CONTRIBUCIONES CIENT¶IFICAS · 2012. 6. 18. · CONTRIBUCIONES CIENT¶IFICAS EN HONOR DE MIRIAN ANDRES G¶ OMEZ¶ (Laureano Lamb¶an, Ana Romero y Julio Rubio, editores), Servicio

CONTRIBUCIONES CIENTIFICASEN HONOR DE MIRIAN ANDRES GOMEZ(Laureano Lamban, Ana Romero y Julio Rubio, editores),Servicio de Publicaciones, Universidad de La Rioja,Logrono, Spain, 2010.

BRIEF ATLAS OF OFFSET CURVES

JUANA SENDRA

Dedicado a la memoria de Mirian Andres Gomez

Resumen. En este artıculo se recuerda algunas de las definiciones basicas ylas principales propiedades de las offsets de curvas algebraicas planas (vease[23]), ası como un algoritmo para parametrizar las componentes de generocero de la curva offset (vease [4]). Ademas, se presenta un breve atlas enel que se obtiene la offset de varias curvas algebraicas planas, se analiza suracionalidad y se dan parametrizaciones.

Abstract. In this paper, we recall some of the basic definitions and mainproperties on offsets to algebraic plane curves (see [23]), as well as an algo-rithm for parametrizing the genus zero components of an offset (see [4]). Inaddition, we present a brief atlas where the offset of several algebraic planecurves are obtained, its rationality analyzed, and parametrizations are pro-vided.

1. Introduction

Let K be an algebraically closed field of characteristic zero (say K = C), and Can irreducible curve in K2; in general, this theory can be developed for irreduciblehypersurfaces in Kn, see e.g. [4], [23]. The offset curve (or parallel curve) to Cat distance d is essentially the envelope of the system of spheres centered at thepoints of C with fixed radius d (see Fig. 1 and, for a formal definition, see Section2). In particular, if C is parametrized by P(t) ∈ K(t)2, the offset to C correspondsto the Zariski closure of the set in K2 generated by the formula

P(t)± dN (t)‖N (t)‖

where N (t) is the normal vector to C associated with P(t). For instance, if C isthe parabola of equation y2 = y2

1 , that can be parametrized as (t, t2), then theoffset at distance d is the Zariski closure of{

(t, t2)± d√1 + 4t2

(−2t, 1)∣∣∣∣ t ∈ C \

{±√−1

2

}}.

The term “parallel” was apparently introduced by Leibniz in [13] for the case ofplane curves. Also, in elementary texts on differential geometry (see [6]) or insome books on algebraic geometry (see for instance [7], [8], [19]) some elementary

Key words and phrases. Offset curve, rationality, genus, computer-aided geometric design.Partially supported by MTM2008-04699-C03-01.

483

484 JUANA SENDRA

–1

0

1

2

3

4

5

y

–3 –2 –1 1 2 3x

Figure 1. Construction of the offset of the parabola.

aspects of parallel curves are studied. Nevertheless, in the 1980s, cagd (ComputerAided Geometric Design) community started to be interested on the topic, andthey began to address problems related to offsets to curves and surfaces, due tothe important role that offsets play in practical applications as tolerance analysis,geometric control, robot path-planning and numerical-control machining problems,etc. [11], [12]. As a consequence of this applicability, many interesting questionsdirectly related to algebraic geometry have been addressed (see, e.g. [1], [2], [3],[4], [5], [9], [10], [14], [15], [16], [17], [18], [21], [22], [23], [24] ) and, currently, thestudy of offsets continue being an active research area.

In this paper, we recall some of the basic definitions and main properties onoffsets to algebraic plane curves, and we present a brief atlas where the offset ofseveral algebraic plane curves are obtained, and its rationality analyzed. Further-more, in case of offset genus zero, a rational parametrization is computed. Theexamples presented in this paper have been executed with the computer algebrasystem Maple, and with the package for constructive algebraic geometry CASA.

2. Basic Notions

In K2 we consider the symmetric bilinear form B((x1, x2), (y1, y2)) = x1y1 +x2y2, which induces a metric vector space with light cone L of isotropy given as(see [20])

L = {(x1, x2) ∈ K2 |x21 + x2

2 = 0}.In this context, the circle of center (a1, a2) ∈ K2 and radius d ∈ K is the planecurve defined by (x1 − a1)2 + (x2 − a2)2 = d2. We will say that the distancebetween the points x, y ∈ K2 is d ∈ K if y is on the circle of center x and radiusd. Notice that the “distance” is hence defined up to multiplication by ±1. Onthe other hand, if x 6∈ L we denote by ‖x‖ any of the elements in K such that‖x‖2 = B(x, x), and if x ∈ L, then ‖x‖ = 0. We usually work with both solutionsof ‖x‖2 = B(x, x). For this reason we use the notation ±‖x‖.

BRIEF ATLAS OF OFFSET CURVES 485

In this situation, let C be the affine irreducible plane curve defined by theirreducible polynomial f(y) ∈ K[y], y = (y1, y2), let d ∈ K∗ be a non-zero fieldelement, x = (x1, x2), and fi = ∂f

∂yi. We assume that C is none of the two lines

defining L.In order to get a formal definition of the offset, one introduces the following

incidence diagram

B(C, d) ⊂ K2 ×K2 ×Kπ1 ↙ ↘ π2

π1(B(C, d)) ⊂ K2 C ⊂ K2

(Incidence Diagram)

where the offset incidence variety is

B(C, d) =

(x, y, λ) ∈ K2 ×K2 ×K

/f(y) = 0x = y + λ(f1(y), f2(y))(x1 − y1)2 + (x2 − y2)2 = d2

and

π1 : K2 ×K2 ×K −→ K2, π2 : K2 ×K2 ×K −→ K2

(x, y, λ) 7−→ x (x, y, λ) 7−→ y.

Then, we define the offset of C at distance d as the algebraic Zariski closure inK2 of π1(B(C, d)), and we denote it by Od(C); i.e.

Od(C) = π1(B(C, d)).

In the above definition, we have considered only irreducible curves. Note thatthe same reasoning can be done for reducible curves, introducing the offset as theunion of the offset of the irreducible components.

Since we have assumed that C is none of the lines of isotropy, then Od(C) isnever empty. Furthermore, with the exception of C being a circle and d its radius,Od(C) has at most two components, all of the them of dimension 1 (see [23]);i.e. with the exception of that particular circle, each component of Od(C) is analgebraic curve.

Another important property of offsets relates the normal vectors to C and toits offsets (see [23]). More precisely: let P ∈ C, and let Q ∈ Od(C) be any of thetwo points on the offset generated by P , then it holds that the normal vectors toC at P and the normal vectors to Od(C) at Q are parallel.

In order to compute the offset, one can proceed as follows: Let I be ideal inK[x, y, λ] generated by the polynomials defining B(C, d). Then, by the ClosureTheorem (see [8] p. 122), one has that Od(C) = V (I ∩ K[x]). Hence eliminationtheory techniques, such as Grobner bases, provide the offset.

