Curvature Singularity of Space Curves and Its Relationship to Computational Mechanics

Curve geometry plays a fundamental role in a wide range of physics and engineering disciplines that include mechanics, transportation, astronomic, and computer sciences [1–6]. Curve geometry is also central in the interpretation of inertia forces, motion description, deformation description, and motion-trajectory analysis. Curvature and torsion of space curves, for example, enter into the definition of strain-energy function of Euler–Bernoulli beams. For this reason, accurate definitions of curvature and torsion, which represent geometric invariants of space curves, are necessary for a wide range of engineering and physics applications. The relationship between mechanics and differential geometry is clear from the fact that curve geometry in its most general form can be represented by two in-plane and out-off-plane bending modes that can be conveniently described using angles. Furthermore, curve frames have been extensively used in mechanics to describe the orientation of beam cross sections and associate the orientation of these cross-section frames with deformation modes.

Space-curve Frenet frame, in particular, has been used widely in mechanics because of its clear geometric interpretation and its direct relationship to kinematic variables and inertia forces that influence motion. Nonetheless, Frenet frame suffers from singularity at curvature-vanishing points; at which two of the frame axes, normal and binormal vectors, cannot be defined if classical differential-geometry approaches are used. This singularity problem motivated researchers to seek alternative frames to define curve geometry [1–6]. This paper addresses this fundamental issue and demonstrates that Frenet-frame singularity problem can be solved using a simple procedure based on curve-coordinate derivatives or alternatively Frenet angles, and consequently, the use of alternative curve frames that may lack the clear Frenet-frame geometric interpretations may not be necessary if the objective is to solve curvature-singularity problem.

Frenet frame defines the geometry of a curve, described in its parametric form by the equation $r(t)=[x(t)y(t)z(t)]T$⁠, where t is the curve parameter [7–14]. Frenet frame is defined by three orthogonal unit vectors that represent curve tangent vector t, normal vector n, and binormal vector b. Transformation matrix that defines the orientation of Frenet frame is written as $Af=[tnb]$⁠. Unit vector tangent to the curve is defined by differentiating curve equation with respect to curve arc length s, that is, $t=dr/ds=rs$⁠, where a_s = da/ds. Norm of curvature vector $rss=dt/ds$ defines curve curvatureκ as $κ=|rss|=1/R$⁠, where R = R(s) is curve radius of curvature. Unit normal vector n normal is defined along curvature vector $rss$ according to $n=rss/|rss|$⁠. While curvature vector can also be written in the form $rss=(rtt−(rttTrt)t)/|rt|2$⁠, this form will not be used in this study since it defines curvature vector as a linear combination of two vectors, one of them is the function of second derivatives of coordinates. It will be shown that a curvature vector can be expressed as a linear combination of two orthogonal vectors expressed in terms of only first derivatives of curve coordinates. First derivative vectors are well-defined at isolated zero-curvature points. Binormal vector b is defined using cross product b = t × n [7–14].

When using computational-mechanics data-driven science (DDS) approaches in the analysis of recorded motion trajectories (RMT), differential-geometry procedure described previously fails, at zero-curvature points, to define normal vector n and curve Frenet frame at these zero-curvature points. Consequently, at these points, binormal vector b and its derivative b_s cannot be defined if a procedure based on the definition of curvature vector is used. Consequently, at points of curvature singularity, Serret-Frenet equations t_s = κn, n_s = −κt + τb, and b_s = −τn, where τ is curve torsion, fail to exist if the classical differential-geometry approach that depends on curvature vector is used; and in this case, unit vectors that define Frenet frame cannot be determined, except for the tangent vector t. Furthermore, curve torsion τ at zero-curvature points cannot be determined as the norm of vector b_s. This curvature singularity creates problems in the analysis of RMT curves which can be determined using experimental measurements or computer simulations of detailed models. Centrifugal-inertia force is always in direction of curve normal vector n, and in most applications, this force is continuous and approaches zero value as curve curvature approaches zero. Discontinuity in the definition of Frenet frame can negatively impact the quality of numerical results that define RMT curves.

