Global constraints in Hermite interpolation problems
Rida Farouki  1@  
1 : University of California , Davis

Hermite interpolation of discrete data --- points, tangents, curvatures, etc.
--- is a common approach to the construction of planar and spatial curves. The
imposition of global (integral) constraints is more difficult, and therefore
less commonly considered. We consider two types of global constraints that can
be exactly achieved by using Pythagorean--hodograph curves. The first is the
imposition of an exact arc length for the interpolant, and it is shown that
this can be achieved for both for planar and spatial $G^1$ end--point data by
use of quintic Pythagorean--hodograph curves \cite{farouki16,farouki19}. The
second constraint involves the construction of a rational adapted orthonormal
frame (comprising the curve tangent and two unit vectors spanning the curve
normal plane) that satisfies prescribed initial/final orientations. Since the
well--known rotation--minimizing frames are solutions of an initial--value
problem, they are incompatible with this constraint. Consequently, the \emph
{minimal--twist frame} is introduced --- an orthonormal frame with prescribed
initial and final instances, with the least possible value for the integral
of the tangent component of its angular velocity. The construction of rational
minimal twist frames on both open and smooth closed--loop Pythagorean--hodograph
curves is demonstrated \cite{farouki18,farouki20}.


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