Reasoning as in Section 2 in [22], one may introduce the notion of generic offset.Let us consider d as a new variable. Now, B(C, d) is seen as an algebraic set in

486 JUANA SENDRA

K2 ×K2 ×K×K; we denote it by B(C, d)G. Then, the generic offset is definedas

Od(C)G = π1(B(C, d)G).Now, if IG is the ideal in K[x, y, λ, d] generated by the polynomials definingB(C, d)G, by the Closure Theorem (see [8] p. 122), one has that Od(C)G =V (IG ∩ K[x, d]). Moreover, reasoning as in Theorem 6 in [22] which is a directconsequence of Exercise 7, p. 283 in [8], one gets that for almost all values ofd ∈ K∗ the generic offset specializes properly.

3. Parametrizing Offsets

In this Section we summarize the results on the rationality of the offsets tocurves, presented in [4], by deriving an algorithm for parametrizing offsets.

The rationality of the components of the offsets is characterized by means of theexistence of parametrizations of the curve whose normal vector has rational norm,and by means of the rationality of the components of an associated curve, that isusually simpler than the offset, as shown in the examples. As a consequence, onededuces that offsets to rational curves behave as follows: they are either reduciblewith two rational components (double rationality), or rational, or irreducible andnot rational.

For this purpose, we first need to introduce two new concepts: rational Pytha-gorean hodographs and the curve of reparametrization. Let

P(t) = (P1(t), P2(t)) ∈ K(t)2

be a rational parametrization of C. Then, P(t) is rph (Rational PythagoreanHodograph) if its normal vector N (t) = (N1(t), N2(t)) satisfies that

N1(t)2 + N2(t)2 = m(t)2,

with m(t) ∈ K(t). For short we will express this fact writing ‖N (t)‖ ∈ K(t).On the other hand, we define the reparametrizing curve of Od(C) associatedwith P(t) as the curve generated by the primitive part with respect to x2 of thenumerator of

x22 P

′1(x1)− P

′1(x1) + 2 x2 P

′2(x1),

where P′i denotes the derivative of Pi. In the following, we denote by GP(C) the

reparametrizing hypersurface of Od(C) associated with P(t).Summarizing the results in [4], one can outline the following algorithm for

offsets.

Algorithm: offset parametrization

Given: a proper rational parametrization P(t) of the plane curve C in K2.Decide: whether the components of Od(C) are rational.Determine: (in the affirmative case) a rational parametrization of eachcomponent of Od(C).

1. [Normal vector computation] Compute the normal vector N (t) of P(t)2. [Checking RPH] If ||N (t)|| ∈ K(t) then return ¿ Od(C) has two rational

components parametrized by P(t)± d||N (t)||N (t) À.


3. [Determination of the reparametrizing curve] Determine GP(C), and decidewhether GP(C) is rational.

4. [Determination of the reparametrizing curve and parametrization] If GP(C) isnot rational then return ¿ no component of Od(C) is rational À else

4.1. Determine a rational parametrization R(t) = (R(t), R(t)) of GP(C) .4.2. Return ¿ Od(C) is a rational curve parametrized by

Q(t) = P(R(t)) +2 dR(t)

N2(R(t))(R(t)2 + 1)N (R(t)),

where N = (N1, N2) À.

4. Atlas of Offsets Curves

In this section we apply the previous algorithm to analyze the rationality of theoffset curve of several classical rational curves, and in the case of rationality wecompute a rational parametrization of the offset. In addition, we also compute theimplicit equation of the corresponding generic offset. The rational curves includedin the atlas are:

1. The Circle (Example 1),2. The Parabola (Example 2),3. The Hyperbola (Example 3),4. The Ellipse (Example 4),5. The Cardioid (Example 5),6. The Three-leafed Rose (Example 6),7. The Trisectrix of Maclaurin (Example 7),8. The Folium of Descartes (Example 8),9. The Tacnode (Example 9),

10. The Conchoid of de Sluze (Example 10),11. The Epitrochoid (Example 11),12. The Ramphoid Cusp (Example 12),13. The Lemniscata of Bernoulli (Example 13),14. Cuspidal curves (Example 14)

Example 1. (Offset of the Circle).Let C be the circle defined by

y21 + y2

2 − r2.

1. Implicit equation: the implicit equation of the generic offset is

(x21 + x2

2 − (d + r)2)(x21 + x2

2 − (d− r)2)

2. Rationality character: Double rational (i.e. the offset has two rational com-ponents)

3. Offset parametrization: its components are parametrized by(

(d± r)2t

t2 + 1,(d± r)(t2 − 1)

t2 + 1

)

488 JUANA SENDRA

4. Remark: Note that if d = r one component of the offset degenerates toa point, namely the center. Indeed, let I be the ideal generated by thepolynomials defining B(C, d), with d = r. Then, computing a Grobnerbasis, one sees that

I ∩ C[x] =< −x2

(−x22 + 4 r2 − x1

2),−x1

(−x22 + 4 r2 − x1

2)

>,

and hence Or(C) is the circle of equation x21 + x2

2 = 4r2 union the origin.

–4

–2

2

4

y

–4 –2 2 4

x

Figure 2. Offset of the unit circle at d = 3 and d = 4 (untraced curves).

Example 2. (Offset of the Parabola). Let C be the parabola defined by

y2 − ay21 , a 6= 0.

1. Implicit equation: The implicit equation of the generic offset is−d2+x2

2+8 x2d2a−2 x2x1

2a−8 x23a−8 d4a2+x1

4a2+16 x24a2−8 d2x2

2a2+32 x1

2x22a2−20 x1

2d2a2+8 x12x2d

2a3−32 x12d2x2

2a4+16 x16a4−32 x2

3a3d2−32 x2

3a3x12−40 x1

4x2a3+16x1

4x22a4+48 x1

2d4a4−48 x14d2a4+32 x2d

4a3+16 d4a4x2

2 − 16 d6a4

2. Rationality character: rational3. Offset parametrization: the generic offset can be parametrized as

((t2 − 1

) (−t2 − 1 + 4 dat)

4at (t2 + 1),t6 − t4 − t2 + 1 + 32 dt3a

16at2 (t2 + 1)

)

4. Details of the computation:P(t) = (t, at2) and N (t) = (−2at, 1).GP is defined by x2

2 − 1 + 4x2ax1.R = (− t2−1

4at , t).


–6

–4

–2

0

2

4

6

8

y2

–8 –6 –4 –2 2 4 6 8

y1

Figure 3. Parabola (untraced curve) and the offsets for several distances.

Example 3. (Offset of the Hyperbola).Let C be the Hyperbola defined over C by the equation

y12

a2− y2

2

b2− 1, a > 0, b > 0, a 6= b.

