Cox's and Piepel's component directions through the design space

When changing the proportion of a component in a mixture to determine its effect on a response, you must make offsetting changes in the other mixture components because the sum of the proportions must always be one. The changes in the component's effect you are assessing along with the offsetting changes in the other components can be considered as a direction through the experimental region.

There are two common trace directions along which the estimated responses are calculated: Cox's direction and Piepel's direction.

When the design is not constrained and the reference point lies at the centroid of the unconstrained experimental region, both Cox's directions and Piepel's directions are the axes of the simplex.
When the design is constrained, the default reference mixture point lies at the centroid of the constrained experimental region that is different than the centroid of the unconstrained experimental region. In this case, Cox's direction is defined in the original design space, whereas, Piepel's direction is defined in the L-pseudo-component space.