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The Successor Transformation of Frenet Curves and its Application to Obtain an Explicit Parametrization of the Frenet Apparatus of the Slant Helix
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The Successor Transformation of Frenet Curves and its Application to Obtain an Explicit Parametrization of the Frenet Apparatus of the Slant Helix

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A helix is a curve that in every point makes a constant angle with a fixed direction, ...

A helix is a curve that in every point makes a constant angle with a fixed direction,
and a slant helix is a curve with a principal normal that has this property.
In this paper, the natural equations of the slant helix are developed and an explicit
parametrization of its Frenet apparatus is given. To this goal, a general method is
presented for transforming a Frenet space curve into a successor curve, which has the
predecessor’s tangent as principal normal. Helices are exactly the successor curves of
plane curves and slant helices are the successor curves of helices. By the same method,
an infinite series of successor curves can be constructed. The new method is not restricted
to curves with positive curvature. To achieve maximal generality, the paper
first gives a generalization of conventional Frenet theory. Bishop frames are examined
along with Frenet frames as a useful tool in the theory of space curves.

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    The Successor Transformation of Frenet Curves and its Application to Obtain an Explicit Parametrization of the Frenet Apparatus of the Slant Helix The Successor Transformation of Frenet Curves and its Application to Obtain an Explicit Parametrization of the Frenet Apparatus of the Slant Helix Document Transcript

    • An Explicit Parametrization of the Frenet Apparatus of the Slant HelixarXiv:submit/0675472 [math.DG] 15 Mar 2013 Toni Menninger∗ March 15, 2013 Abstract Any two continuous functions κ and τ characterize a certain Frenet curve in 3-dimensional space. In principle, the curve can be constructed by solving the Frenet-Serret system of differential equations. Explicit solutions are known for generalized helices. In this paper, explicit characterizations and parametrizations are given for the Frenet apparatus of slant helices. In every point of a general helix, its tangent makes a constant angle with a fixed direction. Similarly, slant helices have a principal normal that has this property and their representations can be deduced by rearranging the Frenet apparatus of a helix. By the same method, a further class of curves can be constructed where the principal normal is the tangent of a slant helix, and so on.1 Keywords: Curve theory; Generalized helix; Slant helix; Frenet equations; Frenet curves 1 Frenet Space Curves 1.1 Regular Curves This paper considers regular curves in oriented three-dimensional euclidian space R3 , endowed with an inner product, u, v = u v cos θ, and a cross product, u × v = u v sin θN, for any u, v ∈ R3 , where θ ∈ [0, π] is the angle between u and v and N is the positively oriented unit vector perpendicular to the plane spanned by u and v. In particular, u ⊥ v ⇔ u, v = 0 and u, v collinear ⇔ u × v = 0. A parametrized curve is a C k map x : I → Rn (where k ≥ 0 denotes the differentiation order) defined on an interval I ⊂ R. We can picture it as the trajectory of a particle moving in ∗ University of Arkansas. E-mail: toni.menninger@sbcglobal.net 1 The results reported here were developed in 1996 as part of a master thesis 1
