Hwan Pyo Moon

Clifford Algebra, Spin Representation, and Rational Parameterization of Curves and Surfaces

The Pythagorean hodograph (PH) curves are characterized by certain Pythagorean n-tuple identities in the polynomial ring, involving the derivatives of the curve coordinate functions. Such curves have many advantageous properties in computer aided geometric design. Thus far, PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts gives rise to different combinations of polynomials that satisfy further complicated identities. We present a novel approach to the Pythagorean hodograph curves, based on Clifford algebra methods, that unifies all known incarnations of PH curves into a single coherent framework. Furthermore, we discuss certain differential or algebraic geometric perspectives that arise from this new approach.

Minkowski pythagorean hodographs

Abstract 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.

New algorithm for medial axis transform of plane domain

Abstract In this paper, we present a new approximate algorithm for medial axis transform of a plane domain. The underlying philosophy of our approach is the localization idea based on the Domain Decomposition Lemma, which enables us to break up the complicated domain into smaller and simpler pieces. We then develop tree data structure and various operations on it to keep track of the information produced by the domain decomposition procedure. This strategy enables us to isolate various important points such as branch points and terminal points. Because our data structure guarantees the existence of such important points—in fact, our data structure is devised with this in mind—we can zoom in on those points. This makes our algorithm efficient. Our algorithm is a “from within” approach, whereas traditional methods use a “from-the-boundary” approach. This “from within” nature of our algorithm and the localization scheme help mitigate various instability phenomena, thereby making our algorithm reasonably robust.

Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves

Abstract 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 R 2,1 . Each simple piece of the domain, called the fundamental domain, corresponds to a space-like curve in R 2,1 . 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 G 1 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.

Algorithms for Minkowski products and implicitly‐defined complex sets

Minkowski geometric algebra is concerned with the complex sets populated by the sums and products of all pairs of complex numbers selected from given complex‐set operands. Whereas Minkowski sums (under vector addition in Rn have been extensively studied, from both the theoretical and computational perspective, Minkowski products in R2 (induced by the multiplication of complex numbers) have remained relatively unexplored. The complex logarithm reveals a close relation between Minkowski sums and products, thereby allowing algorithms for the latter to be derived through natural adaptations of those for the former. A novel concept, the logarithmic Gauss maps of plane curves, plays a key role in this process, furnishing geometrical insights that parallel those associated with the “ordinary” Gauss map. As a natural generalization of Minkowski sums and products, the computation of “implicitly‐defined” complex sets (populated by general functions of values drawn from given sets) is also considered. By interpreting them as one‐parameter families of curves, whose envelopes contain the set boundaries, algorithms for evaluating such sets are sketched.

Equivolumetric offsets for 2D machining with constant material removal rate

We introduce two special types of variable radius planar offsets: equivolumetric offsets. Both types of equivolumetric offsets provide us the machining method with the constant material removal rate. The first type of equivolumetric offsets can be used as semi-finished workpiece boundary with which we are able to perform the final finish milling of constant material removal rate with the uniform feedrate profile. Another type of equivolumetric offsets allows us to devise a machining strategy which has the constant material removal rate and the constant contact trajectory speed. By using this, we are able to achieve the constant cutting force and the fairness of the finished surface simultaneously.

Equivolumetric offset surfaces

We present the equivolumetric offset surface which is a special kind of variable distance offset surfaces. For a given base surface, the volume bounded by the base surface and its equidistance offset may vary depending on the curvature of the base surface. The offset distance function of the equivolumetric offset surface is carefully chosen to compensate this curvature effect and to equalize the volumetric ratio. The explicit formulation of the equivolumetric offset distance function is given by using the Chebyshev cube root.

Bipolar and Multipolar Coordinates

Bipolar or multipolar coordinates offer useful insights and advantages over Cartesian coordinates in certain geometrical problems. In bipolar coordinates (r 1, r 2) the “simplest” curves are the conics, ovals of Cassini, Cartesian ovals, and their special cases, which are characterized by linear or hyperbolic relations in the (r 1, r 2) plane. As a natural extension of these classical examples, we consider the full range of curves characterized by conic (r 1, r 2) loci. A further useful generalization involves the extension of the curve equations to (redundant) multipolar coordinates (r 1,…, r n), taking the n—th roots of unity as “canonical” poles. We survey two key applications of these methods, in geometrical optics and the Minkowski geometric algebra of complex sets, and explore the formulation of geometric design schemes using planar and spatial bipolar or multipolar coordinates.

Minkowski Geometric Algebra and the Stability of Characteristic Polynomials

A polynomial p is said to be Γ-stable if all its roots lie within a given domain Γ in the complex plane. The Γ-stability of an entire family of polynomials, defined by selecting the coefficients of p from specified complex sets, can be verified by (i) testing the Γ-stability of a single member, and (ii) checking that the “total value set” V * for p along the domain boundary ∂Γ does not contain 0 (V * is defined as the set of all values of p for each point on ∂Γ and every possible choice of the coefficients). The methods of Minkowski geometric algebra —the algebra of point sets in the complex plane — offer a natural language for the stability analysis of families of complex polynomials. These methods are introduced, and applied to analyzing the stability of disk polynomials with coefficients selected from circular disks in the complex plane. In this context, V * may be characterized as the union of a one-parameter family of disks, and we show that the Γ-stability of a disk polynomial can be verified by a finite algorithm (a counterpart to the Kharitonov conditions for rectangular coefficient sets) that entails checking that at most two real polynomials remain positive for all t, when the domain boundary ∂Γ is a given polynomial curve γ(t).Furthermore, the “robustness margin” can be determined by computing the real roots of a real polynomial.

Weierstrass-type approximation theorems with Pythagorean hodograph curves

We prove the Weierstrass-type approximation theorem that states every C^1 curve in the 2-dimensional or 3-dimensional Euclidean space or in the 3-dimensional Minkowski space can be uniformly approximated by Pythagorean hodograph curves in the corresponding space. This abundance of PH curves is another theoretical confirmation of the usefulness and the versatility of the PH curves. We also address some algorithmic aspects of proposed PH approximation schemes and their convergence rates.