Define geometric transformation and coordinate transformation

Geometric Transformation:

A geometric transformation is any bijection of a set having some geometric structure to itself or another such set. Specifically, "A geometric transformation is a function whose domain and range are sets of points. Most often the domain and range of a geometric transformation are both R2 or both R3.

Example:

             within transformation geometry, the properties of an isosceles triangle are deduced from the fact that it is mapped to itself by a reflection about a certain line. This contrasts with the classical proofs by the criteria for congruence of triangles.[1]

Co-ordinate Transformation:

                           co-ordinate transformations are no intuitive enough in 2-D, and positively painful in 3-D. This page tackles them in the following order: (i) vectors in 2-D, (ii) tensors in 2-D, (iii) vectors in 3-D, (iv) tensors in 3-D, and finally (v) 4th rank tensor transforms.

A major aspect of coordinate transforms is the evaluation of the transformation matrix, especially in 3-D. This is touched on here, and discussed at length on the next page.

It is very important to recognize that all coordinate transforms on this page are rotations of the coordinate system while the object itself stays fixed. The "object" can be a vector such as force or velocity, or a tensor such as stress or strain in a component. Object rotations are discussed in later sections.

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