1. Implicit equation: The implicit equation of the generic offset is(−2 a2 b6−2 a6 b2+a8+b8−6 b4 a4) d4+(b8+6 b4 a4+6 a2 b6)x1

4+(−6 a6 b2−4 a2 b6 − 8 b4 a4 − 2 b8) d2 x1

2 + (−2 a8 − 8 b4 a4 − 4 a6 b2 − 6 a2 b6) x22 d2 +

(−6 b4 a2 − 2 a4 b2 − 2 a6)x24 x1

2 + (−4 a4 − 2 a2 b2) d2 x26 + (b4 + 6 a4 +

6 a2 b2) d4 x24+(−2 a2 b2−4 b4)x1

6 d2+(a8+6 a6 b2+6 b4 a4) x24+(−2 b8 a2+

2 a6 b4 − 2 a4 b6 + 2 a8 b2) d2 + (−2 b8 a2 − 4 a6 b4 − 6 a4 b6)x12 + (6 a6 b4 +

4 a4 b6+2 a8 b2) x22+(a4+b4+2 a2 b2) d8+(2 b4 a2−2 a4 b2+2 b6−2 a6) d6+

(−2 b6−4 b4 a2) x16 +(−6 b4 a2 +4 a6 +6 a4 b2−4 b6)x1

2 d2 x22 +(−2 a2 b2−

2 b4−6 a4) d2 x12 x2

4+(−10 a4 b2−6 b4 a2−6 a6) x24 d2+(2 a6+4 a4 b2)x2

6+b8 a4 + 2 b6 a6 + a8 b4 + (10 a2 b2 + 6 a4 + 6 b4) x1

2 d4 x22 + (−6 b4 − 2 a4 −

2 a2 b2)x14 d2 x2

2 +(−4 a2 b2 +a4 +b4)x14 x2

4 +b4 x18 +a4 x2

8 +(−6 a2 b2−2 b4− 4 a4) x2

2 d6 + (2 b4− 2 a2 b2)x16 x2

2 + (−6 a2 b2− 2 a4− 4 b4) d6 x12 +

(6 a6 +2 b6 +8 a4 b2 +4 b4 a2)x22 d4 +(−6 a6 b2−6 a2 b6−10 b4 a4) x1

2 x22 +

(2 a4−2 a2 b2)x12 x2

6 +(a4 +6 b4 +6 a2 b2) d4 x14 +(−6 b6−4 a4 b2−2 a6−

8 b4 a2)x12 d4+(6 b6+6 a4 b2+10 b4 a2)x1

4 d2+(2 b4 a2+6 a4 b2+2 b6) x14 x2

2

2. Rationality character: irreducible and non rational

490 JUANA SENDRA

3. Details of the computation:

P(t) = (a(a2+t2b2)−t2b2+a2 ,−2 ab2t

−t2b2+a2 ) and N (t) = ( 2b2a(a2+t2b2)(−t2b2+a2)2 , 4b2a3t

(−t2b2+a2)2 ).GP is defined by x2

2a2x1 − x1a

2 − a2x2 − b2x21x2.

–8

–6

–4

–2

0

2

4

6

8

y2

–8 –6 –4 –2 2 4 6 8

y1

Figure 4. Hyperbola (untraced curve) and the Offsets of theHyperbola for several distances.

Example 4. (Offset of the Ellipse). Let C be the ellipse defined by

y12

a2+

y22

b2− 1, a > 0, b > 0, a 6= b.

1. Implicit equation: The implicit equation of the generic offset is(4 a2 b2 + a4 + b4)x1

4 x24 + (a4− 6 a2 b2 + 6 b4)x1

4 d4 + (10 b4 a2− 6 a4 b2 −6 b6) x1

4 d2+(−2 b8 a2+6 b6 a4−4 a6 b4) x12+(−2 a8 b2+6 a6 b4−4 b6 a4)x2

2+(−2 b8 a2 − 2 a8 b2 + 2 b6 a4 + 2 a6 b4) d2 + (2 b4 a2 − 2 b6 − 6 a4 b2)x1

4 x22 +

(2 a6 b2−6 b4 a4+2 a2 b6+b8+a8) d4+(−2 a2 b2+b4+a4) d8+(−2 b6+6 a6−8 a4 b2 +4 b4 a2)x2

2 d4 +(−2 a6−8 b4 a2 +6 b6 +4 a4 b2)x12 d4 +(−10 b4 a4 +

6 a2 b6+6 a6 b2) x12 x2

2+a8 b4−2 a6 b6+b8 a4+(2 a4+2 a2 b2)x12 x2

6+(6 a4+b4−6 a2 b2) d4 x2

4+(−2 a4+6 a2 b2−4 b4)x12 d6+(6 a2 b2−2 b4−4 a4)x2

2 d6+(−10 a2 b2 + 6 b4 + 6 a4)x2

2 x12 d4 + (6 b4 a4 + b8 − 6 a2 b6)x1

4 + (6 b4 a4 +a8− 6 a6 b2)x2

4 +(−4 a4 b2 +2 a6) x26 +(−6 a4 +2 a2 b2− 2 b4) x1

2 x24 d2 +

(2 a2 b2+2 b4)x16 x2

2+(−6 b4 a2+10 a4 b2−6 a6) d2 x24+(2 a2 b2−4 a4) d2 x2

6+(−2 a8 +4 a6 b2 +6 a2 b6− 8 b4 a4)x2

2 d2 +(2 a2 b2− 6 b4− 2 a4)x14 x2

2 d2 +b4 x1

8 + a4 x28 + (−4 b4 a2 + 2 b6)x1

6 + (2 b4 a2 − 2 b6 + 2 a4 b2 − 2 a6) d6 +(−6 a4 b2−6 b4 a2+4 b6+4 a6) x1

2 x22 d2+(−2 a6−6 b4 a2+2 a4 b2)x1

2 x24+

(6 a6 b2 + 4 a2 b6 − 8 b4 a4 − 2 b8)x12 d2 + (−4 b4 + 2 a2 b2)x1

6 d2


2. Rationality character: irreducible and non rational3. Details of the computation:

P(t) = (a(t2−1)

t2+1 , 2 btt2+1 ) and N (t) = (

2b(−1+t2)(1+t2)2

, −4at(1+t2)2

).GP is defined by −x2

2ax1 + ax1 + x2b− x2bx21.

–4

–2

0

2

4

y2

–4 –2 2 4

y1

Figure 5. Ellipse (untraced curve) and the Offsets of the Ellipsefor several distances.

Example 5. (Offset of the Cardioid). Let C be the Cardioid defined by

(y21 + 4 y2 + y2

2)2 − 16 (y21 + y2

2).

1. Implicit equation: The implicit equation of the generic offset is24 x2

2 x12 d4−36 x2

2 x14 d2−36 x2

4 x12 d2+16 x2

6 x12+12 x2

4 d4−4 d6 x12+

12 d4 x14+4 x1

8+24 x24 x1

4+16 x22 x1

6−12 d2 x16+192 x1

2 x25+64 x2 x1

6−144 d2 x2

5 − 128 x16 − 12 x2

6 d2 − 1024 d2 x23 − 1024 x2

3 x12 − 768 d2 x2

4 +528 d4 x2

2+336 d4 x12−192 d2 x1

4−256 x2 d2 x12−1024 x2 x1

4−960 d2 x22 x1

2+192 x1

4 x23−4 d6 x2

2−16 d6 x2−16 d6+384 x24 x1

2+1024 d4−2048 d2 x12+

1024 x14 + 1280 x2 d4 + 96 d4 x2

3 − 144 d2 x2 x14 + 96 d4 x2 x1

2 + 256 x26 +

4 x28 + 64 x2

7 − 288 d2 x23 x1

2


((−9 + t2) (d t6 − 117 d t4 + 3456 t3 − 1053 d t2 + 729 d)

(243 t2 + 27 t4 + t6 + 729) (t2 + 9),

−18 (d t6 − 16 t5 − 21 d t4 + 864 t3 − 189 d t2 − 1296 t + 729 d) t

(243 t2 + 27 t4 + t6 + 729) (t2 + 9)

)

492 JUANA SENDRA


P(t) = ( −1024 t3

256 t4+32 t2+1, −2048 t4+128 t2

256 t4+32 t2+1) and

N (t) = ( 256 t (48 t2−1)

(16 t2+1) (256 t4+32 t2+1), 1024 t2 (16 t2−3)

(16 t2+1) (256 t4+32 t2+1)).

