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 $A_1\,B_1$ such that $A_1{B}_1\=kAB$, where $k$ is a fixed positive number, is called a similarity (or an equiform transformation), and the particular case where $k\=1$ is called an isometry (or a congruent transformation).

A similarity is said to be direct or opposite depending on whether $\mathrm{\Delta }\stackrel{\&conjugate0;}{\mathrm{ABC}}$ has or does not have the same sense as $\mathrm{\Delta }\stackrel{\&conjugate0;}{A_1{B}_1{C}_1}$ .

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 $P_1$ of S such that $\stackrel{\&conjugate0;}{P{P}_1}$ is equal and parallel to a given directed segment $\stackrel{\&conjugate0;}{AB}$ of space. There are no invariant points under a translation of non-zero vector $\stackrel{\&conjugate0;}{AB}$.

In rotation about an axis, each point P of S is carried into a point $P_1$ 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 $P_1$ of S such that $P{P}_1$ 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 $P_1$ of S such that $P{P}_1$ 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 $P_1$ of S such that $P{P}_1$ 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 $P_1$ of S collinear with P and with a fixed point O of space, such that  $\frac{\stackrel{\&conjugate0;}{{\mathrm{OP}}_1}}{\stackrel{\&conjugate0;}{\mathrm{OP}}}\&equals;k$, 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 $\stackrel{\&conjugate0;}{\mathrm{AB}}$, where $\stackrel{\&conjugate0;}{\mathrm{AB}}$ 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.

In general, to define a transformation without specifying the object to which the transformation is to be applied, one uses the "verb" form of the above transformations.

rotation --> rotate      

translation --> translate               

ScrewDisplacement --> ScrewDisplace
reflection --> reflect   

RotatoryReflection --> RotatoryReflect  

GlideReflection --> GlideReflect

homothety --> dilate     

homology --> StretchRotate

When defining transformations using the "verb" forms, one can do more things relating to the theory of transformations.

When we speak of the resultant of the two transformations $RS$ as their "product", we can make use of the analogy that exists between transformations and numbers.
Let R, S,...  denote transformations, and write $RS$ for the resultant of R and of S in that order.
Pushing the analogy further, let $R^p$ denote the p-fold application of R; for example, if R is a rotation through $\mathrm{\theta }$, $R^p$ is the rotation through $p\mathrm{\theta }$. A transformation R is said to be periodic if there is a positive integer p such that $R^p\&equals;1$ (the identity transformation), then its period is the smallest p in which this happens.
Let $R^{-1}$ denote the inverse of R. Then the general formula for the inverse of a product is easily seen to be ${\left(RST\right)}^{-1}\&equals;T^{-1}{S}^{-1}{R}^{-1}$.
Let x denote any figure to which a transformation is applied. If T transforms x into $x_1$ (so that $T^{-1}$ transforms $x_1$ into x), we write $x_1\&equals;x^T$. This notation is justified by the fact that ${\left(x^T\right)}^S\&equals;x^{TS}$.
If S transforms the pair of figures $\left[x\&comma;x^T\right]$ into $\left[x_1\&comma;x_1^{T_1}\right]$, we say that S transforms T into $T_1$, and write  $T_1\&equals;T^S$; for example, if T is a rotation about an axis l, then $T^S$ is the rotation through the same angle about the transformed axis $l^S$.

In Maple, by using the function inverse, one can compute the inverse of a given product of transformation. The function transprod converts a given transformation or product of transformations into a product of three "primitive" transformations (translation, rotation, and homothety), while the function transform is used to apply the "result" transformation to a specific object.

The following is an example of using transformations in the "verb" forms:

Define t1, which is a homothety with ratio 3, center of homology (0,0,0).

Define the plane oxy, and the directed segment AB.

Define t2, which is a glide-reflection with p as the plane of reflection and AB as the vector of translation.

Define t3 as a screw-displacement, with l3 as the rotational axis and AB as a vector of translation.

Compute q1, which is the product of t2^t1*t3.

Compute the inverse of q1.

Compute the product of q1*q2. One can quickly recognize that this is an identity transformation:

Simple check:

Therefore, the two objects are the same, as expected.