Crucial in the analysis of this paper, and in distinguishing this analysis from previously published work [1], is the fact that curve curvature κ and torsion τ are not, in general, exact differentials and cannot be integrated to determine angles. Resorting to numerical integration of non-exact differentials does not lead to unique solutions or definition of specific curve frames whose geometries are defined in terms of non-commutative finite rotations [19]. Exceptions to this rule are planar curves with zero torsion. In this special case, curvature κ can be written as a derivative of an angle, and in this special case, the curvature can be integrated to determine this angle.

This paper introduces a method based on Frenet angles or curve derivatives to prove that Frenet frame and the Serret-Frenet equations are well defined at curvature-vanishing points without need for integrating curve torsion τ or introducing other framing methods. Results of the analysis of this study show that curvature vector r_ss should be viewed as a vector along a well-defined normal vector n and such a curvature vector has a norm defined by curve curvature κ that varies continuously and can assume zero value without affecting the definition of normal vector n or continuity of Frenet frame. Consequently, all derivatives that appear in Serret-Frenet equations are well-defined at all points at which tangent vector t can be defined, which is the assumption of a regular curve. The provided proof, established using curve derivatives or Frenet angles, which facilitate proving the existence of Frenet frame and Serret-Frenet equations, also demonstrates centrifugal force $ms˙2/R$ continuity of a vehicle or a particle with mass m at curvature-vanishing points [15–18]. The difference between Frenet bank angle and torsion angle used in the definition of Bishop frame is also discussed [1]. Unlike other framing methods, the analysis of this paper is focused on Frenet-frame definition and does not require evaluation and integration of curve torsion. Continuity of normal vector at curvature-vanishing points is demonstrated numerically using two different curve examples: one with zero torsion and the other has non-zero torsion.

Angles ψ and θ are referred to as Frenet horizontal curvature angle and Frenet vertical-development angle, respectively [15–18]. They can be used in a three-angle Euler sequence of rotations to define Frenet frame that does not suffer from discontinuities and also define derivatives that appear in Serret-Frenet equations. The third angle ϕ, called Frenet bank angle, is used in mechanics applications to measure the deviation of centrifugal-force vector from the horizontal plane. The sequence of rotation used in this study for angles ψ, θ, and ϕ is Z, −Y, −X, with negative signs used for consistency with procedures used in the construction of highway roads and railroad tracks [15–18]. Unit-tangent equation shows that space curve is completely defined in terms of two angles, while a planar curve can always be defined in terms of one angle, curvature angle ψ.

Regarding singularity associated with Euler angles, it is to be noted that a transformation matrix written in terms of Euler angles is never singular. Such a matrix is orthogonal and has a well-defined inverse. Euler-angle singularity is associated with the singularity of the matrix that relates angular velocity to angle derivatives. Such a singularity becomes an issue only when interest is in determining angle derivatives. As discussed in the literature [15,16], the Frenet bank angle ϕ for a space curve is dependent on the other two angles and is introduced for purpose of avoiding derivatives in the definition of normal and binormal vectors. As it will be shown in a later section, when θ = π/2, the curve bends vertically with curvature κ = θ_s and Frenet-frame transformation matrix is well-defined and does not suffer from Euler-angle singularity problem because of dependence of bank angle ϕ on Frenet angles ψ and θ. When θ = π/2, ϕ is well-defined and is equal to π/2 for θ_s ≠ 0.

Term at the end of the right side of the preceding equation is the result of the identity $rs×rss=rs×κn$ which shows that $κ=|rs×rss|=|rt×rtt|/|rt|3$⁠. At zero-curvature points, x_tt = y_tt = z_tt = α_c = 0.

