The normal strains (ex' and ey') and the shear strain (ex'y') vary smoothly with respect to the rotation angle q, in accordance with the transformation equations given above. There exist a couple of particular angles where the strains take on special values.
First, there exists an angle qp where the shear strain ex'y' vanishes. That angle is given by,
This angle defines the principal directions. The associated principal strains are given by,
The transformation to the principal directions with their principal strains can be illustrated as: