shape

shape of an object

, used for learning various shapes A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, shape excludes information about the object's position, size, orientation and chirality. A figure is a representation including both shape and size (as in, e.g., figure of the Earth). 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). ==In geometry== , triangle & circle , sphere & cube 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 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 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 multiple 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. 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. 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 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 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 the naming convention of the Greek derived prefix with '-gon' suffix: Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon... See polygon ===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 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 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. ===Similarity classes=== All similar triangles have the same shape. These shapes can be classified using complex numbers , , 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, representing its vertices. Lester and Artzy call the ratio S(u, v, w) = \frac{u - w}{u - v} the shape of triangle . Then the shape of the equilateral triangle is \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 the complex plane, z \mapsto a z + b,\quad a \ne 0,   a triangle is transformed but does not change its shape. Hence shape is an invariant of affine geometry. The shape depends on the order of the arguments of function S, but permutations lead to related values. For instance, 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). Combining these permutations gives S(v,w,u) = (1 - p)^{-1}. Furthermore, 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 a triangle. The shape of a quadrilateral is associated with two complex numbers , . If the quadrilateral has vertices , , , , then and . Artzy proves these propositions about quadrilateral shapes: If p = (1-q)^{-1}, then the quadrilateral is a parallelogram. If a parallelogram has , then it is a rhombus. When and , then the quadrilateral is square. If p = r(1-q^{-1}) and , then the quadrilateral is a trapezoid. A polygon (z_1, z_2,...z_n) has a shape defined by n − 2 complex numbers 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. ==See also== Area Glossary of shapes with metaphorical names Lists of shapes Shape factor Size Skew polygon Solid geometry Region (mathematics)

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