Angle ϕ, called Frenet bank angle, defines super-elevation of curve osculating plane (OP), which contains normal vector as well as velocity, acceleration, and centrifugal-force vectors. For a motion-trajectory curve with zero Frenet bank angle, centrifugal force lies in a plane parallel to the horizontal plane. This is clear from the equation $n=(cosϕ)nh−(sinϕ)nv$⁠, which shows that if ϕ = 0, $n=nh=(1/xt2+yt2)[−ytxt0]T$⁠. Because the general definition of centrifugal-force vector of vehicle or particle with mass m is $Fc=(ms˙2/R)n$⁠, zero Frenet bank angle implies that centrifugal force remains in a plane parallel to the horizontal plane, and therefore, the OP is not Frenet super-elevated. It is important, however, to distinguish between Frenet super-elevation of motion-trajectory curves and railroad super-elevation used in the construction of railway tracks; the former is motion-dependent, while the latter is motion-independent [15].

The fundamental difference between Frenet bank angle ϕ and Bishop-torsion angle will be explained in later sections. Nonetheless, equation $n=(cosϕ)nh−(sinϕ)nv$ should not be confused with an equation used in the development of the Bishop frame [1] in which the normal vector is written as a linear combination of two vectors m₁ and m₂, which have derivatives along tangent vector t as will be discussed in a later section. Derivatives of vectors n_h and n_v do not in general define vectors parallel to tangent vector t. Because vectors n_h and n_v are functions of first derivatives only, their existence is ensured based on the assumption of the existence of tangent vector t, as previously mentioned. Therefore, the existence of angle ϕ at zero-curvature points ensures the existence of normal vector and derivatives that appear in Serret-Frenet equations despite κ = α_h = α_v = 0 at these points. It is also important to note that curvature vector $rss$ can be written in terms of other orthogonal vectors, and consequently, there are an infinite number of choices of such orthogonal vectors as it is clear from Eq. (3). The existence of ϕ and n at zero-curvature points is discussed in the following section.

The existence of normal vector n at curvature-vanishing points can be proven using space-curve coordinates. To this end, L’Hopital rule is used to derive an expression for Frenet bank angle at curvature-vanishing points. While the use of Frenet angles is not necessary, these angles provide a clear interpretation of space-curve geometry as a combination of in-plane and out-of-plane bending modes.

This limit defines Frenet bank angle ϕ at zero-curvature points. It is clear that the limit of the preceding equation is written in terms of first and third derivatives and does not involve second derivatives which appear in curvature expression. At isolated zero-curvature points, curve torsion is not in general equal to zero. Furthermore, the limit of the preceding equation is expressed in terms of derivatives of curve coordinates, and therefore, it is independent of any angle representation. In the case of the RMT curve obtained using computer simulations or measurements, numerical differentiation can be used to define the limit of the preceding equation. Information from previous time-steps can be stored and used in simple derivative expressions to determine this limit.

This equation defines Frenet frame everywhere including at points with curvature singularities. According to this description, Frenet-frame vectors t, n, and b are all differentiable, and therefore, derivatives of tangent, normal, and binormal vectors that appear in Serret-Frenet equations exist at curvature-vanishing points.

Curve angle representation described in this section does not lead to Euler-angle singularity, and therefore, such kinematic singularity is not an issue in Frenet-frame definition. For example, if θ = π/2, one has tangent, normal, and binormal vectors given, respectively, by $t=[001]T,$$n=−[cosψsinψ0]T$⁠, and $b=[sinψ−cosψ0]T$⁠. Derivative of tangent vector, when θ = π/2, shows that curve curvature is defined as κ = θ_s. That is, at θ = π/2 configuration, the curve bends vertically and curvature κ is described by a single bending mode along vector n_v. Furthermore, using equations and identities given in Refs. [15,16], one can show that at this configuration, Frenet bank angle is ϕ = π/2, and Frenet frame is well-defined.

