arXiv:1707.06321v2 [math.DG] 9 Aug 2017

Noname manuscript No. (will be inserted by the editor)

Rotation minimizing frames and spherical curves in simply isotropic and semi-isotropic 3-spaces

arXiv:1707.06321v2 [math.DG] 9 Aug 2017

Luiz C. B. da Silva

Received: date / Accepted: date

Abstract In this work we are interested in the differential geometry of curves in simply isotropic and semi-isotropic 3-spaces. These are examples of CayleyKlein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentz-Minkowski geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of semi-isotropic space, we also discuss on the distinct approaches for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of hyperbolic (or double) numbers, a complex-like system where the square of the imaginary unit is +1. Finally, we also show how to relate isotropic RM and Frenet frames through the use of Galilean trigonometric functions and dual numbers, a complex-like system where the square of the imaginary unit is zero. Keywords Non-Euclidean geometry · Cayley-Klein geometry · isotropic space · semi-isotropic space · spherical curve · plane curve Mathematics Subject Classification (2010) 51N25 · 53A20 · 53A35 · 53A55 · 53B30 L. C. B. da Silva Departamento de Matem´ atica, Universidade Federal de Pernambuco, 50670-901, Recife, Pernambuco, Brazil E-mail: [email protected]

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Luiz C. B. da Silva

1 Introduction The three dimensional (3d) simply isotropic I3 and semi-isotropic SI3 3-spaces are examples of 3d Cayley-Klein (CK) geometries [12,19,23,30]. The basic idea behind a CK geometry is the study of those properties in projective space P3 that preserves a certain configuration, the so called absolute figure. In the spirit of Klein “Erlanger Program” [4,14], a CK geometry is the study of those properties invariant by the action of the subgroup of projectivities that fix the absolute figure. For example, the Euclidean (Minkowski) space E3 (E31 ) is modeled through an absolute figure given in homogeneous coordinates by a plane at infinity identified with x0 = 0 and a non-degenerate quadric of index zero (index one) identified with x20 + · · · + x23 = 0 (x20 + x21 + x22 − x23 = 0, respectively) [12]. In our case of interest, i.e., isotropic space geometries, the absolute figure is given by a plane at infinity, identified with x0 = 0, and a degenerate quadric of index zero or one, identified with x20 + x21 + δ x22 = 0: δ = 1 for the simply isotropic figure and δ = −1 for the semi-isotropic one. Besides its mathematical interest [1,2,13,26,33], see also [24] and references therein, isotropic geometry also finds applications in economics [3,8], elasticity [21], and in image processing and shape interrogation [15,22], just to name a few. Another stimulus to study isotropic geometries comes from the problem of characterizing curves on level set surfaces Σ = F −1 (c). Indeed, an idea to approach such a problem is the introduction of a metric induced by Hess F [10] and, since it may fail to be non-degenerate, one may be led to the study of an isotropic geometry: e.g., for cylindrical quadrics, i.e., a translation surface with a conic as generating curve, the natural geometric framework is that of an isotropic geometry. In fact, here one basically has Hess F = diag(1, δ, 0), which leads to the geometry of isotropic spaces [12,27,31] because such a Hessian may induce in R3 a degenerate metric: simply isotropic space I3 if δ = 1 [24,29]; semi-isotropic space SI3 if δ = −1 [2]; and doubly isotropic space I32 if δ = 0 [7]. Motived by the success of Rotation Minimizing (RM) frames in the study of spherical curves in both E3 and E31 [6,10,11,20], in this work we develop the fundamentals of RM frames in simply isotropic and semi-isotropic spaces, which in combination with an adequate manipulation of osculating spheres allow us to prove that spherical curves can be characterized through a linear equation involving the coefficients that dictate the frame motion1 (in analogy with what happens in both E3 and E31 [6,10]). In addition, for the case of semi-isotropic space we discuss in detail the construction of spheres, moving frames along curves, and the distinct approaches to the study of semi-isotropic space as a CK geometry (with the help of hyperbolic numbers [5,32], we are able to prove that the available approaches are all equivalent). Finally, we also show how to relate RM and Frenet frames, in both I3 and SI3 , by using the Galilean trigonometric functions [32] and dual numbers [23,32]. 1 The characterization of isotropic spherical curves via a Frenet frame is made through a differential equation involving curvature and torsion [29], see also Eq. (7.36) in [24], p. 128.

RM frames and spherical curves in simply isotropic and semi-isotropic 3-spaces

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The remaining of this work is divided as follows. In section 2 we review the concept of RM frames in Euclidean space and its spherical curves. In section 3 we introduce some terminology related to simply isotropic space I3 and, in section 4, we discuss how to introduce moving frames along simply isotropic curves. In section 5 we then study simply isotropic spheres and the characterization of spherical curves. In section 6 we turn our attention to the semi-isotropic space SI3 . In section 7 and 8 we study semi-isotropic spheres and moving frames along semi-isotropic curves, respectively. Finally, in section 9 we present a characterization of semi-isotropic spherical curves. 2 Preliminaries: rotation minimizing frames and spherical curves in Euclidean space Let us denote by E3 the 3d Euclidean space, i.e., R3 equipped with the standard Euclidean metric h·, ·i. The usual way to introduce a moving frame along it is by means of the Frenet frame [16,17]. However, we can also consider any other adapted orthonormal moving frame. For the Frenet frame the equations of motion are given in terms of the curvature function κ and the torsion τ : t′ = κn; n′ = −κt + τ b; and b′ = −τ n. On the other hand, by introducing the notion of a rotation minimizing vector field2 , Bishop considered an orthonormal adapted moving frame {t, n1 , n2 }, where ni ⊥ t, whose equation of motion is [6]      t t 0 κ1 κ2 d    n1 = −κ1 0 0   n1  . (1) ds −κ2 0 0 n2 n2

The basic idea here is that ni rotates only the necessary amount to remain normal to the tangent (then justifying the terminology). In addition, the coefficients κ1 and κ2 relate with the curvature and torsion according to [6]   κ1 = κ cos θ κ2 = κ sin θ . (2)  θ′ = τ

The above relations show that RM frames are not uniquely defined. Indeed, any rotation of n1 and n2 still gives two relatively parallel fields, i.e., there is an ambiguity associated with the group SO(2) acting on the normal plane which specifies an RM frame up to an additive constant3 . Finally, of great interest to us, is that RM frames allow for a simple characterization of spherical curves4 : 2 In fact, Bishop called them relatively parallel moving frames [6]: see da Silva [9] for a discussion concerning terminology. 3 It is possible to show that the prescription of κ , κ still determines a curve up to rigid 1 2 motions and that an RM frame can be globally defined even if the curve is has a curvature κ that vanishes at some points [6]. 4 This characterization also works for curves in Lorentz-Minkowski space E3 , see [10] and 1 references therein.

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Theorem 1 (Bishop [6]) A C 2 regular curve in E3 lies on a sphere if and only if its normal development, i.e., the curve (κ1 (s), κ2 (s)), lies on a line not passing through the origin: the distance of this line from the origin, d, and the radius of the sphere, r, are reciprocals, i.e., r = d−1 . Finally, straight lines passing through the origin characterize plane curves which are not spherical. In the following we shall extend this formalism in order to present a way of building RM frames along curves in both simply isotropic I3 and semiisotropic SI3 3-spaces and then apply them to furnish a unified approach to the characterization of isotropic spherical curves. And, in addition, by employing dual numbers and Galilean trigonometric functions, we will show how to relate (i) a Frenet frame to an RM frame and (ii) an RM frame with another RM frame. 3 Differential geometry in simply isotropic space In this section we introduce some basic terminology in simply isotropic space. We refer the reader to Sachs’ monograph [24] for more details. In the spirit of Klein’s Erlangen Program, simply isotropic geometry is the study of those properties in R3 invariant by the action of the 6-parameter group B6  ¯ = a + x cos φ − y sin φ x y¯ = b + x sin φ + y cos φ , (3)  z¯ = c + c1 x + c2 y + z

where a, b, c, c1 , c2 , φ ∈ R. So, B6 is the group of rigid motions in I3 . Observe that on the xy plane this geometry looks exactly like the plane Euclidean geometry E2 . The projection of a vector u = (u1 , u2 , u3 ) ∈ I3 on the ˜ = (u1 , u2 , 0). xy plane is called the top view of u and we shall denote it by u The top view concept plays a fundamental role in the simply isotropic space I3 , since the z-direction is preserved under the action of B6 (5 ). A line with this direction is called an isotropic line and a plane that contains an isotropic line is said to be an isotropic plane. One may introduce a simply isotropic inner product between two vectors u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ) as hu, viz = u1 v1 + u2 v2 , from which we define a simply isotropic distance as usual6 : p dz (A, B) = hB − A, B − Aiz .

(4)

(5)

5 Maybe, it would be interesting to mention that from a physical perspective such a space is not isotropic. Indeed, the z-direction is a distinguished direction and gives rise to an anisotropy (in the physics jargon). So, anisotropic geometry would be a better name. Anyway, this is a well established nomenclature and we will not attempt to change it. 6 The index z is here just to emphasize that z is the isotropic (degenerate) direction. Note, in addition, that the isotropic inner product in fact induces a semi-distance in R3 , since points in an isotropic line have zero distance.


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Note that the inner product and distance above are just the plane Euclidean counterparts of the top views. In addition, since the isotropic metric is degenerate, the distance from (u1 , u2 , u3 ) to (u1 , u2 , v3 ) is zero. In such cases, one may define a codistance by cdz (A, B) = |b3 − a3 | (the codistance is preserved by B6 and then is an isotropic invariant: it may be used to define angles involving isotropic lines and planes [22,24]). 4 Moving frames along curves in simply isotropic space Now we introduce some terminology related to curves. A regular curve α : I → def ˜ ′ (s)k = 1. I3 , i.e., α′ 6= 0, is parametrized by an arc-length s if kα′ (s)kz = kα In the following we assume that all curves are parametrized by an arc-length s (in particular, this excludes the possibility of an isotropic velocity vector). In addition, a point α(s0 ) in which {α′ (s0 ), α′′ (s0 )} is linearly dependent is an inflection point and a regular unit speed curve α(s) = (x(s), y(s), z(s)) with no inflection point is called an admissible curve if x′ y ′′ − x′′ y ′ 6= 0. Remark 1 The admissible condition implies that the osculating planes, i.e., the planes that have a contact of order 2 with the reference curve7 , can not be isotropic. Moreover, the only curves with x′ y ′′ − x′′ y ′ ≡ 0 are precisely the isotropic lines [24]. 4.1 Simply isotropic Frenet frame The (isotropic) unit tangent, principal normal, and curvature function are defined as usual t(s) = α′ (s), n(s) =

t′ (s) , and κ(s) = kt′ (s)kz = k˜t′ (s)k, κ(s)

(6)

respectively. As usually happens in isotropic geometry, the curvature κ is just the curvature function of its top view α ˜ and then we may write κ(s) = (x′ y ′′ − ′′ ′ x y )(s). To complete the moving trihedron, we define the binormal vector as the (co)unit vector b = (0, 0, 1) in the isotropic direction. The Frenet frame {t(s), n(s), b(s)} is linearly independent for each s: det(t, n, b) =

1 ′ ′′ (x y − x′′ y ′ ) = 1 . κ

(7)

The Frenet equations corresponding to the isotropic Frenet frame {t, n, b} can be written as      t 0 κ 0 t d    n = −κ 0 τ   n  , (8) ds b 0 0 0 b 7 For a level set surface Σ = G−1 (c), a contact of order k with α at α(s ) is equivalent 0 to say that β (i) (s0 ) = 0 (1 ≤ i ≤ k), where β = G ◦ α and c = β(s0 ) = α(s0 ) [16].

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where τ is the (isotropic) torsion [24], p. 110: τ=

det(˜ α′ , α ˜′′ ) det(α′ , α′′ , α′′′ ) ; κ= p . ′ ′′ det(˜ α ,α ˜ ) hα′ , α′ iz 3

(9)

The above expressions for the torsion and curvature are also valid for any generic regular parameter for α and, in addition, they are invariant by rigid motions in I3 . Contrary to the Euclidean space E3 , we can not define the torsion through the derivative of the binormal vector. However, remembering that the idea behind such a definition in E3 is that one can measure the variation of the osculating plane by measuring b′ , we may ask if τ ≡ 0 still characterizes plane curves in I3 . It can be shown that the isotropic torsion is directly associated with the velocity of variation of the osculating plane, see [24], pp. 112-113, and that an admissible curve lies on a non-isotropic plane if and only if its torsion vanishes identically. Observe in addition that, contrary to the isotropic curvature, the torsion is not defined as the torsion of the top view (this would result in τ = 0). The isotropic torsion is an intermediate concept depending on its top view behavior and on how much the curve leaves the plane spanned by α′ and α′′ . 4.2 Rotation minimizing frames in simply isotropic space Let α : I 7→ I3 be an admissible curve parametrized by arc-length s. A normal vector field v is a simply isotropic RM vector field if v′ = µ t, for some function µ. We easily see that the binormal b is an RM vector field: b′ = 0. Except for plane curves, the principal normal fails to be RM: n′ = −κt + τ b. In order to introduce an RM frame in I3 , we just need to look for an RM vector field v in substitution to the principal normal. If v is a normal vector, we may write v = µn + νb , (10) where we suppose µ 6= 0 (otherwise v is just a multiple of b). Now, imposing hv, viz = 1 implies that 1 = hv, viz = µ2 hn, niz ⇒ µ = ±1 .