    • dxspace, with position x(t) and (if k ≥ 1) velocity x(t) = ˙ dt (t) at time t. The image x[I] is calledtrace of the curve. A parametrized curve x is called regular if it has a nonvanishing derivative x(t) at every point. ˙A regular parametrized curve x(t) has a well-defined unit tangent vector T (t) = x(t)/ x(t) ˙ ˙and can be parametrized by arclength. Arclength will be denoted s throughout this paper and aprime denotes differentiation by arclength. The intrinsic geometric properties of regular curves areindependent of regular parameter transformations and any regular curve can always be representedby an arclength (or unit speed) parametrization x(s) with x′ = T . Let x = x(s) be the arclength parametrization of regular C 2 -curve and T = x′ its unittangent vector. Then the continuous function κ+ = T ′ = x′′is called its canonical curvature. A point with κ+ (s) = 0 is called inflection point, otherwisestrongly regular. A curve without inflection points is also called strongly regular. In stronglyregular points, the canonical principal normal vector is defined as N+ = κ−1 T ′ . +Every regular point of a space curve in R3 has a well-defined tangent line, spanned by T , andperpendicular normal plane. Every strongly regular point has a well-defined principal normal linespanned by N+ , osculating plane spanned by T and N+ , and rectifying plane perpendicular N+ .1.2 The Frenet ApparatusDefinition 1 (Frenet Curves and Frenet Apparatus). Given a regular C 2 space curve witharclength parametrization x : I → R3 (I ⊂ R an interval) and unit tangent vector T : I → S 2 ,T (s) = x′ (s).a) A differentiable unit vector field N : I → S 2 is a principal normal of the curve if in every point, T ⊥ N and T ′ and N are collinear.b) A differentiable unit vector field B : I → S 2 is a binormal of the curve if in every point, T ⊥ B and T ′ ⊥ B.c) A positively oriented orthonormal basis of differentiable vector fields (T, N, B) is called a Frenet moving frame of the curve if T is its tangent, N is a principal normal and B is a binormal.d) Given a Frenet moving frame (T, N, B) of the curve, the following are well-defined and continuous: • The curvature κ : I → R, κ = T ′, N • The torsion τ : I → R, τ = N ′, B √ • The Lancret curvature ω+ : I → R, ω+ = κ2 + τ 2 • The Darboux vector field D : I → R3 , D = τ T + κB 2
    • The tuple F = (T, N, B, κ, τ ) is called a Frenet apparatus and the pair (κ, τ ) a Frenet development associated with the curve.e) A curve is called a Frenet curve if it has a Frenet moving frame.Remarks. 1. Given a tangent and either a principal normal or a binormal, the third componentof the moving frame can be constructed as B = T × N or N = B × T , respectively. Theexistence of either a principal normal or a binormal is therefore sufficient for the existence of aFrenet moving frame and apparatus. 2. By definition, a Frenet curve is regular and at least of order C 2 . The components of theFrenet frame are at least of order C 1 and the Frenet development at least of order C 0 . 3. Every strongly regular C 3 -curve is a Frenet curve. It can be assigned a uniquely defined,canonical Frenet apparatus with principal normal N = N+ and curvature κ = κ+ > 0. 4. Whether or not a regular but not strongly regular curve is a Frenet curve depends onwhether the map mapping each strongly regular point to the osculating plane has a continuousextension. Curves that discontinuously ”jump” between planes cannot be Frenet-framed. 5. Frenet curves with inflection points cannot be assigned a canonical Frenet apparatus. If(T, N, B, κ, τ ) is a Frenet apparatus of a Frenet curve, then so is (T, −N, −B, −κ, +τ ). Instrongly regular points, N = ±N+ and κ = ±κ+ . It follows that if a Frenet curve has no or onlyisolated inflection points, then its Frenet apparatus is unique up to the sign of N, B, and κ andin particular, it has a unique and well-defined torsion τ . 6. A Frenet curve containing straight line segments has an infinite family of Frenet appa-ratuses because on a straight line, the curvature always vanishes (T ′ = 0) but the torsion canvary arbitrarily. In general (Wong and Lai (1967)), two Frenet developments (κ, τ ) und (˜ , τ ) κ ˜ 1characterize the same Frenet curve if and only if there is a a C -Funktion θ so that κ sin θ = 0, κ cos θ = κ ˜ and θ′ = τ − τ. ˜ 7. The definition of Frenet curve presented here (cf. Wintner (1956), Wong and Lai (1967))is justified by the fact that important classes of curves can be treated as Frenet curves even if theyhave inflection