Not every planar curve has one of its coordinates constant as demonstrated in numerical-example section. Nonetheless, without any loss of generality, a coordinate transformation can be used to make one of the planar curve coordinates constant. For example, in the special case of the planar curve with constant z coordinate, normal vector n is also well-defined since Frenet vertical-development angle θ is equal to zero, and tangent and normal vectors can be written in this case, respectively, as $t=rs=[cosψsinψ0]T$ and $n=[−sinψcosψ0]T$⁠. As previously shown, angle ψ depends on first derivatives of curve coordinates because $cosψ=xt/xt2+yt2$ and $sinψ=yt/xt2+yt2$⁠. Therefore, in this case of planar curve, normal vector n is well-defined and there is no need for using limit defined by the L’Hopital rule.

Frenet frame, particularly its osculating plane which is plane of motion, has unique geometric properties that can be fully exploited in motion-trajectory analysis. Elements of its skew-symmetric Cartan matrix are curvature and torsion widely used in analytical and computational mechanics. The curvature-singularity problem, however, was the main motivation for introducing other framing methods which lack geometric interpretations offered by Frenet frame [1–6]. Important for analysis and discussion presented in this section is the fact that elements of Cartan matrix defined using orthogonality condition of the matrices that define frame orientations are not in general exact differentials, as it is the case with elements of angular velocity vectors [19].

Unit vector n normal to the curve can always be written as a linear combination of two other orthogonal unit vectors that lie in curve normal plane. The choice of two orthogonal unit vectors in this linear combination is not unique. That is, one can always write $n=c1v1+c2v2$⁠, where c₁ and c₂ can always be expressed in terms of angle β and must satisfy condition $c12+c22=1$ because n is a unit vector. There are an infinite number of choices of orthogonal unit vectors v₁ and v₂, and angle β. To define the unique value of β, a condition must be imposed. In the approach used in this paper to overcome curvature singularity, a normal vector is written as $n=(αh/κ)nh+(αv/κ)nv$ which upon using definition tan ϕ = −α_v/α_h can be written as $n=(cosϕ)nh−(sinϕ)nv$⁠. In this case, the condition imposed on the choice of angle ϕ is ψ_s sin ϕ cos θ + θ_s cos ϕ = 0 [16]. The advantage of using this condition, in addition to its consistency with geometric description of railroad tracks, is avoiding integration of curve torsion τ, as it is required when Bishop frame is used.

In these equations, $κ(s)=k12+k22$⁠. Bishop introduced angle θ_B(s) such that τ(s) = dθ_B/ds and tanθ_B = k₂/k₁. Therefore, angle θ_B can be determined by integrating curve torsion τ as $θB=∫τds$⁠. Depending on initial value of θ_B, orientation of Bishop frame is defined. Nonetheless, the initial value of θ_B does not have an effect on torsion τ due to differentiation. The condition used by Bishop to define angle θ_B is to equate derivative of normal vector as defined by the equation $n=(cosθB)m1+(sinθB)m2$ to second Serret-Frenet equation $ns=−κt+τb$⁠. Noting that $ns=−κt+θBs[(−sinθB)m1+(cosθB)m2]$⁠, one can write θ_Bs = ∂θ_B/∂s = τ. This condition differs from the condition used in this study and previous investigations [15–18], and therefore, Frenet bank angle ϕ differs from Bishop-torsion angle θ_B. Furthermore, the Bishop frame is based on definitions $m1s=−k1t$ and $m2s=−k2t$⁠, which show differences between two vectors m₁ and m₂, and two vectors n_h and n_v whose derivatives are not in general along tangent vector t.

It is important to point out that the Bishop frame, referred to as parallel transport frame, was introduced as an alternative frame well-defined at the vanishing-curvature points. This frame, however, is not unique and does not include a curve normal vector, and therefore, its axes do not define osculating plane which is plane of motion. Definition of Bishop fame also requires integration of curve torsion τ and based on the assumption of smooth θ_B. At curvature-vanishing points, κ = k₁ = k₂ = 0, and torsion τ is not defined according to classical differential geometry.