(11)

The derivative of v is v′ = −µκ t + (µτ + ν ′ ) b . So, if we assume v to be an RM vector field, it follows that Z v′ k t ⇒ ν = −µ τ + constant .

(12)

(13)

Finally, if we impose that {t, v, b} has the same orientation as {t, n, b}, we conclude that 1 = det(t, v, b) = det(t, µn, b) = µ .

(14)


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Remark 2 Using the definition of the Galilean trigonometric functions, i.e., cosg φ = 1 and sing φ = φ [32], we can write an RM vector field v in terms of the Frenet frame as ( v = cosg(θ) n − sing(θ) b . (15) θ′ = τ This is analogous to RM frames in both Euclidean and Lorentz-Minkowski spaces [6,20]. Theorem 2 Let n1 be a unit normal vector field along α : I → I3 . If n1 is RM and {t, n1 , b} has the same orientation as the Frenet frame, then Z s n1 (s) = n(s) − (16) τ (x)dx + τ0 b(s) , s0

Rs where τ0 is a constant and we shall define sing θ(s) = θ(s) = s0 τ (x)dx + τ0 . In addition, a rotation minimizing frame {t, n1 , n2 = b} in isotropic space I3 satisfies      t 0 κ1 κ2 t d    n1 = −κ1 0 0   n1  , (17) ds n2 n2 0 0 0 where the natural curvatures are κ1 = κ and κ2 = κ θ.

Proof The expression for n1 follows from the discussion above. On the other hand, we have for the derivative of n1 and n2 = b n′1 = −κ t + τ b − τ b = −κ t , n′2 = 0.

(18)

Finally, taking into account that n = n1 + θ b, we find t′ = κ n = κ n1 + κθ b .

(19)

From the equalities above we find the desired equations of motion for the RM trihedron {t, n1 , b}. ⊓ ⊔ Using the definition for the Galilean trigonometric functions [32], we can relate the RM frame curvatures κ1 , κ2 to the Frenet ones κ, τ according to    κ1 (s) = κ(s) cosg θ(s) κ2 (s) = κ(s) sing θ(s) . (20)   ′ θ (s) = τ (s)

¯ 1 , b} differ only by This also shows that two RM frames {t, n1 , b} and {t, n an additive constant, θ¯ = θ + θ0 . This issue can be further clarified with the help of the ring dual numbers [23,32] in isotropic plane I2 [23] (note that the normal plane is always isotropic, since b is in it).

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Fig. 1 Unit length and zero divisors hyperbolic (or double) and dual numbers: (a) Dual numbers and their zero divisors, i.e., nonzero numbers p, q such that p q = 0 (dotted black line), and their unit length numbers (solid blue line). Note that a unit length dual number may be represented as (±1, θ) 7→ ±1 + φ ε; (b) Hyperbolic numbers and their zero divisors (dotted black line), and their unit length numbers (solid blue and dashed red lines). Note that there exist four types of unit length hyperbolic numbers, the dashed red lines correspond to (± cosh φ, sinh φ) 7→ ± cosh φ + h sinh φ and the solid blue lines correspond to (sinh φ, ± cosh) 7→ sinh φ ± h cosh φ. The right dashed red line in particular correspond to O ++ (2), the group of vector space and time orientations preserving hyperbolic rotations (see subsection 6.1).

The standard way to write a dual number is p = p1 + p2 ε, where the dual imaginary unit ε satisfies ε2 = 0. The real and imaginary parts are p1 = Re(p) and p2 = Im(p), respectively . The arithmetic operations are defined as (p1 + p2 ε) + (q1 + q2 ε) = (p1 + q1 ) + (p2 + q2 ) ε . (21) (p1 + p2 ε)(q1 + q2 ε) = (p1 q1 ) + (p1 q2 + q1 p2 ) ε The modulus in isotropic plane I2 may be expressed as |p1 + p2 ε| = |p1 |, which may be seen as being induced by the degenerate metric in I2 : u · I2 v = u1 v1 . In analogy with E2 , where we can use unit complex number to describe rotations8 , we can use a unit dual number p = 1 + φ ε = cosgφ + ε singφ in order to describe (Galilean) rotations in I2 through a right multiplication, i.e., p 7→ a p (see Fig. 1). Indeed, if we identify a point (x1 , y1 ) ∈ I2 with the dual number x1 + y1 ε, it then follows that a rigid motion in I2 may be written as a a cosgφ 0 1 0 x1 x1 x2 + + , (22) = = b b y1 y1 singφ cosgφ φ 1 y2 where we used the following linear (matrix) representation for the dual numbers p1 0 . (23) p1 + p2 ε 7→ p2 p1 8 The same applies in Lorentz-Minkowski plane E2 through the use of hyperbolic numbers 1 [5]: see subsection 6.1 below.


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In short, with the help of the ring of dual numbers, we can interpret an isotropic RM frame as a frame that minimizes isotropic (or Galilean) rotations.

4.3 Moving bivectors in simply isotropic space In I3 it is not possible to define a vector product with the same invariance significance as in Euclidean space. However, one can still do some interesting investigations by employing in I3 the usual vector product ×e from Euclidean space E3 . Associated with the isotropic Frenet frame, one introduces a (moving) bivector frame as [24]   T = n ×e b H = b ×e t , (24)  B = t ×e n which satisfies the equation      T 0 κ 0 T d    H = −κ 0 0   H  ds B 0 −τ 0 B

(25)

and [24], Eqs. (7.43a-c), p. 130,

det(T , H, B) = det(t, n, b) = 1 and T = ˜t.

(26)

Analogously, we shall introduce the following (moving) RM bivector frame associated with an RM frame {t, n1 , n2 = b}   T = n1 × e n2 = n × e b N1 = n2 ×e t . (27)  N2 = t ×e n1 Proposition 1 The moving frame {T , N1 , N2 } forms a basis for R3 . In addition, a moving RM bivector frame satisfies the equation      0 κ1 0 T T d  (28) N1  =  −κ1 0 0   N1  , ds N2 −κ2 0 0 N2 where κ1 = κ and κ2 = κ θ.

Proof We have n1 ×e n2 = (n − θ b) ×e b = n ×e b = T .

(29)

N1 = b ×e t = H,

(30)

In addition

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Luiz C. B. da Silva

and N2 = t ×e (n − θ b) = B + θ H .