points. Examples are regular plane curves, generalized helices and slant helices(see section 2 of the present paper) as well as curves that are geodesics or asymptotic curves on aregular surface. In the case of asymptotic curves (this includes plane curves), the surface normalin every point of the curve can be chosen as binormal. In the case of geodesics, the surfacenormal can be chosen as principal normal. The prize to be paid for relaxing the assumption ofstrong regularity and κ > 0 is the loss of uniqueness of the Frenet apparatus.Theorem 1 (Frenet Equations). Any Frenet apparatus (T, N, B, κ, τ ) satisfies the Frenetequations T ′ = κN, N ′ = −κT + τ B, B ′ = −τ N.In matrix notation, we have  ′       T 0 κ 0 T T N  = −κ 0 τ  · N  = D × N  . B 0 −τ 0 B B 3
    • Proof. For any two differentiable unit vector fields V and W keeping a constant angle, it is ′ V, W = 0 ⇒ V ′ , W + V, W ′ = 0 ⇒ V ′ , W = − V, W ′ .For any C 1 unit vector field V , V ′ ⊥ V . It follows that differentiating a moving orthonormalbasis gives rise to a skew symmetric matrix of coefficient functions. By definition 1, they areκ = T ′ , N = − T, N ′ and τ = N ′ , B = − N, B ′ , and T ′ , B = B ′ , T = 0. The right hand side of the Frenet equation suggests that the Darboux rotation vector Dcan be interpreted as the angular momentum vector of the Frenet moving frame. Its directiondetermines the moving frame’s momentary axis of motion (its centrode) and its length the angularspeed D = ω+ . We also note the relationships κ2 = κ2 = T ′, T ′ , + 2 ω+ = N ′ , N ′ , τ 2 = B′, B′ . Given a pair of continuous coefficient functions κ = κ(s) and τ = τ (s) - also known asthe natural equations of a curve - and a set of initial conditions, the Frenet system of lineardifferential equations is guaranteed to have a unique solution. A Frenet development thereforedetermines a Frenet moving frame (unique up to a rotation) which is associated with a unique(up to a translation) Frenet curve, an arclength parametrization of which can be constructed byintegration over the tangent vector contained in the moving frame. Strongly regular C 3 -curveshave a canonical Frenet apparatus and are completely (up to a rigid motion) characterized bytheir canonical curvature and torsion. Two strongly regular space curves are congruent if andonly if their curvature and torsion are identical (in arclength parametrization). This fact is knownas the fundamental theorem of space curves and it is usually stated under the assumption thatκ > 0 and differentiable (e. g. K¨hnel (2006)). When the assumption of strong regularity is urelaxed, the fundamental theorem still essentially holds except that there is no unique, canonicalcurvature. Two different Frenet developments may now characterize the same curve (see remark6). An important question in curve theory is whether explicit solutions to the Frenet equationscan be found given κ and τ . The following theorem describes a method for transforming a Frenetapparatus of a curve to obtain the Frenet apparatus of a related curve. It is useful for findingparametrizations of helices and slant helices.Theorem 2 (Transformation of Frenet Apparatus). Given a Frenet curve x(s) with Frenetapparatus F = (T, N, B, κ, τ ) and the transformation T1 = − sin ϕN + cos ϕB N1 = T B1 = cos ϕN + sin ϕB κ1 = κ sin ϕ τ1 = κ cos ϕ s ϕ(s) = ϕ0 − τ (σ)dσ s0 with some constant ϕ0 ∈ R. Then F1 = (T1 , N1 , B1 , κ1 , τ1 ) is also a Frenet apparatus of aFrenet curve x1 . Its Darboux vector is D1 = κB. 4
