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| {{two other uses|describing the shape of an object|common shapes|list of geometric shapes}}
| | If you've gained a limited pounds in recent years, we may be wondering, "How do I lose weight?" And, we probably like to do thus fast and conveniently. It's equally a popular (plus normally correct) perception which if you would like to get rid of fat, you do thus gradually plus carefully inside order to be healthy about it. There is a way to lose weight, though, where you are able to do it rapidly plus be healthy regarding it -- plus have it be permanent also.<br><br>Whenever I'm striving to lose weight, I eat my last meal at 5:30pm so, by the time 6pm rolls around, I've completed eating for the day. Stopping eating at 6pm gives your body time to burn off the calories you've consumed during the day. But, if you eat following 6pm, almost all of those calories can not be burned off plus may be turned into fat while we sleep. If you stop eating at 6pm, you'll even find we can eat somewhat more during the day plus still lose weight fast.<br><br>One last question: where does fat disappear first and last? If you lose weight by doing exercises, you lose fat initially inside the trunk (shoulders, back and abdomen) plus upper limbs (arms especially). Next finally the buttocks and thighs.<br><br>Many persons join gyms because of the structure plus ambiance that a gym delivers. There are expensive machines, good classes, and trainers accessible at gyms that could help you remain on track. But if you are just going to the gym to employ the equipment, why not cut costs and make a gym at home? You do not have to have a treadmill and fat machines. There are many aerobic exercises that you can do at house, plus you are able to constantly purchase dumbbells to utilize. Buy some good strolling boots or a jump rope to take full benefit of the benefits of these exercises and lose pounds.<br><br>Eat more usually. This is possibly the number one tip of the "7 ways [http://safedietplansforwomen.com/how-to-lose-weight-fast how to lose weight fast for women] and easy". By eating little plus usually throughout the day, you will naturally accelerate your metabolism, meaning you'll burn calories more effectively. Of course, it doesn't indicate eat a 3-course meal, 6 times a day?snacks including raisins, an apple, celery or carrot sticks are wise choices plus because you may be eating continuously, you're less probably to stray towards foods you need to be limiting.<br><br>Weight Training: Weight training exercises help you improve we muscles which results in toned and sculpted body. It moreover helps in dealing with particular issue fat regions of the body. Strength training workout may come to the rescue whenever you need to target a specific body piece.<br><br>It is something to get rid of weight swiftly and effectively plus another thing to do it quick and unsuccessfully. The majority of individuals, who wish rapid results create a huge calorie deficit, by cutting their calories. Although this system will appear to be functioning fairly well, at the beginning, but it usually backfires. It is better to cut your calories by 20% or thus plus then burn a great deal of calories with exercise. |
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| [[File:Congruent_non-congruent_triangles.svg|thumb|370px|An example of the different definitions of '''shape'''. The two triangles on the left are congruent, while the third is [[Similarity (geometry)|similar]] to them. The last triangle is neither similar nor congruent to any of the others, but it is homeomorphic.]]
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| The term '''shape''' is commonly used to refer to the geometric properties of an object or its external boundary (outline, external surface), as opposed to other properties such as color, texture, material composition. There are several ways to compare the shape of two objects:
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| * [[Congruence (geometry)|Congruence]]: Two objects are '''congruent''' if one can be transformed into the other by a sequence of rotations, translations, and/or reflections.
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| * [[Similarity (geometry)|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.
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| * [[Homotopy#Isotopy|Isotopy]]: Two objects are '''isotopic''' if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it.
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| 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, an 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.
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| Simple shapes can often be classified into basic [[geometry|geometric]] objects such as a [[Point (geometry)|point]], a [[line (geometry)|line]], a [[curve]], a [[plane (geometry)|plane]], a [[plane figure]] (e.g. [[square (geometry)|square]] or [[circle]]), or a solid figure (e.g. [[cube]] or [[sphere]]). However, 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.
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| == Rigid shape definition ==
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| In geometry, two subsets of a [[Euclidean space]] have the same shape if one can be transformed to the other by a combination of [[translation (geometry)|translations]], [[rotation]]s (together also called [[rigid transformation]]s), and [[Scaling (geometry)|uniform scaling]]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]], 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.