GP is defined by 32 x22 x3

1 − 6 x22 x1 − 32 x3

1 + 6 x1 − 48 x2 x21 + x2.

R = ( −3 t2 (−9+t2)

, −t (−27+t2)

9 (−3+t2)).

–10

–5

5

y2

–10 –5 5 10y1

Figure 6. Cardioid (untraced curve) and the Offsets of the Car-dioid for several distances.

Example 6. (Offset of the Three-leafed Rose). Let C be the Three-leafed Rosedefined by

(y21 + y2

2)2 + r y1 (3 y22 − y2

1) = 0, r ∈ C, r 6= 0.

1. Implicit equation: The implicit equation of the generic offset for r = 1 is−63191384064x1x

62d

2 + 32212254720d4x82x

21 + 106451435520x4

1d8 −

163980509184d6x22x

21 + 128043712512d4x1x

82 + 18345885696x2

2d4 −

48318382080x1x102 d2 + 23187161088x4

1d4 + 95806291968x2

1d8 +

106904420352x62d

4x1 − 17179869184x61x

22d

6 − 17179869184x21x

62d

6 +3221225472x2

1x42d

8 + 3221225472x41x

22d

8 + 94220845056x1x42d

8 +62813896704x3

1x22d

8+67947724800x22x1d

8+3623878656x101 +764411904x6

1+764411904x8

1 + 3057647616x61x

22 + 4586471424x4

1x42 − 14495514624x8

1x22−


7247757312x61x

42+43486543872x4

1x62−3567255552x6

1d2−22649241600x8

1d2+

7644119040x41x

22d

2 − 76101451776x41d

2x42 + 88785027072x6

1d2x2

2 −4586471424x4

1x22 − 6879707136x4

1d2 + 6879707136x2

1x42 + 3057647616x2

1x62 −

13759414272x21x

22d

2 − 22932357120x21x

42d

2 + 95806291968x22d

8 −137933881344d6x2

2x1+764411904x82+18345885696x2

1d4−57076088832x2

1d6+

64424509440d4x61x

42 + 32212254720d4x8

1x22 − 1528823808d2x6

2 −6879707136d2x4

2 − 21743271936x1d10x2

2 + 46374322176x21x

22d

4 −156732751872x1d

6x42 − 104488501248x3

1d6x2

2 + 23187161088d4x42 −

57076088832d6x22 + 256087425024x5

1x42d

4 − 52881784832x51d

6x22 −

177167400960x31x

82d

2 + 10737418240x61x

62 + 178174033920x3

1x42d

4 +16106127360x9

1d2x2

2 + 64424509440d4x41x

62 + 37580963840x8

1x62 +

10737418240x91x

42 + 22548578304x10

1 x42 − 15300820992d2x10

1 +7851737088d2x9

1− 12079595520d2x102 − 1811939328x9

1− 3221225472x101 x2

2 +21063794688x7

1d2 + 7644119040x5

1d2 − 190253629440x6

2x21d

2 −2717908992x8

2d2 − 6115295232x3

1d4 − 81990254592d6x4

2 −65229815808x5

1d4 + 26046627840d4x6

2 + 41901096960x61d

4 −81990254592x4

1d6 + 341449900032d4x3

1x62 − 16986931200x4

1x22d

4 +220830105600x2

1x42d

4 + 18345885696x22x1d

4 + 130459631616x31x

22d

4 +195689447424x1x

42d

4 + 35634806784x51d

4x22 + 7516192768x12

1 x22 +

22548578304x41x

82 − 158645354496x6

2d6x1 − 2147483648x1

13 +1073741824x12

1 − 15288238080x31x

22d

2 − 22932357120x1x42d

2 −105318973440x3

1x42d

2 + 16307453952x71x

22 − 48922361856x5

1x42 +

48922361856x31x

62 + 32614907904x8

2x21 + 10871635968x7

1x42 +

25367150592x51x

62+19931332608x3

1x82+5435817984x1x

102 −12230590464d6−

1811939328x111 + 1073741824x1

14 − 4294967296x81d

6 + 6442450944d4x101 −

42681237504d4x91 + 6442450944d4x10

2 − 225485783040x51x

62d

2 +1528823808x2

2x51 +7644119040x4

2x31 +4586471424x6

2x1−35634806784x71d

4−22649241600x3

1d8 − 31406948352x5

1d8 + 1073741824d8x6

2 +1073741824x6

1d8 − 4294967296x8

2d6 − 264408924160x4

2x31d

6 +36691771392d8 + 12230590464d12 + 52244250624x5

1d6 + 45977960448x3

1d6 +

7247757312x31d

10 + 52881784832x71d

6 + 16106127360x111 d2 +

212902871040x21x

22d

8 − 47110422528x51x

42d

2 + 328564998144x41d

4x42 +

196494753792x61d

4x22 − 25769803776x4

2x41d

6 − 1811939328x91x

22 −

21063794688x51d

2x22−1528823808x7

1−36691771392d10−4294967296x111 x2

2−6442450944x8

1x42 − 25769803776d2x10

1 x22 − 85899345920d2x6

1x62 −

64424509440d2x41x

82 + 241591910400x6

2x21d

4 + 56371445760x81d

4 +106451435520d8x4

2 − 57076088832d10x22 − 57076088832d10x2

1 +53150220288x8

2d4−103750303744d6x6

1−4294967296d2x121 +1073741824x1

24−25769803776d2x2

1x102 − 301587234816d6x4

1x22 − 317693362176d6x2

1x42 −

102676561920d6x62 − 4294967296d2x12

2 − 64424509440d2x81x

42 −

159450660864d2x41x

62 − 62813896704d2x3

1x62 − 23555211264d2x1x

82 −

89389006848d2x82x

21 − 96636764160x7

1d2x4

2 − 47513075712d2x81x

22 −

114353504256d2x61x

42 + 6442450944x1x

122 + 7516192768x2

1x122 +

30064771072x31x

102 + 22548578304x4

1x102 + 53687091200x5

1x82 +

37580963840x61x

82 + 42949672960x7

1x62 + 9663676416x2

1x102

494 JUANA SENDRA


P(t) = ( (1−3 t2)(1+t2)2 , (1−3 t2) t

(1+t2)2 ) and N (t) = (− (−12 t2+3 t4+1)(1+t2)3 , 2 t (−5+3 t2)

(1+t2)3 ).GP is defined by −5 x2

2 x1 +3 x22 x3

1 +5x1−3 x31−12 x2 x2

1 +3 x2 x41 +x2.

–3

–2

–1

1

2

3

y2

–3 –2 –1 1 2 3

y1

Figure 7. Three-leafed Rose (untraced curve) and the Offsets ofthe Three-leafed Rose for several distances.

Example 7. (Offset of the Trisectrix of Maclaurin). Let C be the Trisectrix ofMaclaurin defined by

y1 (y21 + y2

2)− a (y22 − 3 y2

1), a ∈ C, a 6= 0.