(31)

Then, we find det(T , N1 , N2 ) = det(T , H, B) = 1, which shows that the RM bivectors form a basis. For the equations of motion, we have T ′ = n′1 ×e n2 + n1 ×e n′2 = κ1 N1 ,

(32)

N1′ = n′2 ×e t + n2 ×e t′ = n2 ×e (κ1 n1 + κ2 n2 ) = −κ1 T ,

(33)

N2′ = t′ ×e n1 + t ×e n′1 = (κ1 n1 + κ2 n2 ) ×e n1 = −κ2 T .

(34)

and ⊓ ⊔

5 Simply isotropic spherical curves 5.1 Isotropic osculating spheres Due to the degeneracy of the isotropic metric, some geometric concepts can not be defined just using h·, ·iz . This is the case for spheres. Definition 1 We define simply isotropic spheres as connected and irreducible surfaces of degree 2 given by the 4-parameter family9 (x2 + y 2 ) + 2c1 x + 2c2 y + 2c3 z + c4 = 0 ,

(35)

where ci ∈ R. In addition, up to a rigid motion (in I3 ), we can express a sphere in one of the two normal forms below 1. (sphere of parabolic type) z=

1 2 (x + y 2 ) with p 6= 0; 2p

(36)

2. (sphere of cylindrical type) x2 + y 2 = r2 with r > 0.

(37)

It can be shown that the quantities p and r are isotropic invariants. Moreover, spheres of cylindrical type are precisely the set of points equidistant from a given center hx − P, x − P iz = r2 . (38) Note however, that the center P of a cylindrical sphere is not defined. More ˜ = P˜ , would precisely, any other point Q with the same top view as P , i.e., Q 9 Rigorously, isotropic spheres are connected and irreducible surfaces of degree 2 in P3 that contain the absolute figure (in fact, this definition applies to any CK geometry). One then shows that in I3 this condition leads to the 4-parameter family in Eq. (35) [24], p. 66.


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do the same job. We can remedy this by assuming the center located on the z = 0 plane. An osculating sphere of an admissible curve α at a point α(s0 ) is the (isotropic) sphere that has a contact of order 3 with α. The position vector x of an osculating sphere satisfies, Eq. (7.18) of [24], λhx − α0 , x − α0 iz + hu, x − α0 i = 0 ,

(39)

where α0 = α(s0 ), h·, ·i is the usual inner product in Euclidean space E3 , and λ ∈ R and u ∈ R3 are constants to be determined.

5.2 Characterization of spherical curves in simply isotropic space Our approach to spherical curves is based on order of contact. More precisely, we investigate osculating spheres in I3 by using RM frames and their associated bivector frames. Then, we use that a curve is spherical when its osculating spheres are all equal to the sphere that contains the curve. We refer to [9] for a similar approach in the simpler setting of Euclidean spherical curves. Defining a function F (x) = λhx−α0 , x−α0 iz +hu, x−α0 i, where α0 = α(s0 ) and λ, u are constants to be determined, we have for the derivatives of (F ◦α)(s)  ′ F = 2λhα − α0 , tiz + hu, ti,    P P F ′′ = 2λht, tiz + 2λhα − α0 , i κi ni iz + hu, i κi ni i, . (40)   P P  ′′′ F = 2λhα − α0 , −κ21 t + i κ′i ni iz + hu, −κ21 t + i κ′i ni i

Imposing the condition (F ◦α)′ (s0 ) = (F ◦α)′′ (s0 ) = (F ◦α)′′′ (s0 ) = 0 (contact of order 3) gives  hu, t(s0 )i = 0   P (41) 2λ = −hu, i κi (s0 )ni (s0 )i .  P ′  hu, i κi (s0 )ni (s0 )i = 0

From the first and third equations above, we find that

u = ρ [t ×e (κ′1 n1 + κ′2 n2 )](s0 ) = ρ [κ′1 N2 − κ′2 N1 ](s0 ),

(42)

for some constant ρ 6= 0. On the other hand, from the second equation we find that 2λ + ρ [κ′1 κ2 hn2 , N2 i − κ1 κ′2 hn1 , N1 i](s0 ) = 0. (43) The reader can easily verify that hni , Ni i = det(t, n1 , n2 ) = 1, and then we can rewrite the expression above as 2λ = ρ [κ1 κ′2 − κ′1 κ2 ](s0 ) = ρ τ (s0 )κ2 (s0 ),

(44)

where in the last equality one should use the expressions form (κ1 , κ2 ) in terms of (κ, τ ), see Eq. (20).

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In short, the equation for the isotropic osculating sphere (39), with respect to an RM frame and its associated bivector frame, can be written as 2 κ′1 N2 − κ′2 N1 α ˜0 κ′2 N1 − κ′1 N2 2 ˜ − 2 x, α = 0, |s0 + 2 |s0 − α0 , x ˜0 + τ κ2 2 τ κ2 (45) ˜ 2 = hx, xiz . where x Theorem 3 An admissible regular curve α : I → I3 lies on the surface of a sphere if and only if its normal development, i.e., the curve (κ1 (s), κ2 (s)), lies on a line not passing through the origin. In addition, α is a spherical curve of cylindrical type with radius r if and only if κ is constant and equal to r−1 . Proof The condition of being spherical implies that the isotropic osculating spheres are all the same and equal to the sphere that contains the curve. This condition demands κ′ N1 κ′ N2 d α ˜+ 2 2 − 1 2 =0 (46) ds τκ τκ

and

2 d α α ˜ d ˜ κ′ N2 − κ′2 N1 κ′1 N2 − κ′2 N1 α, = = 0. − α, 1 − ds 2 τ κ2 ds 2 τ κ2 The first condition gives ′ ′ ′ ′ κ2 κ′ κ′ κ1 0 = ˜t + N − N2 + 22 (−κ1 T ) − 12 (−κ2 T ) 1 2 2 τκ τκ τκ τκ ′ ′ ′ ′ κ1 κ2 N1 − N2 , = τ κ2 τ κ2

(47)

(48)

which, by taking into account the linear independence of {N1 , N2 }, implies a1 := −

κ′1 κ′2 = constant; a := = constant. 2 τ κ2 τ κ2

On the other hand, condition (47) implies !+ * + * α ˜ X ˜ X d α + t, − ai Ni ai Ni − 0 = α, ds 2 2 i i = (1 + a1 κ1 + a2 κ2 )hα, tiz ,

(49)

(50)

where we used that h˜ α, ti = hα, ˜ti = hα, tiz to obtain the second equality. If the curve is not of cylindrical type, we can not have hα, αiz = constant, which is equivalent to hα, tiz = 0, and then we conclude that for a parabolic spherical curve the normal development (κ1 , κ2 ) lies on a line not passing through the origin. On the other hand, if the curve is of cylindrical type hα(s)−P, α(s)−P iz = r2 , taking the derivative gives ht, α − P iz = 0 .

