In geometry, two sets of points are of the same shape precisely if one can be transformed to another by dilating (i.e., multiplying all distances by the same factor) and then, if necessary, rotating and translating. Dilation changes the size but not the shape; rotation and translation preserve both size and shape.
The shape of an object can be characterized by basic geometry such as points, line, curves, plane, and so on. 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.
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.
See also