1. Implicit equation: The implicit equation of the generic offset for a = 1 is−8x1x

62d

2+296d4x31−8d2x7

1−840x22d

4−1296x41+864x2

1x22−144x4

2+210x41d

4+4320x3

1d2 − 1728x5

1 + 192x1x42 − 648x6

1 + 47x81 + 44x6

1x22 − 54x4

1x42 − 6x6

1x42 −

4x41x

62 − 164x6

1d2 − 84x4

1x22d

2 + 18x41d

2x42 − 8x1x

22d

6 + 5d2x81 − 1080x4

1x22 +

1440x41d

2+296x21x

42−52x2

1x62+2048x2

1x22d

2+132x21x

42d

2+3456x21d

2−40x62−

x82−116x2

1d6 +52d2x6

2 +480d2x42 +36x2

1x22d

4−78d4x42 +4d6x2

2 +16d6x21x

22−

2304d4 − 24x51x

22d

2 − 216x1x22d

4 + 12d4x51 − 136d6x1 + 24d4x3

1x22 − 8d6x3

1 +2d8x1−184d2x5

1−2688d4x1+8x62x

21d

2+x82d

2−4d4x62−18x2

1x42d

4+12x1x42d

4−4d8x2

2 − 5d8x21 + 10d6x4

1 + 6d6x42 + d10 + 1632x1d

2x22 − x1

10 + 264x1x42d

2 −24x3

1x42d

2 +8x31x

62 +2x1x

82−x8

2x21−760d4x2

1−10d4x61−376x5

1x22−312x3

1x42 +

88x1x62− 32d6 + 2x9

1 + 1152x22d

2 + 23d8− 4x81x

22 + 24x7

1 + 8x71x

22 + 12x5

1x42−

24d4x41x

22 + 16d2x6

1x22 + 592x3

1x22d

2



P(t) = ( (t2−3)1+t2 , t (3−t2)

1+t2 ) and N (t) = ( (6t2+t4−3)(1+t2)2 , 8 t

(1+t2)2 ).GP is defined by 4x1x

22 − 4x1 + 6x2x

21 + x2x

41 − 3x2.

–10

–5

5

10

y2

–8 –6 –4 –2 2 4 6

y1

Figure 8. Trisectrix of Maclaurin (untraced curve) and the Off-sets of Trisectriz of Maclaurin for several distances.

Example 8. (Offset of the Folium of Descartes). Let C be the Folium of Descartesdefined by

y31 + y3

2 − 3 a y1 y2, a 6= 0.

