The Helmert transition segment is a special case of a fourth order spiral where the curvature rate of change is a quadratic function and the terms are dependent on the length L measured from the inflection point. The parameter value is defined as the deflection i.e. bearing angle &Theta.

The terms for the first half of the segment:
QuadraticTerm = L/√2
rest are 0.

Terms for the second half of the segment:
QuadraticTerm = -L/√2
LinearTerm = L/4
Constant = -1

SegmentStart is the bearing angle at start and Segment length is the bearing angle at the end of the segment.

The following diagram shows the generic classes and relationships used when applying this concept.