Shape

For SHAPE, see Supreme Headquarters Allied Powers Europe.

In geometry, two sets have the same shape if one can be transformed to another by a combination of translations, rotations and uniform scalings. In other words, the shape of a set is all the geometrical information that is invariant to location, scale and rotation. Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above.

Shape can also have a looser definition as the appearance of something, especially its outline ([[wikt:Shape|Wiktionary]]). Such a definition agrees with the above in that the shape does not depend on its position, size or orientation. However it does not always imply an exact mathematical transformation. For example it is common to talk of star-shaped objects even though the number of points of the star is undefined. Typically the shape of an object can be characterized by basic geometry such as points, line, curves, plane, and so on.

Objects which are geometrically similar either have the same shape or one has the same shape as the other's mirror image (or both if they are themselves symmetric). Congruent objects are similar and also have the same size. Thus one object has the same size and shape as either the other object or its mirror image.

green square, tilted, illustrated w/ cross & 1 diagonal

A square

For an object of greater than 2 dimensions, one can always reduce the dimensions of the shape by considering the shape of a cross-section or a projection.

An elliptical ring

The cross-section of a spherical object, for example, will be circular. More complex shapes would, however, generate various curvatures depending on the type of cross-section (e.g. horizontal, vertical). Because of the variation possible in taking cross-section, the orientation of the object is critical.

The shape does not depend on changes in orientation/direction. However, a mirror image could be called a different shape. Shape may change if the object is scaled differentially. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal axis. In other words, preserving axis of symmetry is important for preserving shapes.

Shape analysis

The modern definition of shape has arisen in the field of statistical shape analysis. In particular Procrustes analysis, which is a technique for analysing the statistical distributions of shapes. These techniques have been used to examine the alignments of random points.

Shape

Shape analysis

See also