1. Implicit equation: The implicit equation of the generic offset for a = 1 is

496 JUANA SENDRA

−248714388x1d8 + 38263752d8x5

2 − 248714388x1x62d

2 + 6377292d2x2x111 +

19131876x41d

8 − 841802544d6x22x

21 + 191318760x2

2d4 + 86093442x2

1x22 −

35075106x41d

4 − 420901272x1d6 − 229582512x1d

4 + 172186884x21d

8 +108413964d4x5

2 + 1256326524d4x41x2 − 229582512x6

1x2 − 229582512x21x

32 −

229582512x31x

22 − 57395628x1x

42 + 44641044x10

1 + 38263752d4x21x2 +

9565938x61 + 34012224x8

1 − 714256704x41x

32 + 741891636x6

1x22 +

1109648808x41x

42 − 114791256x8

1x22 − 153055008x6

1x32 − 6377292x6

1x42 +

178564176x61x

52 − 6377292x4

1x62 + 382637520x4

1x72 − 563327460x6

1d2 −

165809592x81d

2 − 25509168x101 x3

2 − 38263752x81x

52 − 867311712d4x1x

32 −

867311712d4x31x2 − 605842740x4

1x22d

2 + 669615660d4x1x2 −1135157976x4

1x32d

2−51018336x41d

2x42−44641044x6

1d2x3

2−184941468x41d

2x52+

165809592x61d

2x22 − 25509168x6

1x72 − 6377292x4

1x92 + 459165024x4

1x22 −

353939706x41d

2 + 459165024x21x

42 − 918330048x2

1x52 + 741891636x2

1x62 +

1128780684x21x

32d

2−765275040x21x

22d

2−605842740x21x

42d

2+172186884x22d

8−86093442x2

1d2 + 12754584x7

2x71 + 8503056x9

2x51 + 2125764x11

2 x31 +

1428513408d4x21x

32 + 86093442d4 + 9565938x6

2 − 25509168x72 +

34012224x82 + 191318760x2

1d4 + 459165024x2

1d6 + 229582512x3

2d2 −

563327460d2x62 − 353939706d2x4

2 + 593088156d2x52 − 1071385056x2

1x22d

4 −35075106d4x4

2 + 459165024d6x22 − 25509168x5

1x72 − 25509168x2x

111 −

146677716d4x32 + 6377292x1d

12 + 76527504d8x21x

32 − 51018336x3

1x92 −

25509168x1x112 − 44641044x6

1x62 + 5314410x8

1x62 − 6377292x9

1x42 +

6377292x101 x4

2 − 9565938d2x101 + 14880348d2x9

1 − 9565938d2x102 −

420901272d6x2 − 886443588d4x51x2 − 51018336x9

1 + 76527504d2x52x

61 −

337996476d4x52x

21 + 204073344x7

1x32d

2 + 9565938x101 x2

2 + 25509168x101 x2 +

2125764x111 x3

2 + 76527504x81x

32d

2 − 624974616x41d

4x32 − 57395628x4

1x2 −822670668x3

2x1d2− 1403004240d4x3

1x32 +280600848x7

1d2 +593088156x5

1d2 +

165809592x62x

21d

2 − 165809592x82d

2 − 146677716x31d

4 − 242337096d6x42 +

108413964x51d

4 − 89282088x31d

6 + 165809592d4x62 + 165809592x6

1d4 −

242337096x41d

6 +414523980x41x

22d

4 +414523980x21x

42d

4 +38263752x22x1d

4 +1428513408x3

1x22d

4 + 1256326524x1x42d

4 + 4251528x121 x2

2 − 6377292x121 x2 +

51018336x91x2 − 12754584x4

1x82 + 286978140x2

2x1d2 + 44641044x10

2 +1128780684x3

1x22d

2 + 612220032x1x42d

2 − 1135157976x31x

42d

2 −127545840x7

1x22 − 216827928x5

1x42 − 153055008x3

1x62 − 318864600x1x

82 −

114791256x82x

21 + 382637520x7

1x42 + 178564176x5

1x62 + 25509168x1x

102 −

886443588d4x52x1 − 8503056x11

1 + 1062882x114 − 280600848x1x

52d

2 +289103904x3

1x32d

2 − 51018336x92 − 8503056x11

2 + 153055008x52x1 +

707879412x31x

32 − 216827928x5

2x41 − 127545840x7

2x21 + 382637520x7

2x1 +374134464x5

2x31 + 374134464x3

2x51 − 242337096x7

1x32 − 688747536x5

1x52 −

242337096x31x

72 + 51018336x1x

92 + 170061120x2

1x92 + 280600848d2x7

2 −401769396x2

1x52d

2 − 86093442x22d

2 − 918330048x22x

51 − 714256704x4

2x31 −

229582512x62x1 +153055008d6 +105225318d8 +25509168d10−3188646d12−

2125764d14 + 200884698x21x

22d

8 + 44641044x1x62d

4 − 102036672d6x61x

22 −

184941468x51x

42d

2+31886460d8x41x2−146677716d8x2

1x2−178564176d8x1x32−

178564176d8x31x2+440033148d8x1x2+1192553604x4

1d4x4

2−51018336x41d

6x32


−51018336x61d

4x32 + 656861076x6

1d4x2

2 + 153055008x51x2 + 170061120x9

1x22−

401769396x51d

2x22 + 229582512x3

1d2 − 25509168x7

1 − 25509168x71x

52 +

1836660096x31x

52d

2 + 586710864x1x72d

2 + 1836660096x51d

2x32 −

51018336x91x

32 − 318864600x8

1x2 + 14880348x92d

2 − 235959804x72x

21d

2 +25509168x7

2x41d

2 + 127545840x92x1d

2 + 204073344x72x

31d

2 + 8503056x91x

52 +

25509168d4x72 + 153055008x5

1x52d

2 − 440033148d6x22x1 + 382637520x7

1x2 −12754584x8

1x42 + 25509168d2x10

1 x2 − 25509168d2x101 x2

2 − 34012224d2x61x

62 −

35075106d2x41x

82−63772920d8x5

1x2+182815704x31x

52d

6−114791256x71x

32d

4+23383404x9

1x32d

2+182815704x51d

6x32−125420076d8x3

1x32+656861076x6

2x21d

4+25509168d2x1x

102 + 76527504d2x3

1x82 + 76527504d2x5

1x62 + 25509168d2x7

1x42 +

95659380x41x

22d

8 +95659380x21x

42d

8− 146677716x22x1d

8 +76527504x31x

22d

8 +31886460x1x

42d

8 − 624974616x31x

42d

4 − 63772920d8x52x1 + 35075106x8

1d4 +

11691702x21d

12 + 19131876d8x42 + 11691702d12x2

2 + 85030560d8x32 +

22320522d4x101 + 6377292d4x9

1 + 22320522d4x102 − 28697814d10x4

2 −25509168d10x3

2 + 3188646d10x22 − 44641044d10x2 − 44641044d10x1 +

3188646d10x21 − 25509168d10x3

1 − 28697814d10x41 + 6377292d12x2 +

25509168x71d

4 + 35075106x82d

4 + 85030560x31d

8 + 38263752x51d

8 +41452398d8x6

2 + 41452398x61d

8 − 89282088d6x32 − 72275976d6x5

1 −44641044d6x6

1 − 72275976d6x52 − 7440174d2x12

1 + 1062882x124 +

38263752x71x

52d

2−25509168d2x21x

102 −127545840d6x4

1x42−631351908d6x4

1x22−

631351908d6x21x

42 − 102036672d6x2

1x62 + 299732724d6x4

1x2 −76527504d6x2

2x51− 51018336d6x4

2x31 +280600848d6x5

1x2 +63772920d6x71x2 +

89282088d4x41x

62 − 31886460d4x9

1x2 + 25509168d4x71x

22 − 51018336d4x3

1x62 −

31886460d4x1x82 +66961566d4x8

2x21− 31886460d4x8

1x2− 255091680d4x71x2 +

31886460d10x1x32 − 25509168d10x1x

22 + 51018336d10x1x2 −

51018336d10x21x

22−25509168d10x2

1x2+31886460d10x31x2−6377292d12x1x2−

191318760d6x31x2−440033148d6x2

1x2+44641044d4x61x2+66961566d4x8

1x22+

89282088d4x61x

42 − 38263752d6x8

1 − 44641044d6x62 − 38263752d6x8

2 −25509168d6x7

1 − 25509168x72d

6 + 6377292x92d

4 − 248714388d8x2 −7440174d2x12

2 + 584585100d6x32x

21− 191318760d6x3

2x1 + 584585100x31d

6x22 +

299732724x1d6x4

2−337996476x51d

4x22−35075106d2x8

1x42−382637520x3

1x52d

4−255091680x1x

72d

4 + 408146688x31d

6x32 + 280600848x1d

6x52 −

382637520x51d

4x32 + 25509168x7

2x21d

4− 76527504x52d

6x21 + 6377292x11

2 x1d2 +

38263752x72x

51d

2 +23383404x92x

31d

2− 31886460x92x1d

4− 114791256x72x

31d

4 +63772920x7

2d6x1−165809592x5

1x52d

4−567578988d2x41x

62+612220032d2x4

1x2+127545840d2x9

1x2−235959804d2x71x

22−44641044d2x3

1x62−178564176d2x1x

82−

239148450d2x82x

21 − 280600848d2x5

1x2 − 178564176d2x81x2 +

586710864d2x71x2+765275040d6x1x2−229582512d4x2+286978140d2x2

1x2−248714388d2x6

1x2− 239148450d2x81x

22− 567578988d2x6

1x42− 6377292x1x

122 +

4251528x21x

122 − 25509168x3

1x102 + 6377292x4

1x102 − 38263752x5

1x82 +

5314410x61x

82 − 25509168x7

1x62 − 822670668d2x3

1x2 + 9565938x21x

102


P(t) = ( 3 t1+t3 , 3 t2

1+t3 ) and N (t) = ( 3 t (−2+t3)(1+t3)2 , −3 (−1+2 t3)

(1+t3)2 ).

498 JUANA SENDRA

GP is defined by x22 − 2 x2

2 x31 − 1 + 2 x3

1 + 4 x2 x1 − 2 x2 x41.

–10

–5

5

10

y2

–10 –5 5 10

y1

Figure 9. Folium of Descartes (untraced curve) and the Offsetsof the Folium of Descartes for several distances.

Example 9. (Offset of the Tacnode). Let C be the Tacnode defined by

2 y41 − 3 y2

1 y2 + y22 − 2 y3

2 + y42 .

1. Implicit equation: The implicit equation of the generic offset is a polynomialof degree 20 with 493 terms (we do not write here for space reasons)


P(t) = ( 18 t4+21 t3−7 t−218 t4+48 t3+64 t2+40 t+9 , 36 t4+84 t3+73 t2+28 t+4

18 t4+48 t3+64 t2+40 t+9 ) and

N (t) = (−2 (108 t6+990 t5+2340 t4+2520 t3+1410 t2+401 t+46)(18 t4+48 t3+64 t2+40 t+9)2

,

486 t6+2304 t5+3882 t4+3144 t3+1303 t2+256 t+17(18 t4+48 t3+64 t2+40 t+9)2

).

GP is defined by486x2

2x61+2304x2

2x51+3882x2

2x41+3144x2

2x31+1303x2

2x21+256x2

2x1+17x22−

486x61 − 2304x5

1 − 3882x41 − 3144x3

1 − 1303x21 − 256x1 − 17 + 432x2x

61 +

3960x2x51 + 9360x2x

41 + 10080x2x

31 + 5640x2x

21 + 1604x2x1 + 184x2..

Example 10. (Offset of the Conchoid of de Sluze). Let C be the Conchoid of deSluze defined by

(y1 − 1) (y21 + y2

2) + y21 .


–4

–2

0

2

4

6

8

y2

–6 –4 –2 2 4 6

y1

Figure 10. Tacnnode (untraced curve) and the Offsets of theTacnode for several distances.