(51)


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Then α − P = a1 n1 + a2 n2 . We have that a1 = hα − P, n1 iz and, therefore, a′1 = ht, n1 iz + hα − P, −κ1 tiz = 0 and a1 is a constant. Taking the derivative of Eq. (51) gives 0 = ht, tiz + hκ1 n1 + κ2 n2 , α − P iz = 1 + hκ1 n1 + κ2 n2 , a1 n1 + a2 n2 iz = 1 + a1 κ.

(52)

Hence, the curvature κ = κ1 is a constant and, in addition, r2 = hα − P, α − P iz = ha1 n1 + a2 n2 , a1 n1 + a2 n2 iz = a21 . Reciprocally, if κ is a (non-zero) constant, define P = α + κ−1 n1 . Taking the derivative gives P ′ = t + κ−1 (−κt) = 0 and then P is a constant. Clearly we have hα − P, α − P iz = 1/κ2 . ⊓ ⊔ Remark 3 In the discussion above, we can also use the Frenet frame instead of an RM one. In this case, spherical curves may be characterized by κ′ /(κ2 τ ) = constant. Proposition 2 An admissible regular curve α : I → I3 lies on a plane if and only if its normal development (κ1 (s), κ2 (s)) lies on a line passing through the origin. Proof It is known that α is a plane curve if and only if all its osculating planes are equal to the plane that contains the curve. Define a function F (x) = hx − α0 , ui, where α0 = α(s0 ) and hu, ui = 1 (the idea here is that a plane can be represented through a unit vector in the Euclidean sense). Taking the derivatives of F ◦ α twice and demanding a contact of order 2, we have ( (F ◦ α)′ (s0 ) = ht(s0 ), ui =0 . (53) (F ◦ α)′′ (s0 ) = h[κ1 n1 + κ2 n2 ]|s0 , ui = 0 From these equations we deduce that u = u(s0 ) = ρ(s0 ) [t ×e (κ1 n1 + κ2 n2 )]|s0 = ρ(s0 ) [κ1 N2 − κ2 N1 ]|s0 ,

(54)

where, by applying the definition of the Frenet and RM bivectors, we can write ρ = (κ1 kBk)−1. The condition of being a plane curve is equivalent to du/ds = 0. This leads to κ1 τ hB, Hi κ1 κ′2 − κ′1 κ2 κ2 τ hB, Hi 1 du N1 + + =− N2 ds κ1 kBk κ1 hB, Bi hB, Bi τ θhB, Hi hB, Hi =− 1+ N1 + (55) N2 , kBk hB, Bi hB, Bi where we used that τ κ2 = κ1 κ′2 − κ′1 κ2 , N1 = H, hH, Hi = 1, and hT , T i = 1. Finally, it is easy to see that the planarity condition, i.e., u′ = 0, is equivalent to τ = 0 ⇔ (κ2 /κ1 )′ = κ1 κ′2 − κ′1 κ2 = 0. We then deduce that it is equivalent to κ2 /κ1 = constant and then (κ1 , κ2 ) lies on a line passing through the origin. ⊓ ⊔

14

Luiz C. B. da Silva

6 Differential geometry in semi-isotropic space Following the Cayley-Klein paradigm, we must specify an absolute figure in order to build the semi-isotropic space. Here, the semi-isotropic absolute is composed by a plane, identified with x0 = 0, and a degenerate quadric of index one, identified with x20 + x21 − x22 = 0 (10 ). Equivalently, we may say that in homogeneous coordinates the semi-isotropic absolute figure is composed by a plane ω, x0 = 0, and a pair of lines f1 , 0 = x0 = x1 + x2 , and f2 , 0 = x0 = x1 − x2 . Observe in addition that the point F = [0 : 0 : 0 : 1] ∈ P3 lies in the intersection f1 ∩ f2 and, therefore, should be preserved. Hence, the semi-isotropic absolute figure is alternatively given by {ω, f1, f2 , F }. Let us denote a projectivity in P3 by  x ¯0 = a00 x0 + a01 x1 + a02 x2 + a03 x3    x ¯1 = a10 x0 + a11 x1 + a12 x2 + a13 x3 , det(aij ) 6= 0. (56) x ¯2 = a20 x0 + a21 x1 + a22 x2 + a23 x3    x ¯3 = a30 x0 + a31 x1 + a32 x2 + a33 x3

Imposing that ω and F should be preserved leads to a01 = a02 = a03 = 0 and a13 = a23 = 0, respectively. A projectivity that preserves the absolute figure is said to be a direct projectivity if it takes fi to fi , i.e., x1 ± x2 = 0 goes in x¯1 ± x ¯2 = 0, and an indirect projectivity if it takes fi to fj (i 6= j), i.e., x1 ± x2 = 0 goes in x¯1 ∓ x ¯2 = 0. The coefficients aij of a direct projectivity should satisfy the following relations a11 − a12 + a21 − a22 = 0 . (57) a11 + a12 − a21 − a22 = 0 Adding and subtracting the equations above leads to a11 = a22 and a12 = a21 . Going to affine coordinates and denoting a := a10 /a00 , b := a20 /a00 , c := a30 /a00 , p cosh φ := a11 /a00 , p sinh φ := a12 /a00 , and ci := a3i /a00 (i = 1, 2, 3), defines the group SG8 of semi-isotropic direct similarities  ¯ = a + p(x cosh φ + y sinh φ) x y¯ = b + p(x sinh φ + y cosh φ) . (58)  z= c + c1 x + c2 y + c3 z Let us introduce a metric in SI3 = P3 /ω according to hu, viz,s = u1 v1 − u2 v2 .

(59)

If we apply a transformation from SG8 to A, B ∈ SI3 , then the norm kvks = p |hv, viz,s | induced by the metric above satisfies ¯ − A¯ ks = p kB − A ks . kB

(60)

10 There are other choices for the semi-isotropic absolute figure. We discuss it in the next subsection.


15

For p = 1, the semi-isotropic metric h·, ·iz,s is an absolute invariant. Note in addition that, as happens in the simply isotropic space, the distance between two points with the same top view11 A = (a1 , a2 , a3 ), B = (a1 , a2 , b3 ) is zero. In such case one may introduce a semi-isotropic codistance as cdz,s (A, B) = |b3 − a3 | .

