B 2 ]

1173-424: Is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the other axis. Another widely used coordinate system is the polar coordinate system , which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray. In Euclidean geometry ,

1224-480: 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

1275-477: 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

1326-406: 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

1377-1596: 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

1428-415: 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

1479-422: The position of each point . It is an affine space , which includes in particular the concept of parallel lines . It has also metrical properties induced by a distance , which allows to define circles , and angle measurement . A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane . The set R 2 {\displaystyle \mathbb {R} ^{2}} of

1530-620: 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

1581-404: 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

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1632-425: The discovery. Both authors used a single ( abscissa ) axis in their treatments, with the lengths of ordinates measured along lines not-necessarily-perpendicular to that axis. The concept of using a pair of fixed axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify

1683-437: The distinction between a visually perceived object and its surroundings Arts [ edit ] Figurine , a miniature statuette representation of a creature Action figure , a posable jointed solid plastic character figurine Figure painting , realistic representation, especially of the human form Figure drawing Model figure , a scale model of a creature Writing [ edit ] figure, in writing,

1734-462: The endpoints of C and a < b {\displaystyle a<b} . For a vector field F  : U ⊆ R → R , the line integral along a piecewise smooth curve C ⊂ U , in the direction of r , is defined as where · is the dot product and r : [a, b] → C is a bijective parametrization of the curve C such that r ( a ) and r ( b ) give the endpoints of C . A double integral refers to an integral within

1785-399: The endpoints of the curve γ. Let C be a positively oriented , piecewise smooth , simple closed curve in a plane , and let D be the region bounded by C . If L and M are functions of ( x , y ) defined on an open region containing D and have continuous partial derivatives there, then where the path of integration along C is counterclockwise . In topology , the plane

1836-617: The ideas contained in Descartes' work. Later, the plane was thought of as a field , where any two points could be multiplied and, except for 0, divided. This was known as the complex plane . The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot

1887-454: 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

1938-448: 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

1989-502: The ordered pairs of real numbers (the real coordinate plane ), equipped with the dot product , is often called the Euclidean plane or standard Euclidean plane , since every Euclidean plane is isomorphic to it. Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas , parallelism,

2040-407: 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

2091-427: The positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin . They are usually labeled x and y . Relative to these axes, the position of any point in two-dimensional space

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2142-542: The same unit of length . Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat , although Fermat also worked in three dimensions, and did not publish

2193-404: The same vertex arrangements of the convex regular polygons. In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols { n / m } for all m such that m < n /2 (strictly speaking { n / m } = { n /( n − m )}) and m and n are coprime . The hypersphere in 2 dimensions is a circle , sometimes called a 1-sphere ( S ) because it

2244-422: 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

2295-482: 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}}}

2346-409: The same term [REDACTED] This disambiguation page lists articles associated with the title Figure . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Figure&oldid=1241199454 " Category : Disambiguation pages Hidden categories: Short description

2397-454: 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

2448-403: 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

2499-900: The size and shape of the Earth in geodesy Sports [ edit ] Figure (horse) , a stallion who became the foundation sire of the Morgan horse breed Figure skating Compulsory figures Figure competition , a form of physique competition for women, related to bodybuilding Beyer Speed Figure , a statistic in Thoroughbred racing People [ edit ] Figure (musician) , stage name of American electronic musician Josh Gard Michael Figures (1947–1996), American politician Shomari Figures (born 1985/1986), American politician Thomas Figures (1944–2007), American attorney and judge Vivian Davis Figures (born 1957), American politician See also [ edit ] Figure 8 (disambiguation) Topics referred to by

2550-432: The sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called Cartesian coordinate system , a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates , which are the signed distances from the point to two fixed perpendicular directed lines, measured in

2601-461: The way natural shapes vary. There is also clear evidence that shapes guide human attention . Plane (geometry) In mathematics , a Euclidean plane is a Euclidean space of dimension two , denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E} ^{2}} . It is a geometric space in which two real numbers are required to determine

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