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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 location, scale, orientation and reflection.[1]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 non-Euclidean two-dimensional space).

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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 is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares, etc.

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 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.[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 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.

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.

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 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.

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 (see Procrustes superimposition for details). However, a mirror 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, 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 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.

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 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,[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 you coordinates on and 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 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).

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:

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) called geons.[9] Others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their segmentability, compactness and spikiness.[10] When comparing shape similarity, however, at least 22 independent dimensions are needed to account for the way natural shapes vary. [7]

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Definition of insideness: A point is considered to lie inside a Shape if and only if: it lies completely inside theShape boundary or it lies exactly on the Shape boundary and the space immediately adjacent to the point in the increasing X direction is entirely inside the boundary or it lies exactly on a horizontal boundary segment and the space immediately adjacent to the point in the increasing Y direction is inside the boundary. The contains and intersects methods consider the interior of a Shape to be the area it encloses as if it were filled. This means that these methods consider unclosed shapes to be implicitly closed for the purpose of determining if a shape contains or intersects a rectangle or if a shape contains a point.Since:1.2See Also:PathIterator, AffineTransform, FlatteningPathIterator, GeneralPathMethod SummaryAll Methods Instance Methods Abstract Methods Modifier and TypeMethod and Descriptionbooleancontains(double x, double y)Tests if the specified coordinates are inside the boundary of the Shape, as described by the definition of insideness.booleancontains(double x, double y, double w, double h)Tests if the interior of the Shape entirely contains the specified rectangular area.booleancontains(Point2D p)Tests if a specified Point2D is inside the boundary of the Shape, as described by the definition of insideness.booleancontains(Rectangle2D r)Tests if the interior of the Shape entirely contains the specified Rectangle2D.RectanglegetBounds()Returns an integer Rectangle that completely encloses the Shape.Rectangle2DgetBounds2D()Returns a high precision and more accurate bounding box of the Shape than the getBounds method.PathIteratorgetPathIterator(AffineTransform at)Returns an iterator object that iterates along the Shape boundary and provides access to the geometry of the Shape outline.PathIteratorgetPathIterator(AffineTransform at, double flatness)Returns an iterator object that iterates along the Shape boundary and provides access to a flattened view of the Shape outline geometry.booleanintersects(double x, double y, double w, double h)Tests if the interior of the Shape intersects the interior of a specified rectangular area.booleanintersects(Rectangle2D r)Tests if the interior of the Shape intersects the interior of a specified Rectangle2D.Method DetailgetBoundsRectangle getBounds()Returns an integer Rectangle that completely encloses the Shape. Note that there is no guarantee that the returned Rectangle is the smallest bounding box that encloses the Shape, only that the Shape lies entirely within the indicated Rectangle. The returned Rectangle might also fail to completely enclose the Shape if the Shape overflows the limited range of the integer data type. The getBounds2D method generally returns a tighter bounding box due to its greater flexibility in representation. Note that the definition of insideness can lead to situations where points on the defining outline of the shape may not be considered contained in the returned bounds object, but only in cases where those points are also not considered contained in the original shape. 2351a5e196