(61)

3

Applying a transformation from SG8 to A, B ∈ SI leads to ¯ B) ¯ = p cdz,s (A, B) . cdz,s (A,

(62)

Definition 2 The group of dz,s -length and cdz,s -colength preserving direct projectivities forms the group of semi-isotropic (rigid) motions SB6 . The semiisotropic geometry is the study of (SI3 , SB6 ). Remark 4 In SI3 one has A = B ⇔ dz,s (A, B) = 0 and cdz,s (A, B) = 0. In short, semi-isotropic geometry is the study of those properties in R3 invariant by the action of the 6-parameter group SB6  ¯ = a + x cosh φ + y sinh φ x y¯ = b + x sinh φ + y cosh φ , (63)  z¯ = c + c1 x + c2 y + z

where a, b, c, c1 , c2 , φ ∈ R. Note that on the top view, i.e., on the xy plane, the semi-isotropic geometry behaves like the geometry in a Minkowski plane E21 . Indeed, up to translations, the action of SB6 on the top view corresponds to the action of O1++ (2) [18], the group of (hyperbolic) rotations in E21 that preserves both the orientation of R2 as a vector space, i.e., O1++ (2) ⊂ SO1 (2), and the time-orientation of E21 , i.e., O1++ (2) ⊂ O1+ (2). Remark 5 The group of isometries of E21 has in fact four components: O1++ (2), O1+− (2), O1−+ (2), and O1−− (2) (a + sign in the 1st upper position means that the vector space orientation is preserved, while a + sign in the 2nd upper position means that the time-orientation is preserved; a minus sign means that orientation is not preserved) [18]. Choosing “− cosh φ” for the x-coefficient in Eq. (58) would lead to the action of O1+− (2) on the top view. On the other hand, the study of indirect projectivities, would lead to  ¯ = a ± x cosh φ − y sinh φ x y¯ = b + x sinh φ ∓ y cosh φ . (64)  z = c + c1 x + c2 y + z These projectivities correspond to the action of O1−± (2) on the top view. In addition, observe that O1−+ (2) ∪ O1−− (2) does not form a group (the identity is not an indirect projectivity).

11 As in I3 , we may define the top view as the projection on the xy plane, (semi-)isotropic direction as (0, 0, z), which is preserved by SB6 , (semi-)isotropic lines as those lines with isotropic direction, and (semi-)isotropic planes as those planes containing an isotropic line.

16

Luiz C. B. da Silva

6.1 Equivalent descriptions of semi-isotropic geometry In the classical literature [24,28], the absolute figure of the semi-isotropic (also called pseudoisotropic) space is given in homogeneous coordinates by the plane x0 = 0 together with the pair or real lines x0 = x1 = 0 and x0 = x2 = 0, which leads to the 6 parameter group   x¯ = a + p x y¯ = b + p−1 y , (65)  z¯ = c + c1 x + c2 y + z

where a, b, c, c1 , c2 , p ∈ R [28], Eqs. (1), (2), (4), and (10), pp. 136-137; or [24], Eqs. (1.68), (1.70), p. 24. This choice however furnishes a geometry equivalent to that described by SB6 , Eq. (63). Indeed, this can be made clear with the help of hyperbolic (also known as double or Lorentz) numbers [5,32]: the standard way to write a hyperbolic number is p = p1 + ph h, where the hyperbolic imaginary unit h satisfies h2 = 1. The real and imaginary parts are p1 = Re(p) and ph = Im(p), respectively. The arithmetic operations are defined as usual: (p1 + ph h) + (q1 + qh h) = (p1 + q1 ) + (ph + qh )h and (p1 + ph h)(q1 + qh h) = (p1 q1 + ph qh ) + (p1 qh + q1 ph )h. The inner product in E21 may be expressed as hp, qi1 = Re(p q¯), where p¯ = p1 + ph h = p1 − ph h denotes hyperbolic conjugation (see Fig. 1). Instead of working with the (canonical) basis {1, h}, we can describe a hyperbolic number p in terms of the light cone basis e± = (1 ± h)/2: the number e± is lightlike, i.e., he± , e± i1 = 0, and writing p = p+ e+ + p− e− , we have   p q = (p+ q+ ) e+ + (p− q− ) e− p¯ = p− e+ + p+ e− . (66)  e2± = e± , e± e∓ = 0 Now, using that a unit hyperbolic number in the basis {e± } is written as p e+ + p−1 e− and that a (hyperbolic) rotation in E21 may be described through a right multiplication by a unit length hyperbolic number a, i.e., p 7→ a p [5], it follows that a rotation may be written in the basis {1, h} or {e+ , e− } as x2 cosh φ sinh φ x1 x2 p 0 x1 = or = , (67) sinh φ cosh φ 0 p−1 y2 y1 y2 y1 respectively. In short, the above discussion shows that our expression for SB6 in Eq. (63) and that of Strubecker [28] in Eq. (65) are all equivalent, the choice between them being just a matter of convenience. Finally, let us mention that the group SB6 was already considered by Aydin [2] along with some investigations of semi-isotropic curves and surfaces (in his approach however, it is neither discussed how SI3 can be seen as a CK geometry nor how it relates to Strubecker’s absolute figure).


17

7 Semi-isotropic spheres A semi-isotropic sphere is a connected and irreducible surface of degree 2 that contains the semi-isotropic absolute figure. As we will see below, the semiisotropic spheres are given by the 4-parameter family (x2 − y 2 ) + 2c1 x + 2c2 y + 2c3 z + c4 = 0 .

(68)

In addition, up to a rigid motion (in SI3 ), we can express a sphere in one of the two normal forms below 1. (sphere of parabolic type) z=

1 2 (x − y 2 ) with p 6= 0; 2p

(69)

2. (sphere of cylindrical type) x2 − y 2 = ±r2 with r > 0.

(70)

Remark 6 The equations above in Euclidean space define the surface of a hyperbolic paraboloid and a hyperbolic cylinder, respectively, then justifying the name for the normal forms. A degree 2 surface in P3 may be written as Q :

3 X

cij xi xj = 0 ,

(71)

i,j=0

where cij ∈ R. If Q contains the absolute figure, it means that doing x0 = x1 ± x2 = 0 in fact vanishes the equation above. This leads to (c11 + 2c12 + c22 )x22 + c33 x23 + 2(c23 + c13 )x2 x3 = 0 . (72) (c11 − 2c12 + c22 )x22 + c33 x23 + 2(c23 − c13 )x2 x3 = 0 Since the above equation must be satisfied for all [x2 : x3 ] ∈ P1 , we conclude that   c11 ± 2c12 + c22 = 0 c12 = c13 = c23 = 0 c23 ± c13 = 0 ⇔ . (73) c11 + c22 = 0  c33 = 0 Going to affine coordinates gives

c11 (x2 − y 2 ) + 2c01 x + 2c02 y + 2c03 z + c00 = 0 ,

(74)

where we must have c11 6= 0. If c03 6= 0, we may write the equation above as z = R(x2 − y 2 ) + ax + by + cz + d

(75)

It is not difficult to show that the sphere above can be written in the parabolic normal form in Eq. (69) after a convenient semi-isotropic rigid motion. On the other hand, if c03 = 0 we can write, after a convenient semi-isotropic rigid motion, a sphere in the cylindrical normal form in Eq. (70).

