Last update: July 17, 2002

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References for Computational Geometry on Hodography:

• Rida T. Farouki et al. :
  • Structural invariance of spatial Pythagorean hodographs, 2002.
• Hyeong In Choi et al.,
  • Rational Parametrization of Canal Surface by 4 Dimensional Minkowski Pythagorean Hodograph Curves, 2000.
  • Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves, 1999.
• Hwan Pyo Moon:
  • Minkowski Pythagorean Hodographs, 1999.

BibTeX references

The structural invariance of the four-polynomial characterization for three-dimensional Pythagorean hodographs introduced by Dietz et al. (1993), under arbitrary spatial rotations, is demonstrated. The proof relies on a factored-quaternion representation for Pythagorean hodographs in three-dimensional Euclidean space¯¯a particular instance of the "PH representation map" proposed by Choi et al. (2002)¯¯and the unit quaternion description of spatial rotations. This approach furnishes a remarkably simple derivation for the polynomials , u(t), v(t), p(t), q(t), that specify the canonical form of a rotated Pythagorean hodograph, in terms of the original polynomials u(t), v(t), p(t), q(t) and the angle theta and axis n of the spatial rotation. The preservation of the canonical form of PH space curves under arbitrary spatial rotations is essential to their incorporation into computer-aided design and manufacturing applications, such as the contour machining of free-form surfaces using a ball-end mill and real-time PH curve CNC interpolators.

Author Keywords: Pythagorean-hodograph curves; Spatial rotations; Quaternions.

Lorentzian geometry with Minkowski Pythagorean Hodograph (MPH) formalism in R3, 1 gives us a new and insightful method of rational parametrization of canal surfaces. Our previous works about MPH curves in R3,1 [11] shows that a curve (t)=(x(t); y(t); z(t); r(t)) in R3,1 can be represented by the PH representation map in Cl(3, 1) which avoids the complex root finding algorithm.

Our parametrization method gives us the flexibility to represent the canal surfaces within their fiber ambiguities. This paper constitutes the first step of our ongoing work, which deals with the issues for canal surfaces in a truly new and intriguing manner such as finding rotation minimizing frames. We believe this is just the tip of the iceberg and the further work will yield many valuable applications in the area of canal surfaces.

We present a new approach to medial axis transform and offset curve computation. Our algorithm is based on the domain decomposition scheme which reduces a complicated domain into a union of simple subdomains each of which is very easy to handle. This domain decomposition approach gives rise to the decomposition of the corresponding medial axis transform which is regarded as a geometric graph in the three dimensional Minkowski space R2,1. Each simple piece of the domain, called the fundamental domain, corresponds to a space-like curve in R2,1. Then using the new spline, called the Minkowski Pythagorean hodograph curve which was recently introduced, we approximate within the desired degree of accuracy the curve part of the medial axis transform with a G1 cubic spline of Minkowski Pythagorean hodograph. This curve has the property of enabling us to write all offset curves as rational curves. Further, this Minkowski Pythagorean hodograph curve representation together with the domain decomposition lemma makes the trimming process essentially trivial. We give a simple procedure to obtain the trimmed offset curves in terms of the radius function of the MPH curve representing the medial axis transform.

Keywords: Domain decomposition; Medial axis; Offset curves; Minkowski Pythagorean hodographs.

We introduce the Minkowski Pythagorean hodograph (MPH) curve as a polynomial curve whose speed measured under the Minkowski metric is polynomial. It is a generalization of the Pythagorean hodograph (PH) curve. The MPH curve is well adapted to the representation of the medial axis transform of a planar domain. In fact, if the smooth curve segment of the medial axis transform is written in the MPH form, the boundaries of the corresponding domain are easily computed as rational curves of the MPH curve parameter. Furthermore, just subtracting a constant value from the radius, the offset curves can be obtained as rational curves. We also give the characterization of MPH curves which is invariant under the Lorentz transform.

Keywords: Pythagorean hodograph; Minkowski Pythagorean hodograph; Medial axis transform; Rational offset; Lorentz transform

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