In the case of Frenet frame, if a sequence of finite rotations is used to define frame geometry, such finite rotations cannot all be directly associated with bending and twisting modes as discussed in the literature [19]. This fact is clear from the preceding equation since curvature and torsion of a space curve are not, in general, exact differentials. If θ_Bs = τ = ψ_s sin θ − ϕ_s were an exact differential, one must be able to write θ_Bs = (∂θ_B/∂ψ)ψ_s + (∂θ_B/∂ϕ)ϕ_s, with (∂θ_B/∂ψ) = sin θ and (∂θ_B/∂ϕ) = −1. These two equations show that (∂²θ_B/∂ψ∂ϕ) ≠ (∂θ_B/∂ϕ∂ψ), demonstrating that θ_Bs = τ = ψ_s sin θ − ϕ_s is not an exact differential and cannot be integrated as is the case with the angular velocity vector of rigid bodies in space. Condition θ_Bs = τ = ψ_s sin θ − ϕ_s resembles non-integrable nonholonomic constraints encountered in mechanics. These nonholonomic constraints impose restrictions on derivatives but do not impose restrictions on coordinates. That is, the number of independent derivatives is less than the number of independent coordinates.

Nonetheless, in computational mechanics, some investigators used finite rotations that follow a sequence of three Euler angles to describe bending and torsion modes. When these angles are used, a distinction needs to be made between twist of a beam centerline which is due to out-of-plane bending and torsion due to shear. In the case of Euler–Bernoulli beams, centerline geometry can be described by only two independent angles, and therefore, a condition must be imposed on the angles as previously discussed in this paper and in the literature [19].

This equation shows that torsion is not defined at curvature-vanishing points at which x_tt = y_tt = z_tt = 0, leading to zero values for the numerator and denominator in the preceding equation. Therefore, at curvature-vanishing points, angle θ_B cannot be defined unless a procedure similar to the L’Hopital rule, adopted to define curvature, is also used to define torsion. An alternative to applying the L’Hopital rule to define torsion is to use Frenet angles to define torsion using equation τ = ψ_s sin θ − ϕ_s [16]. Since ϕ is a well-defined smooth function, the use of Frenet bank angle ϕ to define torsion can be more convenient. It is also to be noted that ϕ can have non-zero values at curvature-vanishing points and can oscillate around a non-zero constant nominal value [18].

In the case of zero torsion, ϕ_s = ψ_s sin θ, and the curve is a planar curve that can lie in a plane rotated with respect to the global system. Even for such planar curves that lie in planes rotated with respect to the global system, zero Frenet bank angle (ϕ = 0) implies that normal vector lies in a plane parallel to the horizontal plane and normal vector n, which defines the direction of centrifugal force, is the intersection of this horizontal plane with the osculating plane.

Having mechanics interpretation of curve geometry contributes to meaningful analysis of motion trajectories recorded experimentally or obtained using computer simulations of detailed vehicle models. Frenet bank angle ϕ defines super-elevation of osculating plane (OP), which contains absolute velocity, acceleration, and centrifugal-force vectors, as previously mentioned. Projection of gravity force onto OP defines the amount of centrifugal-force balancing. Therefore, the definition of Frenet bank angle can play important role in RMT analysis and accurate definitions of balance speeds that ensure safe vehicle operations [18].

This equation shows the difference between two vectors m₁ and m₂, and two vectors n_h and n_v.

Based on the analysis presented in this section and preceding sections, the following observations can be made:

1. Condition of Eq. (15) is of nonholonomic type due to the fact that curve curvature and torsion, when written in terms of angle derivatives, are not exact differentials. Therefore, Eq. (15) imposes constraint on derivatives, but does not impose constraint on angles; that is, angle θ_B can assume any value. Non-integrability of torsion explains the non-uniqueness of Bishop frame.

2. Torsion angle θ_B used in the definition of Bishop frame has nothing to do with super-elevation of Frenet-frame osculating plane defined solely by Frenet bank angle ϕ. This fact will be demonstrated by an example which shows that angle θ_B remains constant or zero based on assuming arbitrary initial value, while Frenet bank angle ϕ is oscillatory.