18

Luiz C. B. da Silva

8 Moving frames along curves in semi-isotropic space A curve α : I → SI3 is said to be regular if α′ 6= 0. As in I3 , α′ (t0 ) is an inflection point if {α′ (t0 ), α′′ (t0 )} is linearly dependent, i.e., ∃ σ ∈ R such that α′′ (t0 ) = σα′ (t0 ). We leave as an exercise to the reader to show that being regular is a (semi-isotropic) geometric concept, i.e., ∀ T ∈ SB6 , α′ (t) 6= 0 ⇒ (T ◦ α)′ (t) 6= 0. The same for an inflection point. In order to find the osculating plane at an inflection point α(t0 ), let us employ the inner product in Lorentz-Minkowski space E31 given p by hu, vi1 = u1 v1 − u2 v2 + u3 v3 . Define F (x) = hx, ui1 , where kuk1 = |hu, ui1 | = 1: F (x) = 0 is the equation of a plane. Taking the first and second derivative of F ◦ α gives ′ F = hα′ , ui1 . (76) F ′′ = hα′′ , ui1 Now, imposing the order 2 contact (F ′ = F ′′ = 0 at α(t0 )) leads to u = ρ (α′ ×1 α′′ ), ρ 6= 0 , E31

(77)

12

where ×1 in the vector product in ( ). The above expression for u tell us that the osculating plane is spanned by the velocity and acceleration vector of α, in completely analogy with what happens in E3 (and also in E31 and I3 ). So, we may write the position vector x of the osculating plane at α(t0 ) as h x − α(t0 ), α′ (t0 ) ×1 α′′ (t0 ) i1 = 0 .

(78)

3

Definition 3 A regular curve α : I → SI without inflection points is an admissible curve if all the osculating planes are not semi-isotropic. This is equivalent to say that (x′ y ′′ − x′′ y ′ )|t 6= 0 for all t ∈ I, where α(t) = (x(t), y(t), z(t)): span{α′ , α′′ } is semi-isotropic if and only the third coordinate of α′ ×1 α′′ vanishes. The concept of reparametrization and arc-length parameter are defined as usual. Note however that curves in SI3 may have distinct casual characters: a vector v is said to be spacelike if hv, viz,s > 0 or v = 0, timelike if hv, viz,s < 0, and lightlike if hv, viz,s = 0 and v 6= 0. The casual character of a curve α is defined according to the casual character of its velocity vector α′ . Remark 7 A lightlike curve α gives rise to a top view curve α ˜ in E21 whose image must be contained on a straight line, the light cone in E21 is x = ±y. Note in addition that these curves are not admissible: the light cone in SI3 is the set of semi-isotropic planes (µ, ±µ, ν), µ, ν ∈ R. So, in our study we shall restrict ourselves to space- and timelike curves13 . Finally, since in a LorentzMinkowski plane a vector (x, y) is spacelike (timelike) if and only if (y, x) ⊥ (x, y) is timelike (spacelike), we do not have non-lightlike curves with a lightlike acceleration vector. 12

Mnemonically, it can be computed as u ×1 v = det[(i, u1 , v1 ), (−j, u2 , v2 ), (k, u3 , v3 )]. In principle, the curve may change its casual character. We shall not consider this possibility in this work, but the interested reader may consult [25]: please, observe that their notation for the metric and curvature in E21 is slightly distinct from ours. 13


19

8.1 Semi-isotropic Frenet frame Let α : I → SI3 be a unit speed curve (we can reparametrize a curve by an arcRt p length parameter through s(t) = t0 |hα′ (u), α′ (u)iz,s | du). The unit tangent is defined as t(s) = α′ (s). We introduce in addition ǫ = ht, tiz,s ∈ {−1, +1}. If t′ 6= 0, we define the semi-isotropic principal normal vector and curvature function, respectively, as n(s) =

t′ (s) , and κ = η kα′′ (s)kz,s = −ǫkα ˜ ′′ (s)k1 , η κ(s)

(79)

where η = hn, niz,s = −ǫ (note that t′ is not lightlike): if α is not parametrized by an arc-length, κ = ηkα ˜ ′ ×1 α ˜ k1 kα ˜ ′ k−3 1 . Note that the curvature function is just the curvature of the top view curve α ˜ in E21 . For the binormal vector we define b = (0, 0, 1). Clearly we have b′ = 0 and ′ t = ηκ n = −ǫκn. For the derivative of the principal normal, let us write n′ = a t + b n + c b .

(80)

Since hn, niz,s = η = ±1, we necessarily have b = 0. On the other hand, for the first coefficient a = ǫhn′ , tiz,s = −ǫhn, t′ iz,s = −ǫκ.

(81)

From the third coefficient we define the semi-isotropic torsion c = −ǫητ = τ , in analogy with the definition of torsion in E31 [10,17]. In short, we have the following semi-isotropic Frenet equations for curves in SI3         t 0 ηκ 0 t 0 −ǫκ 0 t d    n = −ǫκ 0 −ǫητ   n  . =  −ǫκ 0 τ   n  . (82) ds b 0 0 0 b 0 0 0 b

Proposition 3 An admissible curve α : I → SI3 is a plane curve if and only if τ = 0. Proof As we saw above, from a semi-Euclidean viewpoint, the osculating plane has a normal vector given by u=

α′ ×1 α′′ . kα′ ×1 α′′ k1

(83)

The condition of being a plane curve is equivalent to u′ ≡ 0, i.e., the osculating planes are always the same. Taking the derivative of u and using that α′ ×α′′ = ηκ t ×1 n in combination with the semi-isotropic Frenet equations gives b ht ×1 n, t ×1 bi du − n . (84) = τ t ×1 ds kt ×1 nk1 kt ×1 nk31 Clearly u′ ≡ 0 if and only if τ ≡ 0, and this completes the proof.

⊓ ⊔

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Luiz C. B. da Silva

8.2 Semi-isotropic spherical image and moving bivectors Let α : I → SI3 be an admissible curve and Σsi be the parabolic sphere of radius p = 1 1 (85) z = (x2 − y 2 ) . 2 Definition 4 For each s, let α∗ (s) be the point on Σsi such that the tangent plane Πs to Σsi at α∗ (s) is parallel to the osculating plane πs of α at α(s). The curve α∗ is the spherical image 14 of α. The equation of the tangent plane to Σsi at α∗ is z = x∗ x − y ∗ y − z ∗ .