    • Proof. We need to show (T1 , N1 , B1 ) is a positively oriented moving frame of orthogonal unitvectors satisfying the Frenet equations with curvature κ1 and torsion τ1 . Let         0 − sin ϕ cos ϕ 0 κ 0 T T1 A= 1 0 0  , F = −κ 0 τ  , B = N  and B1 = N1  . 0 cos ϕ sin ϕ 0 −τ 0 B B1A is orthogonal with det A = +1, therefore A−1 = At , and we have B1 = AB, B′ = F B, andB1 = A′ B + AB′ = (A′ + AF )B = (A′ + AF )At B1 . Denoting F1 = A′ At + AF At , calculation ′confirms that   0 κ sin ϕ 0 F1 = −κ sin ϕ 0 κ cos ϕ . 0 −κ cos ϕ 0By construction, therefore, (T1 , N1 , B1 ) is a positively oriented moving frame satisfying the Frenetequations B1 = F1 B1 . ′Remark. This transformation (considered in similar form in Bilinski (1963)) can be derived fromconverting a curve’s Frenet frame to its Bishop frame (which consists of tangent and two normalvectors with tangential derivatives, see Bishop (1975) for details), then rearranging it into a newFrenet frame. x1 has a Frenet apparatus with the tangent of x as its principal normal. Moreover,its Darboux vector is D1 = κB so the Frenet frame of x1 is momentarily rotating around thebinormal of x at angular speed ω1 = |κ|. If is interesting to consider the case when the Frenet apparatus is periodic, F (s + L) = F (s)for all s, which is necessary but not sufficient for the curve to be closed. The transformed Lapparatus F1 is also periodic if and only if the total torsion of F , defined as 0 τ (s)ds, is arational multiple of 2π.2 Helices and Slant Helices2.1 Plane CurvesDefinition 2 (Constant Slope, Fixed Normal). A vector field v : I → Rn is said to haveconstant slope if it makes a constant angle θ ∈ (0, π/2] with a fixed direction, represented bya unit vector V0 called slope vector. If the angle is π/2, then V0 is a fixed normal of the vectorfield.Remarks. A unit vector field V has constant slope θ if and only if there is a unit vector V0 sothat V, V0 ≡ cos θ. Obviously, a unit vector field with constant slope traces part of a circle onthe unit sphere, and if θ = π/2, it traces part of a great circle. A vector field with a fixed normaltraces a plane curve.Lemma 1. A differentiable unit vector field V : I → Rn has constant slope if and only if theslope vector is a fixed normal of V ′ .Proof. V has constant slope with V0 ⇔ V, V0 = const. ⇔ V, V0 ′ = V ′ , V0 + V, V0′ =0 ⇔ V ′ , V0 = 0. 5
    • Theorem 3 (Plane Curves). For a regular C 2 space curve x(s) with unit tangent vector T ,the following conditions are equivalent: i.) x lies on a plane. ii.) T has a fixed normal.iii.) x has a constant binormal.iv.) x has a Frenet apparatus with vanishing torsion. Given continuous functions κ = κ(s) and τ ≡ 0 defined on an interval s0 ∈ I ⊂ R. Let sΩ(s) = s0 κ(σ)dσ and       − sin Ω(s) − cos Ω(s) 0 TP (s) =  cos Ω(s) , NP (s) =  − sin Ω(s)  , BP = 0 . 0 0 1Then FP = (TP , NP , BP , κ, τ ) is a Frenet apparatus of a plane curve. Its tangent and principalnormal trace parts of great circles, its binormal is constant and its axis of motion is fixed, withDarboux vector DP = κBP .Proof. (i) ⇒ (ii): Let x(s) be a regular plane curve with plane normal B0 . Then if x(s0 ) is acurve point, B0 is a fixed normal of the vector field x(s) −x(s0 ) ⇒ x′ (s), B0 = T (s), B0 = 0. (ii) ⇒ (iii): T has a fixed normal B0 with T ⊥ B0 ⇒ T ′ (s), B0 = 0 (lemma 1). Bydefinition 1, B0 is a binormal. (iii) ⇒ (iv): Given a constant binormal B0 , τ 2 = B0 , B0 ≡ 0. ′ ′ (iv) ⇒ (i): If a Frenet development κ = κ(s) and τ ≡ 0 is given, the binormal must beconstant, B = B0 . x(s) − x(s0 ), B0 ′ = x′ (s), B0 + x(s) − x(s0 ), B0 = 0. It follows that ′the curve must lie on a plane. Finally, it is easy to verify that FP satisfies the definition of a Frenet apparatus. 6
    • 2.2 Generalized HelicesDefinition 3 (Generalized Helix, Slope Line). A regular C 2 -curve is called a generalized helix(also slope line or curve of constant slope) if at every point it makes a constant angle θ ∈ (0, π/2)with a fixed direction.Remark. The excluded cases correspond to straight lines (θ = 0) and plane curves (θ = π/2).Theorem 4 (Generalized Helices). For a regular C 2 space curve x with unit tangent vector T ,the following conditions are equivalent: i.) x is a generalized helix. ii.) T has constant slope θ.iii.) x has a principal normal that has a fixed normal.iv.) x is a Frenet curve and has a Frenet apparatus with τ = cot θκ.Proof. (i) ⇔ (ii): By definition. 