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On the Implicit Equation of Conics and Quadrics Offsets mathematics Article On the Implicit Equation of Conics and Quadrics Offsets Jorge Caravantes 1 , Gema M. Diaz-Toca 2,* , Mario Fioravanti 3 and Laureano Gonzalez-Vega 4 1 Departamento de Física y Matemáticas, Universidad de Alcalá, 28805 Madrid, Spain; [email protected] 2 Departamento de Ingeniería y Tecnología de Computadores, Universidad de Murcia, 30100 Murcia, Spain 3 Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005 Santander, Spain; mario.fi[email protected] 4 Departamento de Métodos Cuantitativos, CUNEF Universidad, 28040 Madrid, Spain; [email protected] * Correspondence: [email protected] Abstract: A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in “A Treatise on the Analytic Geometry of Three Dimensions” by G. Salmon for offsets to non-degenerate conics and central quadrics. Keywords: offsets; conics and quadrics; implicit equations; discriminant 1. Introduction Citation: Caravantes, J.; Fioravanti, Let d > 0 be a fixed real number. Given a curve C in R2 or a surface S in R3, their M.; Diaz-Toca, GM.; Gonzalez-Vega, respective d-offset, denoted by Cd and Sd, is the locus of the points that are at constant L. On the Implicit Equation of Conics Euclidean distance d, measured along the normal line, from the curve C or surface S. and Quadrics Offsets. Mathematics Offsets have a variety of useful applications in computer-aided design, such as tool 2021, 9, 1784. https://doi.org/ path control, geometric modeling, NC milling, design of surfaces with homogeneous thick- 10.3390/math9151784 ness and tolerance analysis in manufacturing (see, for example, Ref. [1,2]). This leads to the fact that they have been widely studied by several authors; see, for example, Ref. [3–10]. Academic Editor: Valery G. Moreover, the notion of offset has been generalized in different ways, as reflected in [11–13]. Romanovski Dealing with offset curves is easier when the arc-length of the given curve is polyno- mial. Thus, Pythagorean hodograph (PH) curves, spatial PH curves, rational PH curves Received: 17 June 2021 and Minkowski PH curves have been thoroughly studied (see [14–16] and the bibliogra- Accepted: 25 July 2021 Published: 28 July 2021 phy in [13]). Regarding surfaces, Pythagorean normal (PN) surfaces were introduced by Pottmann [17]. C S Publisher’s Note: MDPI stays neutral It is well known that the offset of a rational curve or a rational surface is an with regard to jurisdictional claims in algebraic curve or surface, but it is not rational in most cases. In fact, there are numerous published maps and institutional affil- articles where conditions for curves and surfaces are studied so that their offsets are iations. rational (see, for example, [18–21]). In addition, the implicit equation of the offset is typically of much higher degree than the initial curve or surface, has many terms and coefficients of large size and its calculation involves using elimination methods that are not always effective. However, G. Salmon proved in his classical texts [22,23] that the implicit equation of Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. the offset to non-degenerate conics and to central quadrics can be characterized in terms of This article is an open access article the discriminant of a univariate polynomial. We show here that this representation works distributed under the terms and for the implicit equation of Cd and Sd when C is any conic and S is any quadric. These conditions of the Creative Commons implicit equations are of determinantal type and free of extraneous components; they pro- Attribution (CC BY) license (https:// vide a very compact expanded form and they are very useful when dealing with geometric creativecommons.org/licenses/by/ queries such as point positioning for offsets or solving intersection purposes involving 4.0/). offsets. This is one of the main motivations for deriving these new representations for the Mathematics 2021, 9, 1784. https://doi.org/10.3390/math9151784 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 1784 2 of 19 offset implicit equations that can be considered a complement to the new lower-degree rational parametrizations of regular quadric offsets introduced in [18] when manipulating offsets to quadrics. Finally, we should mention that in [24], a determinantal form for