1. Implicit equation: The implicit equation of the generic offset is6 d4 x1

4 − d2 x26 − 32 d2 x2

2 + 3 x16 x2

2 + 3 x14 x2

4 + x12 x2

6 − 12 x24 x1

3 −2 x2

6 x1−18 x22 x1

5+x18−18 d4 x2

2 x1−9 d2 x14 x2

2−6 d2 x12 x2

4−32 x13 x2

2+9 d4 x2

2 x12 + 16 x1

6 + d8 − 3 d6 x22 + 3 d4 x2

4 − 40 x1 x24 + 16 x2

4 + x26 −

4 d6 x12 + 48 x1

4 x22 + 33 x1

2 x24 + 24 d2 x1

5 + 12 d4 x22 − 60 d2 x2

2 x12 −

4 x16 d2+16 d4−8 x1

7+8 d6 x1−24 d4 x13+8 x2

2 d2 x1−32 x13 d2+32 d4 x1−

24 d2 x14 − 21 d2 x2

4 + 8 d6 + 12 x24 d2 x1 + 36 x2

2 x13 d2


(− t (d t5 − t5 − 6 t4 + 6 d t4 + 15 d t3 − 14 t3 + 20 d t2 − 16 t2 + 15 d t− 8 t + 6 d)

2 + 15 t4 + 20 t3 + 15 t2 + 6 t5 + 6 t + t6,

− t8 + 8 t7 + 26 t6 + 44 t5 + 2 d t4 + 40 t4 + 16 t3 + 8 d t3 + 12 d t2 + 8 d t + 2 d

(2 + 15 t4 + 20 t3 + 15 t2 + 6 t5 + 6 t + t6) (1 + t)

)


P(t) = (1− 11+t2

, (1− 11+t2

) t) and N (t) = (−t2 3+t2

(1+t2)2, 2 t

(1+t2)2).

GP is defined by x22 − 1 + 3 x2 x1 + x2 x3

1.

R = (−t (t+2)t+1

, −13 t2+3 t+1+t3

).

500 JUANA SENDRA

–8

–6

–4

–2

2

4

6

8

y2

–4 –2 2 4 6

y1

Figure 11. Conchoid of de Sluze (untraced curve) and the Off-sets of Conchoid of de Sluze for several distances.

Example 11. (Offset of the Epitrochoid). Let C be the defined by

y42 + 2 y2

1 y22 − 34 y2

2 + y41 − 34 y2

1 + 96 y1 − 63.

1. Implicit equation: The implicit equation of the generic offset is63504−288792x1+72x1x

62d

2−64800d2−72d6x22x

21+542025x2

1+104265x22−

316128x1x22 +16758x2

2d4− 537312x3

1 +294652x41 +349688x2

1x22 +55036x4

2 +916x4

1d4+178632x1d

2+216x1d6−14472x1d

4+9x21d

8−80400x51−160800x3

1x22−

80400x1x42 + 9x10

1 + 3574x61 − 596x8

1 − 2384x61x

22 − 3576x4

1x42 + 45x8

1x22 +

90x61x

42+90x4

1x62+436x6

1d2−36x8

1d2+1308x4

1x22d

2−216x41d

2x42−144x6

1d2x2

2+15330x4

1x22−34460x4

1d2+19938x2

1x42−2384x2

1x62−57400x2

1x22d

2+1308x21x

42d

2+9x2

2d8 − 200916x2

1d2 + 1296d4 + 8182x6

2 − 596x82 + 17334x2

1d4 − 756x2

1d6 +

436d2x62− 22940d2x4

2 + 1832x21x

22d

4 +916d4x42− 756d6x2

2− 24x91 +72x7

1d2 +

3000x51d

2 − 144x62x

21d

2 − 36x82d

2 − 6576x31d

4 − 36d6x42 − 72x5

1d4 + 24x3

1d6 +

54d4x62+54x6

1d4−36x4

1d6+162x4

1x22d

4+162x21x

42d

4−6576x22x1d

4−144x31x

22d

4−72x1x

42d

4 +117048x22x1d

2 +9x102 +6000x3

1x22d

2 +3000x1x42d

2 +216x31x

42d

2−96x7

1x22 − 144x5

1x42 − 96x3

1x62 − 24x1x

82 + 45x8

2x21 − 78804x2

2d2 + 10080x2

2x51 +

10080x42x

31 + 3360x6

2x1 + 216x51d

2x22 + 117048x3

1d2 + 3360x7

1 + 24d6x22x1

2. Rationality character: irreducible and non rational


3. Details of the computation:P(t) = (−7 t4+288 t2+256

t4+32 t2+256 , −80 t3+256 tt4+32 t2+256 ) and

N (t) = ( −16 (5 t4−288 t2+256)(t2+16) (t4+32 t2+256) ,

−1024 t (t2−8)(t2+16) (t4+32 t2+256) ).

GP is defined by32 x2

2 x31 − 256 x2

2 x1 − 32 x31 + 256 x1 − 5 x2 x4

1 + 288 x2 x21 − 256 x2.

–8

–6

–4

–2

2

4

6

8

y2

–8 –6 –4 –2 2 4 6 8

y1

Figure 12. Epitrochoid (untraced curve) and the Offsets of theEpitrochoid for several distances.

Example 12. (Offset of the Ramphoid Cusp). Let C be the Ramphoid Cusp definedby

y41 + y2

1 y22 − 2 y2

1 y2 − y1 y22 + y2

2 .

1. Implicit equation: The implicit equation of the generic offset is a polynomialof degree 20 with 933 terms (we do not write here for space reasons)


P(t) = ( t2−2 t+12 t2+2 , t4−4 t3+6 t2−4 t+1

6 t4+8 t2+2 ) and

N (t) = (−2 (−7 t5−2 t3+3 t6+5 t4+t2+t−1)(3 t4+4 t2+1)2 , t2−1

(t2+1)2 ).GP is defined by9 x2

2 x51 − 9 x5

1 + 12 x2 x51 − 16 x2 x4

1 + 9 x22 x4

1 − 9 x41 + 4 x2 x3

1 − 6 x31 +

6 x22 x3

1 − 4 x2 x21 − 6 x2

1 + 6 x22 x2

1 + x22 x1 − x1 + 4 x2 + x2

2 − 1.

Example 13. (Offset of the Lemniscata of Bernoulli). Let C be the Lemniscata ofBernoulli defined by

(y21 + y2

2)2 − 2a2(y21 − y2

2).

502 JUANA SENDRA

–6

–4

–2

0

2

4

6

y[2]

–4 –2 2 4 6

y[1]

Figure 13. Ramphoid Cusp (untraced curve) and the Offsets ofthe Ramphoid Cusp for several distances.

