Projections and Canonical Coordinates

An m-dimensional subspace within the original n-dimensional space can be represented as the intersection of (n - m) hyperplanes of dimension (n - 1), 

The most convenient way to plot a projection of an w-dimensional system in an m-dimensional linear projective subspace is to use multiple orthogonal projection, in which the directions of projection rays are parallel to the normal vectors (qi, q ,. .., qn-m) defining the projective subspace. Under this projection, a point is first projected in the direction of qi, then of q/, and so on. A convenient set of canonical coordinates describing the projective subspace is given by 

A special case of orthogonal projection in homogeneous space is the frequently used Janecke projection, where one reference component is neglected and the composition of the remaining species are renormalized. As a special case of multiple orthogonal projection, we define multiple Janecke projection, where the dimensionality is reduced directly to m. If components (m + 1) to (C - 1) are taken to be the reference components, the elements of the projection ray vectors (gi, qi, qc-m-1 are 

This projection is equivalent to neglecting components (m + 1) to (C - 1), and normalizing the remaining mole fractions. 

Another special case is the reactive projection. When chemical reactions are present, possible compositions are confined to a lower dimensional subspace. Consider a system involving C components and R independent reactions described as 

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