the implicit equation for offsets to non-degenerate conics and quadrics was introduced. Here, we refine these formulas, analyze the degenerate cases, provide explicit proofs and include more applications. This paper is divided into six sections. Following this introduction, the second section introduces the usual formula defining offsets to plane curves and surfaces. In the third one, the implicit equation of the offset to a conic is presented as the determinant of a 4 × 4 or a 3 × 3 matrix, the proof being fully computational. The fourth one introduces the implicit equation of the offset to a quadric as the determinant of a 6 × 6 or a 4 × 4 matrix, the proof being fully computational too. The fifth section shows some geometric queries involving offsets to quadrics where the introduced implicit equations are used. The last section is devoted to the conclusions of the paper. 2. Preliminaries The next subsections provide some basic notions and the equations describing the offsets to curves and surfaces. 2.1. Conics and Quadrics Representation Let A be a conic in R2. Let X = (X, Y, 1). It is well known that the implicit equation of A in R2 can be written as 2 2 a(X, Y) = a11X + a22Y + 2a12XY + 2a13X + 2a23Y + a33 = 0, or in matricial form A : X AX t = 0, where A is the symmetric matrix 0 1 a11 a12 a13 A = @a12 a22 a23A. a13 a23 a33 Continuing in the same way, let A be a quadric in R3 and X = (X, Y, Z, 1). The implicit equation of a quadric A in R3 can be written as 2 2 2 a(X, Y, Z) = a11X + a22Y + a33Z + 2a12XY + 2a13XZ + 2a23YZ +2a14X + 2a24Y + 2a34Z + a44 = 0, or in matricial form A : X AX t = 0, where X = (X, Y, Z, 1) and A is the symmetric matrix 0 1 a11 a12 a13 a14 Ba a a a C A = B 12 22 23 24C. @a13 a23 a33 a34A a14 a24 a34 a44 2.2. Describing the Implicit Equation of an Offset 2 2 Given a curve C in R , the d-offset to C, Cd, is the locus of the points in R , which are at constant Euclidean distance d from the initial curve C along its normal line. The offset of a curve is generally not rational (even when C is rational) but it is always an algebraic curve. The implicit equation of the d-offset Cd is usually a very dense polynomial of higher degree than the one of the original curve. If the curve C is presented by its implicit equation f (x, y) = 0, then the computation of the implicit equation of Cd requires the elimination of a and b from these three equations: 2 2 2 f (a, b) = 0, (x − a) + (y − b) − d = 0, (x − a) fy(a, b) − (y − b) fx(a, b) = 0. Mathematics 2021, 9, 1784 3 of 19 3 3 Similarly, given a surface S in R , the d-offset to S, Sd, is the locus of the points in R , which are at constant Euclidean distance d from the initial surface S (along its normal line). As in the case of curves, Sd is generally not rational. If the surface S is given by its implicit equation f (x, y, z) = 0, then the computation of the implicit equation of Sd requires the elimination of a, b and g from these four equations: f (a, b, g) = 0, (x − a) fy(a, b, g) − (y − b) fx(a, b, g) = 0, (x − a)2 + (y − b)2 + (z − g)2 − d2 = 0, (y − b) fz(a, b, g) − (z − g) fy(a, b, g) = 0. Consequently, the implicit equation of offsets can be obtained using elimination techniques but this can be a time-consuming calculation and, even so, we do not always obtain a closed formula without extraneous factors. 2.3. Discriminant of a Polynomial Let f (x) and g(x) be two univariate polynomials with coefficients in a field: m n i i f (x) = ∑ aix , g(x) = ∑ bix . i=0 i=0 Then, the Sylvester matrix of f (x) and g(x) with respect to x, denoted by Syl( f , g, x), is defined as n+m z }| { 0 1 am ... a0 9 B .. .. C = B . C n B C ; B am ... a0 C Syl( f , g, x) = B C 9 . B b ... b C B n 0 C = B .. .. C m @ . A ; bn ... b0 The resultant of f (x) and g(x) with respect to x, denoted by Res( f , g, x), is the deter- minant of the Sylvester matrix. It is well known that Res( f , g, x) = 0 if and only if f (x) and g(x) have a common nontrivial factor. Moreover, the discriminant of f (x) is defined to be m(m−1) (−1) 2 disc( f ) = Res( f , f 0, x), am and it is widely known that f (x) has a multiple root if and only if its discriminant vanishes. 3. On the Implicit Equation for the d-Offset to a Conic Given two conics A : X AX t = 0 and B : X BX t = 0, their characteristic equation is defined as f (l) = det(lB + A) = det(B)l3 + ..
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