1 (ii) ⇔ (iii): Let D0 be the slope vector, then N = sin θ D0 × T is a unit vector with fixednormal D0 . T ′ ⊥ D0 (lemma 1) and T ′ ⊥ T imply that T ′ is collinear N, so N is a principalnormal. Conversely, if N is a principal normal collinear T ′ , and N ⊥ D0 , then is also T ′ ⊥ D0and by lemma 1, T has constant slope. (ii) ⇒ (iv): Let D0 be the slope vector. Define B = csc θD0 − cot θT so that D0 =cos θT + sin θB. By lemma 1, T ′ ⊥ D0 . It is also T ′ ⊥ T so T ′ ⊥ B. By construction,T ⊥ B. By definition 1, B is a binormal. By construction, B has constant slope π/2 − θ withD0 . Differentiating B gives B ′ = − cot θT ′ = − cot θκN ⇒ τ = cot θκ. (iv) ⇒ (ii): We can construct the vector D0 = cos θT + sin θB. D0 = cos θκN − sin θτ N = ′0 ⇒ D0 = const. By construction, T has slope θ and B has slope π/2 − θ with D0 . The slope vector D0 is the direction of the axis of rotation: D = τ T + κB = κ/ sin θD0 , ω = csc θκTheorem 5 (Frenet Apparatus of Generalized Helix). Given a constant θ ∈ (0, π/2) andcontinuous functions κ, κH = sin θ κ, τH = cot θ κH = cos θ κ : I → R, s0 ∈ I. Let sΩ(s) = s0 κ(σ)dσ and       sin θ cos Ω(s) − sin Ω(s) − cos θ cos Ω(s) TH (s) =  sin θ sin Ω(s)  , NH (s) =  cos Ω(s) , BH (s) =  − cos θ sin Ω(s)  . cos θ 0 sin θ Then FH = (TH , NH , BH , κH , τH ) is a Frenet apparatus of a generalized helix with slopeangle θ. Its tangent has constant slope θ, its binormal has constant slope π/2 − θ and itsprincipal normal lies on a great circle. Its axis of motion is fixed in the direction of the slopevector, with angular speed κ. 7
    • Proof. This is a direct application of theorem 2 to the Frenet apparatus of a plane curve (theorem3) with ϕ = θ = const.2.3 Slant HelicesDefinition 4 (Slant Helix). A Frenet curve is called a slant helix if it has a principal normalthat has constant slope (Izumiya and Takeuchi (2004)). A principal normal vector of constant slope is exactly a tangent vector of a generalized helix.Applying the transformation of theorem 2 to the Frenet apparatus of a generalized helix gives aFrenet apparatus of a slant helix.Theorem 6 (Frenet Apparatus of Slant Helix). A Frenet curve is a slant helix if and only if ithas a Frenet development s κSH = ω sin ϕ, τSH = ω cos ϕ, ϕ(s) = ϕ0 − cot θ ω(σ)dσ s0(ω : I → R a continuous function, s0 ∈ I ⊂ R, ϕ0 ∈ R, θ ∈ (0, π/2)). Given such a Frenet development and let λ1 := 1 − cos θ, λ2 := 1 + cos θ. Then the tangentvector of the slant helix thus characterized can be parametrized as follows:   λ1 cos λ2 Ω − λ2 cos λ1 Ω s 1 ϕ0 1 TSH =  λ1 sin λ2 Ω − λ2 sin λ1 Ω  , Ω(s) = + ω(σ)dσ. 2 cos θ sin θ s0 2 sin θ cos(cos θ · Ω) The regular arcs traced by T are spherical helices and the slant helix has a Darboux vectorwith constant slope π/2 − θ.Proof. Assume that a curve is a slant helix. It has a principal normal N making constantslope θ with some unit vector D0 . We can write D0 = cos θN + sin θ(cos ϕT + sin ϕB) withsome differentiable function ϕ. We then have D0 = − cos θκT + cos θτ B + ϕ′ sin θ(− sin ϕT + ′cos ϕB) + sin θ(cos ϕκN − sin ϕτ N). From D0 = 0 derive three conditions: ′ κ = − tan θ ϕ′ sin ϕ, τ = − tan θ ϕ′ cos ϕ, cos ϕ κ = sin ϕ τ.Setting ω = − tan θ ϕ′ , we get κ = ω sin ϕ, τ = ω cos ϕ, and ϕ′ = − cot θ ω as claimed. Now the transformation described in theorem 2 is applied to the Frenet apparatus of thegeneralized helix (theorem 4), with ω = κH . The result is a Frenet apparatus of a slant helix,because the tangent of the helix becomes the principal normal of the slant helix, and its Frenetdevelopment is exactly as stated in the premise. The calculation of TSH is shown; parametrizationsof principal normal and binormal are obtained in the same way but are omitted here. 8
    • By theorem 2, we have TSH = − sin ϕNH + cos ϕBH , ϕ = ϕ0 − cot θ ω. Therefore   + sin ϕ sin Ω − cos θ cos ϕ cos Ω 1 TSH = − sin ϕ cos Ω − cos θ cos ϕ sin Ω  , Ω = Ω0 + ω. sin θ sin θ cos ϕThe integration constant Ω0 can be set arbitrarily because it only rotates the moving frame. Bysetting Ω0 = −ϕ0 / cos θ, we get ϕ = − cos θΩ and evaluation yields the expression above. The regular arcs of the curve described by the unit tangent T are slope lines because T ′ = κNhas constant slope. By construction, the unit Darboux vector of the slant helix is the binormalof a helix, which has constant slope (see theorem 2).Remarks. 