Affine transformations

Back to Transformations

An affine transformation is a function from a vector space to another such that it preserves affine combinations.

Theory

Let `V` and `W` be vector spaces over the reals. Let `f : V -> W`. `f` is called affine if it holds that

`\forall x, y in V: \forall t in R: f((1 - t) x + t y) = (1 - t) f(x) + t f(y)`

When `V` and `W` are finite dimensional, say of dimensions `n` and `m` respectively, we can identify them with `RR^n` and `RR^m` by choosing a basis in each. A linear transformation between `RR^n` and `RR^m` can be described with an `RR^{m xx n}` matrix. Similarly, an affine transformation can be described with functions of the form `f(x) = Ax + b`, where `A` is an `RR^{m xx n}` matrix, and `b` is a vector in `RR^m`.

Practice

Pastel implements the AffineTransformation class to model affine transformations in `RR^n` as well as functions for forming common affine transformations from various inputs, and inverting a transformation.

Learn more

Remark 1.7.5 - Page generated 16.06.2017 01:36.