Shear Transformations

Shear transformations occur in a variety of contexts, for example when converting ordinary type into italics.

We can think of this transformation as resulting from holding the base of these characters fixed and sliding the top by a certain vector parallel to the base line (the shear line) of the characters.

To describe such a shear transformation \(S\from \RNrSpc{n}\to \RNrSpc{n}\) mathematically, we adopt the following setup:

• Let \(\Vect{n}\) be a unit vector of \(\RNrSpc{n}\).
• Let \(H\) denote the hyperspace of \(\RNrSpc{n}\) perpendicular to \(\Vect{n}\). This is the hyperspace parallel to which we shear.
• Let \(\Vect{s} \DefEq S(\Vect{n}) - \Vect{n}\) denote the shear vector; i.e. \(\Vect{s}\) is the vector by which \(\Vect{n}\) must be slanted parallel to \(H\) so as to transform it into \(S(\Vect{n})\).

How to read the picture above:

1. The effect of \(S\) on \(\Vect{n}\) is \(S(\Vect{n}) = \Vect{n} + \Vect{s}\).
2. The effect of \(S\) on a stretched copy \(t\cdot \Vect{n}\) of \(\Vect{n}\) is \(S(t\cdot \Vect{n}) = t\cdot \Vect{n} + t\cdot \Vect{s}\).
3. The effect of \(S\) on an arbitrary vector \(\Vect{x}\) with \(\OrthoPrjctn{\Vect{n}}{\Vect{x}} = t\cdot \Vect{n}\) is \[S(\Vect{x}) = \Vect{x} + t\cdot \Vect{s} = \Vect{x} + (\DotPr{\Vect{x}}{\Vect{n}})\cdot \Vect{s}\]