1. The natural equations for slant helices derived above are of unrestricted generality.Any choice of a continuous function ω(s) and constants θ and ϕ0 gives rise to a well-definedslant helix and its Frenet apparatus, with Frenet development κSH = ω sin ϕ = − tan θ ϕ′ sin ϕ, τSH = ω cos ϕ = − tan θ ϕ′ cos ϕ, ϕ′ = − cot θ ω.Inflection points and even straight line segments are not excluded. The function ω(s) is thesigned Lancret curvature of the curve and has a geometric interpretation as the Frenet frame’sspeed of precession. 2. It is easy to verify that under the assumption κ = 0, the natural equations are equivalentto the condition κ2 τ ′ = cot θ = const. (κ2 + τ 2 )3/2 κThis expression is equivalent to the geodesic curvature of the spherical image of the principalnormal (see Izumiya and Takeuchi (2004)), which is part of a circle. 3. The differential equation −ϕ′ sin ϕ = cot θκ can be solved for ϕ. Setting m = cot θ, weget − sin ϕ dϕ = m κ ds ⇒ cos ϕ(s) = m κ ds. Further, we have τ = κ cot ϕ = κ √ cos ϕ 2 1−cos ϕ(here we must assume ϕ ∈ (0, π)). It follows m κ(s)ds τ (s) = κ(s) . 1 − m2 ( κ(s)ds)2This condition is due to Ali (2012). Given an arbitrary curvature κ, a torsion function can beconstructed to create a slant helix, provided that the domain of s is approriately restricted. Oneapplication is the construction of a Salkowski curve (for details see Ali (2012)) with ms κ(s) ≡ 1, τ (s) = √ , s ∈ (−1/m, +1/m). 1 − m2 s2 4. The case ω = const. gives rise to the class of curves of constant precession (Scofield(1995)). Setting µ = cot θ ω and α = ω 2 + µ2 , and omitting ϕ0 as it only effects a phaseshift, we can substitute cos θ = µ/α, sin θ = ω/α, λ1 = (α − µ)/α, λ2 = (α + µ)/α, Ω = αsin the formulae for the slant helix. 9
    • Theorem 7 (Curves of Constant Precession). A curve of constant precession has curvature,torsion and tangent parametrization as follows : κCP = −ω sin µs, τCP = ω cos µs (ω, µ = 0 arbitrary constants),   (α − µ) cos(α + µ)s − (α + µ) cos(α − µ)s 1  TCP = (α − µ) sin(α + µ)s − (α + µ) sin(α − µ)s  . 2α 2ω cos µs An explicit arclength parametrization of the curve is obtained by elementary integration. Theresulting curve of constant precession lies on a hyperboloid of one sheet, and it is closed if andonly if µ/α is rational (for details see Scofield (1995)).Remark. Curves of constant precession arise by transformation of the Frenet apparatus of acircular helix with curvature ω and torsion µ, which is periodic with period L = 2π/α (althoughthe helix itself is not closed). The total torsion of the helix is 2πµ/α and so the transformedFrenet apparatus is periodic if and only if if µ/α is rational. In this case, we obtain a closedcurve. Under what conditions a periodic Frenet apparatus is associated with a closed curve is anopen problem, known as the closed curve problem (Scofield (1995)).OutlookIn this paper, a certain transformation was applied first to a the Frenet apparatus of a genericplane curve to obtain a generic helix, and then again to obtain a slant helix. This procedure couldbe repeated ad infinitum to give rise to new, yet unexplored classes of curves.ReferencesAli, A. T. (2012). Position vectors of slant helices in euclidean 3-space. Journal of the Egyptian Mathematical Society, 20:1–6, DOI: 10.1016/j.joems.2011.12.005. ¨Bilinski, S. (1963). Uber eine Erweiterungsm¨glichkeit der Kurventheorie. Monatshefte f. Math- o ematik, 67:289–302.Bishop, R. L. (1975). There is more than one way to frame a curve. Amer. Math. Monthly, 82:246–251.Izumiya, S. and Takeuchi, N. (2004). New special curves and developable surfaces. Turk J Math, 28:153–163.K¨hnel, W. (2006). Differential Geometry: Curves - Surfaces - Manifolds, Second Edition. u American Mathematical Society.Scofield, P. D. (1995). Curves of constant precession. Amer. Math. Monthly, 102:531–537.Wintner, A. (1956). On frenet’s equations. Amer. J. Math., 78:349–355.Wong, Y.-C. and Lai, H.-F. (1967). A critical examination of the theory of curves in three dimensional differential geometry. Tˆhoku Math. J., 19:1–31. o 10