A mapping of a set A onto a set B in which distinct elements of A have distinct images in B is called a transformation (or a one-to-one mapping) of A onto B.
A point transformation of the unextended space onto itself which carries each pair of points A,B into a pair such that , where is a fixed positive number, is called a similarity (or an equiform transformation), and the particular case where is called an isometry (or a congruent transformation).
A similarity is said to be direct or opposite depending on whether has or does not have the same sense as .
In Maple, the following isometries and similarities are implemented and are applicable to polyhedra.
Direct isometries: rotation, translation, ScrewDisplacement
Opposite isometries: reflection, RotatoryReflection, GlideReflection
Non-isometric similarities: homothety, homology
In translation, the set S of all points of unextended space is mapped onto itself by carrying each point P of S into a point of S such that is equal and parallel to a given directed segment of space. There are no invariant points under a translation of non-zero vector .
In rotation about an axis, each point P of S is carried into a point of S by rotating P about a fixed line in space through a given angle. The fixed line is called the axis of rotation, and the points of the axis are the invariant points of the rotations.
In reflection in a point, each point P of S is carried into the point of S such that is bisected by a fixed point O of space. The fixed point O is the only invariant point of the transformation.
In reflection in a line, each point P of S is carried into the point of S such that is perpendicular bisected by a fixed line l of space.
In reflection in a plane, each point P of S is carried into the point of S such that is perpendicularly bisected by a fixed plane p of space, and the points of p are the invariant points of the transformation.
In homothety, each point P of S is carried into the point of S collinear with P and with a fixed point O of space, such that , where k is a nonzero real number. If k <> 1, the point O is the only invariant point of the transformation.
A ScrewDisplacement is the product of a rotation and a translation along the axis of rotation.
A GlideReflection is the product of a reflection in a plane and a translation of vector , where lies in the plane.
A RotatoryReflection is the product of a reflection in a plane and a rotation about a fixed axis perpendicular to the plane.
A homology is the product of a homothety and a rotation about an axis passing through the center of the homothety.