1. Implicit equation: The implicit equation of the generic offset is

−6 d10 x12−20 d6 x1

6+13 a4 x18+16 a8 d4+4 a8 x1

4−12 a6 x16+6 a2 x2

10+44 a6 x1

4 d2−16 a8 x12 d2−46 a4 d2 x1

6+24 a2 d2 x18−18 a2 x2

2 x18+18 a2 x2

8 x12+

12 a2 x26 x1

4− 12 a2 x24 x1

6 + 12 a6 x26 + 4 a8 x2

4 + 13 a4 x28− 6 d8 a2 x1

2−36 d4 x1

6 a2−4 d6 a4 x12−36 a2 x2

2 d4 x14+6 x1

10 x22+20x1

6 x26−30 x1

2 x28 d2−

60 x16 x2

4 d2−60 x14 x2

6 d2−6 x110 a2−30 x1

8 d2 x22 +x2

12 +4 a4 x22 x1

6−18 a4 x2

4 x14+4 a4 x2

6 x12−16 a8 x2

2 d2−4 a4 x22 d6−8 a8 x2

2 x12−20 a6 x2

4 x12+

20 a6 x22 x1

4 + 24 d6 x14 a2 − 40 d4 a6 x1

2 + 45 d4 a4 x14 + 90 x1

4 d4 x24 +

60 x26 x1

2 d4−6 x210 d2+x1

12−42 a4 x24 d2 x1

2−42 a4 x22 d2 x1

4+36 a2 x24 d4 x1

2+42 a4 x2

2 d4 x12 − 24 a2 x2

8 d2 + 36 a2 x26 d4 + 45 a4 x2

4 d4 − 24 a2 x24 d6 +

40 a6 x22 d4+48 a2 x2

2 d2 x16+d12−48 a2 x2

6 d2 x12+15 d4 x1

8−20 d6 x26+

60 x16 d4 x2

2 + 15 d8 x14 + 6 x1

2 x210 + 15 x1

4 x28 + 15 d4 x2

8 + 6 a2 x22 d8 −

44 a6 x24 d2−46 a4 x2

6 d2+30 d8 x22 x1

2+15 x24 x1

8+15 d8 x24−6 d10 x2

2−60 d6 x2

4 x12 − 8 d8 a4 − 60 d6 x1

4 x22 − 6 d2 x1

10


P(t) = (a√

2 (t+t3)1+t4 , a

√2 (t−t3)1+t4 ) and

N (t) = (−a√

2(1−3 t4−3 t2+t6)(1+t4)2

,−a√

2(−1+3 t4−3 t2+t6)(1+t4)2

).GP is defined by x2

2−3x22x

41 +3x2

2x21−x6

1x22−1+3x4

1−3x21 +x6

1 +2x2−6x2x

41 − 6x2x

21 + 2x6

1x2.


–6

–4

–2

0

2

4

6

y2

–10 –5 5 10

y1

Figure 14. Lemniscata of Bernoulli (untraced curve) and theOffsets of the Lemniscata of Bernoulli for several distances.

Example 14. (Offset of Cuspidal Curves). Let F be a family of rational affinecurves, called Cuspidal Curves, defined by

bn−11 (y1 − a0)n − an

1 (y2 − b0)n−1, a1 6= 0, b1 6= 0, n > 1


(P(ϕ) +

2 d t

(n− 1) a1 ϕn−2 (t2 + 1)N (ϕ)

), with ϕ =

a1(1− n)(t2 − 1)

2b1n t


P(t) = (a1tn−1 + a0, b1t

n + b0) and N (t) = (−nb1tn−1, (n− 1)a1t

n−2).GP is defined by 2 n b1 x1 x2 + (n− 1) a1 (x2

2 − 1).

R = (a1(1−n)(t2−1)2b1n t

, t).

4. Remark: Parabolas {(t, kt2)}, and the cubics {(t2, kt3)} belong to this family.5. Particular case of a cuspidal curve of degree 5: we consider the cubics y5

1 − y42

paramentrized by {(t4, t5)}. The implicit equation of the generic offset is

504 JUANA SENDRA

−300781250x42x

51d

2 + 65536x82 − 1600000x8

2x1 + 9765625x82x

21 − 262144x6

2d2 +

3200000x42x

61−19531250x6

2x51 +6272000x6

2x1d2−1228800x4

2x31d

2−19531250x42x

71 +

9765625x102 + 97656250x4

2x31d

4− 131072x51d

4− 41015625x82d

2 + 30200000x42x

41d

2−36250000x6

2x21d

2 + 393216x42d

4 − 50200000x42d

6 − 195312500x62x

31d

2 +66250000x6

2d4 − 97656250x6

2d4x1 − 58593750x10

1 d2 + 64800000x42x

21d

4 −9792000x4

2d4x1 + 9765625d12 − 1600000d10 + 409600x3

1d6 − 2048000x1d

8 +65536x10

1 + 19328000x61x

22d

2 + 65536d8 + 9765625x101 x2

2 − 409600x81d

2 +9765625x12

1 −786432x51x

22d

2−1600000x111 +222656250x4

2d6x1−48828125x8

1x22d

2 +48828125x2

2x21d

8 − 150000000x22x1d

8 + 97656250x61x

22d

4 − 97656250x41x

22d

6 +819200x3

1x22d

4 − 18400000x41x

22d

4 − 120000000x71x

22d

2 + 90000000x51x

22d

4 +180000000x3

1x22d

6 − 59200000x21x

22d

6 + 7168000x1x22d

6 + 4288000x61d

4 −11200000x4

1d6 + 20800000x2

1d8 + 9800000x9

1d2 − 44800000x7

1d4 + 91600000x5

1d6 −

80000000x31d

8 + 25000000x1d10 − 58593750x2

1d10 − 195312500x6

1d6 +

146484375x41d

8 + 16800000x22d

8 − 262144x22d

6 + 146484375x81d

4 − 9765625x22d

10 −131072x4

2x51,

and the generic offset can be parametrized as

((16 t8 − 32 t6 + 32 t2 − 16 + 625 d t4) (t2 − 1)

625 t4 (t2 + 1),

2 (−16 t12 + 64 t10 − 80 t8 + 80 t4 − 64 t2 + 16 + 3125 d t6)

3125 t5 (t2 + 1))

–10

–5

5

10

y[2]

–4 –2 2 4 6 8 10

y[1]

Figure 15. Cuspidal curve (t4, t5) (untraced curve) and the Off-sets of the curve for several distances.

References

[1] J. G. Alcazar. Good Global Behavior of Offsets to Plane Algebraic Curves. Journal ofSymbolic Computation 43, 659–680, 2008.

[2] J. G. Alcazar. Good Local Behavior of Offsets to Regularly Parametrized Surfaces. Journalof Symbolic Computation 43, 845–857, 2008.


[3] J. G. Alcazar, R. Sendra. Local Shape of Offsets to Algebraic Curves. Journal of SymbolicComputation 42, 338–351, 2007.

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1995.[15] M. Peternell. Rational two-parameter families of spheres and rational offset surfaces.

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Models and Image Processing 58, 544–552, 1996.[18] H. Pottmann. Rational Curves and Surfaces with Rational Offsets. Comput. Aided Geom.

Des. 12, 175–192, 1995.[19] G. Salmon. A Treatise an the Higher Plane Curves. Chelsea, New York ,1960.[20] E. Snapper, E. Troyer. Metric Affine Geometry. Academic Press, 1971.[21] F. San Segundo, J. R. Sendra. Degree Formulae for Offset Curves. Journal of Pure and

Applied Algebra 195, 301–335, 2005.[22] F. San Segundo, J. R. Sendra. Partial Degree Formulae for Plane Offset Curves. Journal

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Departamento de Matematica Aplicada a la Ingenierıa Tecnica de Telecomuni-cacion, Universidad Politecnica de Madrid, Spain

E-mail address: [email protected]

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