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| Mathematician and statistician [[David George Kendall]] writes:<ref>{{cite journal|
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| doi = 10.1112/blms/16.2.81|
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| author = Kendall, D.G.|
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| title = Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces|
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| journal = Bulletin of the London Mathematical Society|
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| year = 1984|
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| volume = 16|
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| issue = 2|
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| pages = 81–121}}</ref>
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| <blockquote>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<ref>Here, scale means only [[uniform scaling]], as non-uniform scaling would change the shape of the object (e.g., it would turn a square into a rectangle).</ref> and rotational effects are filtered out from an object.’</blockquote> | |
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| 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 "'''<small>d</small>'''" and a "'''<big>p</big>'''" have the same shape, as they can be perfectly superimposed if the "'''<small>d</small>'''" 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 "'''<big>b</big>'''" and a "'''<big>p</big>'''" 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, 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.{{Citation needed|date=March 2011}}
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| ===Congruence and similarity===
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| {{Main|Congruence (geometry)|Similarity (geometry)}}
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| Objects that can be transformed into each other by rigid transformations and mirroring are [[Congruence (geometry)|congruent]]. An object is therefore congruent to its [[mirror image]] (even if it is not symmetric), but not to a scaled version.
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| Objects that have the same shape or one has the same shape as the other's mirror image are called [[Similarity (geometry)|geometrically similar]].
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| 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.
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| == Homeomorphism ==
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| {{Main|Homeomorphism}}
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| 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.
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| One way of modeling non-rigid movements is by [[homeomorphism]]s. Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a [[square (geometry)|square]] and a [[circle]] are homeomorphic to each other, but a [[sphere]] and a [[torus|donut]] are not. An often-repeated [[mathematical joke]] is that topologists can't tell their coffee cup from their donut,<ref>{{cite book|title=Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems|first1=John H.|last1=Hubbard|first2=Beverly H.|last2=West|publisher=Springer|series=Texts in Applied Mathematics|volume=18|year=1995|isbn=978-0-387-94377-0|page=204|url=http://books.google.com/books?id=SHBj2oaSALoC&pg=PA204&dq=%22coffee+cup%22+topologist+joke#v=onepage&q=%22coffee%20cup%22%20topologist%20joke&f=false}}</ref> 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.
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| == Classification of simple shapes ==
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| {{Main|Lists of shapes}}
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| [[File:Polygon types.svg|thumb|right|300px|A variety of [[polygon|polygonal]] shapes.]]
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| For simple shapes, there are other classifications than those mentioned above. For instance, [[polygon]]s are classified according to their number of edges as [[triangle]]s, [[quadrilateral]]s, [[pentagon]]s, etc. Each of these is divided into smaller categories; triangles can be [[equilateral]], [[isoceles]], [[obtuse triangle|obtuse]], [[Triangle#By internal angles|acute]], [[Triangle|scalene]], etc. while quadrilaterals can be [[rectangle]]s, [[rhombi]], [[trapezoids]], [[squares]], etc.
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| Other common shapes are [[Point (geometry)|point]]s, [[line (geometry)|line]]s, [[plane (geometry)|plane]]s, and [[conic sections]] such as [[ellipse]]s, [[circle]]s, and [[parabola]]s.
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| Among the most common 3-dimensional shapes are [[polyhedra]], which are shapes with flat faces; [[ellipsoid]]s, which are egg-shaped or sphere-shaped objects; [[cylinder (geometry)|cylinder]]s; and [[cone]]s.
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| 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 circle, because it is approximately the same geometric object as an actual geometric circle.
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| == Shape analysis ==
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| {{main|Statistical shape analysis}}
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| The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of [[statistical shape analysis]]. In particular [[Procrustes analysis]], which 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]]).
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| ==Similarity classes==
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| All [[similar triangles]] have the same shape. These shapes can be classified using [[complex number]]s in a method advanced by J.A. Lester<ref>J.A. Lester (1996) "Triangles I: Shapes", ''Aequationes Mathematicae'' 52:30–54</ref> and [[Rafael Artzy]]. For example, an [[equilateral triangle]] can be expressed by complex numbers 0, 1, (1 + i √3)/2. Lester and Artzy call the ratio
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| :S(''u,v,w'') = (''u'' −''w'')/(''u'' − ''v'') the '''shape''' of triangle (''u, v, w''). Then the shape of the equilateral triangle is
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| :(0–(1+ √3)/2)/(0–1) = ( 1 + i √3)/2 = cos(60°) + i sin(60°) = exp(i π/3).
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| For any [[affine transformation]] of the Gaussian plane, ''z'' mapping to ''a z + b, a'' ≠ 0, a triangle is transformed but does not change its shape. Hence shape is an [[invariant (mathematics)|invariant]] of [[affine geometry]].
