Rational framing motions and spatial rational Pythagorean hodograph curves

Optimal interpolation with spatial rational Pythagorean hodograph curves

“…In [16,17,[20][21][22] this problem was studied for F(t) := A(t)iA * (t). In particular in [22] extensive results were obtained for generic cases.…”

“…As already explained in [21,22], it is no loss of generality to assume that the field F(t) is polynomial and without real polynomial factors. Indeed, multiplying any given rational field by the common denominator and factoring out any common factor of the numerator does not influence condition (1) and therefore the space R will remain the same.…”

“…The issue with formula (3) is due to the fact that not all λ(t) provide rational curves as integration may produce irrational functions (logarithms and arctangents). This lead to alternative approaches, namely the dual (envelope based) approach of [16,17,20] and the kinematics based approach of [21,22].…”

“…For this reason in two recent papers [21,22] a new approach was developed to describe a natural basis of the linear space of PH curves with given tangent field. In [21], the theory of rational PH curves was developed from so called framing rational motions.…”

“…For this reason in two recent papers [21,22] a new approach was developed to describe a natural basis of the linear space of PH curves with given tangent field. In [21], the theory of rational PH curves was developed from so called framing rational motions. Representing the spherical part of the motion by a quaternion polynomial A ∈ H[t] and fixing the PH curve's denominator its numerator was computed by solving a modestly sized and wellstructured system of linear equations.…”