(86)

On the other hand, from Eq. (78), the equation for the osculating plane is z=

ǫ ′ ′′ ǫ (y z − y ′′ z ′ )x − (x′ z ′′ − x′′ z ′ )y + w, κ κ

(87)

where we used that κ = −ǫ (x′ y ′′ − x′′ y ′ ): the value of w is not important for all purposes. The condition Πs k πs leads to ǫ x′ z ′ ǫ y ′ z ′ ∗ ∗ (88) x = ′′ ′′ , y = ′′ ′′ . κ y z κ x z

Finally, in order to find z ∗ , one may use that x′2 − y ′2 = ǫ (⇒ x′ x′′ − y ′ y ′′ = 0) and κ2 = η(x′′2 − y ′′2 ) (η = −ǫ). Then, z∗ =

1 ∗2 ǫ (x − y ∗ 2 ) = 2 (κ2 z ′2 − z ′′2 ) . 2 2κ

(89)

The spherical image will be used to describe the semi-isotropic moving bivectors. In fact, using the vector product in E31 , one introduces the moving bivectors associated with the Frenet frame {t, n, b} as   T = n ×1 b H = b ×1 t . (90)  B = t ×1 n Proposition 4 The Frenet bivectors satisfy f∗ , H = η n ˜ , T = ǫ ˜t . B = b−α

(91)

14 In E3 one can associate with every curve three other curves on the unit sphere S2 : the tangent, normal, and binormal spherical images (or indicatrices). In (semi-)isotropic space one can define a tangent and normal indicatrices, they are curves on the unit sphere of cylindrical type, but not a binormal indicatrix.


21

Proof We have B = t ×1 n = −ǫα′ ×1 α′′ /κ and so ǫ B = − (y ′ z ′′ − y ′′ z ′ , x′ z ′′ − x′′ z ′ , x′ y ′′ − x′′ y ′ ) κ x′ z ′′ − x′′ z ′ y ′ z ′′ − y ′′ z ′ , −ǫ , 1) = (−ǫ κ κ f∗ = b−α

(92)

On the other hand, since x′ x′′ − y ′ y ′′ = 0, we have x′′ /(x′ y ′′ − x′′ y ′ ) = ǫy ′ and y ′′ /(x′ y ′′ − x′′ y ′ ) = ǫx′ . Then, T = −ǫα′′ ×1 k/κ = ǫ˜t. Analogously, H = k ×1 α′ = −ǫ˜ n. ⊓ ⊔

8.3 Rotation minimizing frames in semi-isotropic space As in I3 , the semi-isotropic binormal b = (0, 0, 1) is an RM vector field, b′ = 0. Thus, we just need to introduce an RM vector field in substitution to the principal normal n. Let us write v = µn + νb ,

(93)

where µ 6= 0 (otherwise v k b). Now, imposing hv, viz,s = hn, niz,s = ±1 implies that µ = ±1. The derivative of v is v′ = −µκ t + (µτ + ν ′ ) b .

(94)

In addition, if we impose that ǫ det(t, v, b) = det(t, n, b) = − (x′ y ′′ − x′′ y ′ ) = 1, κ

(95)

we conclude that µ = 1 and then the condition v′ k t leads to Theorem 4 Let n1 be a unit normal vector field along α : I → SI3 . If n1 is RM then Z s n1 (s) = n(s) − (96) τ (x)dx + τ0 b(s) , s0

Rs where τ0 is a constant and we shall define sing θ(s) = θ(s) = s0 τ (x)dx+τ0 . In addition, an RM frame {t, n1 , n2 = b} in semi-isotropic space SI3 satisfies15      t t 0 ηκ1 κ2 d    n1 = −ǫκ1 0 0   n1  , (97) ds 0 0 0 n2 n2 where the natural curvatures are κ1 = κ and κ2 = η κ θ = −ǫ κ θ. 15

It may be instructive to compare this equation of motion with Eq. (16) from [10].

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Luiz C. B. da Silva

We now introduce the moving bivectors associated with an RM frame   T = n1 × 1 b N1 = b ×1 t , (98)  N2 = t ×1 n1

which satisfies the following equation of motion      T 0 ǫκ1 0 T d  N1  =  −ηκ1 0 0   N1  . ds N2 −κ2 0 0 N2

(99)

9 Semi-isotropic spherical curves In previous sections we introduced all the necessary ingredients to describe semi-isotropic spherical curves. All the results to be described below are analogous to their simply isotropic versions and, therefore, we will not go through the details. We may write a semi-isotropic osculating sphere at α0 = α(s0 ) as F1 (x) = λhx − α0 , x − α0 iz,s + hu, x − α0 i1 = 0,

(100)

where the constants λ ∈ R, u ∈ E31 will be determined by the contact of order 3 condition. Taking the derivatives (F ◦ α)(k) (s), k = 1, 2, 3, at s = s0 gives  hu, ti1 = 0  2λǫ + hu, ηκ1 n1 + κ2 n2 i1 = 0 . (101)  hu, ηκ′1 n1 + κ′2 n2 i1 = 0 By using the RM moving bivectors, we deduce that ( u = ρ[ηκ′1 N2 − κ′2 N1 ]|s0 2λ ǫ = ρ τ κ2

,

(102)

for some constant ρ 6= 0. In short, the equation for a semi-isotropic osculating sphere, with respect to an RM frame and its associated bivector frame, can be written as 2 ηκ′1 N2 − κ′2 N1 κ′ N1 − ηκ′1 N2 α ˜0 ˜ 2 −2 x, α = 0. | | − α , +2 x ˜0 + 2 s0 s0 0 ǫτ κ2 2 ǫτ κ2 1 1 (103) The condition of being spherical implies that the semi-isotropic osculating spheres are all the same. This condition demands d κ′ N1 ηκ′ N2 α ˜+ 2 2 − 1 2 =0 (104) ds ǫτ κ ǫτ κ and

2 ˜ d α ηκ′ N2 − κ′2 N1 = 0, − α, 1 ds 2 ǫτ κ2

(105)


23

The first equation above leads to a1 :=

κ′1 κ′2 = constant, a := − = constant , 2 τ κ2 τ κ2

(106)

while the second equation gives (ǫ + a1 κ1 + a2 κ2 )hα, tiz,s = 0 .

(107)

Theorem 5 An admissible regular curve α : I → SI3 lies on the surface of a sphere if and only if its normal development (κ1 (s), κ2 (s)) lies on a line not passing through the origin. In addition, α is a plane curve if and only if the normal development lies on a line passing through the origin. Acknowledgements The author would like to thank S. A. V. Silva and G. G. Carvalho for useful discussions and also the financial support by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico - CNPq (Brazilian agency).

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