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| The shape ''p'' = S(''u,v,w'') depends on the order of the arguments of function S, but [[permutation]]s lead to related values. For instance,
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| :<math>1 - p = 1 - (u-w)/(u-v) = (w-v)/(u-v) = (v-w)/(v-u) = S(v,u,w).</math> Also <math>p^{-1} = S(u,w,v).</math>
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| Combining these permutations gives <math>S(v,w,u) = (1 - p)^{-1}.</math> Furthermore,
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| :<math>p(1-p)^{-1} = S(u,v,w)S(v,w,u)=(u-w)/(v-w)=S(w,v,u). </math> These relations are "conversion rules" for shape of a triangle.
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| 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 <math> p=(1-q)^{-1},</math> then the quadrilateral is a [[parallelogram]].
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| #If a parallelogram has |arg ''p''| = |arg ''q''|, then it is a [[rhombus]].
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| #When ''p'' = 1 + i and ''q'' = (1 + i)/2, then the quadrilateral is [[square]].
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| #If <math>p = r(1-q^{-1})</math> and sgn ''r'' = sgn(Im ''p''), then the quadrilateral is a [[trapezoid]].
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| A [[polygon]] <math> (z_1, z_2,...z_n)</math> has a shape defined by ''n'' – 2 complex numbers <math>S(z_j,z_{j+1},z_{j+2}), \ j=1,...,n-2.</math> The polygon bounds a [[convex set]] when all these shape components have imaginary components of the same sign.<ref>[[Rafael Artzy]] (1994) "Shapes of Polygons", ''Journal of Geometry'' 50(1–2):11–15</ref>
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| == See also ==
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| * [[Solid geometry]]
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| * [[Glossary of shapes with metaphorical names]]
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| * [[List of geometric shapes]]
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| == References ==
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| {{Reflist}}
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| == External links ==
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| {{wiktionary}}
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| [[Category:Elementary geometry]]
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| [[Category:Geometric shapes| Shape]]
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| [[Category:Morphology]]
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| [[Category:Structure]]
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If you've gained a limited pounds in recent years, we may be wondering, "How do I lose weight?" And, we probably like to do thus fast and conveniently. It's equally a popular (plus normally correct) perception which if you would like to get rid of fat, you do thus gradually plus carefully inside order to be healthy about it. There is a way to lose weight, though, where you are able to do it rapidly plus be healthy regarding it -- plus have it be permanent also.
Whenever I'm striving to lose weight, I eat my last meal at 5:30pm so, by the time 6pm rolls around, I've completed eating for the day. Stopping eating at 6pm gives your body time to burn off the calories you've consumed during the day. But, if you eat following 6pm, almost all of those calories can not be burned off plus may be turned into fat while we sleep. If you stop eating at 6pm, you'll even find we can eat somewhat more during the day plus still lose weight fast.
One last question: where does fat disappear first and last? If you lose weight by doing exercises, you lose fat initially inside the trunk (shoulders, back and abdomen) plus upper limbs (arms especially). Next finally the buttocks and thighs.
Many persons join gyms because of the structure plus ambiance that a gym delivers. There are expensive machines, good classes, and trainers accessible at gyms that could help you remain on track. But if you are just going to the gym to employ the equipment, why not cut costs and make a gym at home? You do not have to have a treadmill and fat machines. There are many aerobic exercises that you can do at house, plus you are able to constantly purchase dumbbells to utilize. Buy some good strolling boots or a jump rope to take full benefit of the benefits of these exercises and lose pounds.
Eat more usually. This is possibly the number one tip of the "7 ways how to lose weight fast for women and easy". By eating little plus usually throughout the day, you will naturally accelerate your metabolism, meaning you'll burn calories more effectively. Of course, it doesn't indicate eat a 3-course meal, 6 times a day?snacks including raisins, an apple, celery or carrot sticks are wise choices plus because you may be eating continuously, you're less probably to stray towards foods you need to be limiting.
Weight Training: Weight training exercises help you improve we muscles which results in toned and sculpted body. It moreover helps in dealing with particular issue fat regions of the body. Strength training workout may come to the rescue whenever you need to target a specific body piece.
It is something to get rid of weight swiftly and effectively plus another thing to do it quick and unsuccessfully. The majority of individuals, who wish rapid results create a huge calorie deficit, by cutting their calories. Although this system will appear to be functioning fairly well, at the beginning, but it usually backfires. It is better to cut your calories by 20% or thus plus then burn a great deal of calories with exercise.