3. Therefore, angle θ_B does not enter into the definition of absolute velocity, acceleration, and centrifugal-force vectors, while Frenet bank angle ϕ enters into the definition of curve geometry. Condition θ_B = 0 does not necessarily imply that normal vector n lies in a plane parallel to the horizontal plane.

4. Because torsion τ can be written in terms of Frenet–Euler angles as τ = ψ_s sin θ − ϕ_s, the definition of angle ϕ at curvature-vanishing points ensures proper definition of torsion τ at those points.

The fundamental difference between Frenet bank angle ϕ and Bishop-torsion angle θ_B can be further made clear by considering helix curve which has constant torsion. For the helix curve, Frenet bank angle ϕ is identically zero, while Bishop-torsion angle θ_B is a linear function of curve arc length s, and therefore, θ_B increases as the curve arc length increases.

In this section, two curve examples are considered to demonstrate the application of the equations developed in this paper. In the first example, the curvature and Frenet angles are computed and presented. As previously discussed, Frenet bank angle defines super-elevation of the osculating plane which contains tangent and normal vectors.

Curve curvature results of Fig. 2 show repeated zero-curvature points. Figure 2 also shows curvature components α_h and α_v, which are oscillatory and assume positive and negative values. While curve curvature κ is always positive, that is, κ ≥ 0, curvature components α_h and α_v can assume negative values. At curvature-vanishing points, curvature vector $rss$ is a zero vector. The results shown in Figs. 3–5, however, show that all Frenet angles are well defined, and therefore, normal vector $n=[n1n2n3]T$ is defined at zero-curvature points. Components of the normal vector, defined by Eq. (12), are shown in Fig. 6. The results shown in this figure and Eq. (12) demonstrate that components of normal vector n are smooth, and binormal vector $b=t×n$ is well-defined and smooth, demonstrating existence of Frenet frame and derivatives of vectors $t,n$⁠, and b at points of zero curvature. The existence of derivatives of these vectors provides proof of the existence of Serret-Frenet equations at curvature-singularity points.

Using this equation, one has $cosψ=−sint,sinψ=cost,$$cosθ=1/|rt|$⁠, and $sinθ=bt/|rt|$⁠. The torsion τ is θ_Bs = τ = ψ_s sin θ − ϕ_s. This equation shows that the torsion is not, in general, an exact differential.

Curve geometry plays a fundamental role in analytical and computational mechanics, particularly in the analysis of motion trajectories that can be recorded experimentally or using computer simulations of detailed vehicle models. RMT curves and their geometry are central in new DDS computer algorithms that can be used for developing safety and operation guidelines of transportation systems. Furthermore, curvature and torsion of space curves are widely used in deformation analysis of structural systems. For these reasons, addressing singularity problems of RMT curves is necessary for proper interpretation of experimentally or numerically recorded motion trajectories.

Nonetheless, because curve curvature and torsion are not, in general, exact differentials, they cannot be integrated to define angles. Consequently, an approach different from what has been previously proposed in the literature to alleviate curvature singularity is needed. This paper introduces a procedure to establish the existence of Frenet frame and Serret-Frenet equations at curvature-vanishing points for a general three-dimensional curve with arbitrary parameterization. The paper demonstrates that it is not necessary to determine a curve normal vector from a curvature vector which can approach zero at vanishing-curvature points. Frenet curvature, vertical-development, and bank angles are used to define curve geometry. Particular attention is given to the definition of Frenet bank angle, used to prove the existence of curve normal and binormal vectors at curvature-vanishing points. Difference between Frenet bank angle and torsion angle used to define Bishop frame is discussed. Because of the unique features of Frenet frame and its clear geometric interpretation, the analysis of this study is focused on addressing and providing a solution to the curvature-singularity problem. The procedure described in this paper for the definition of Frenet frame at zero-curvature points is developed for arbitrary curve parameters and does not require evaluation and integration of curve torsion. Numerical differentiation can be used to determine L’Hopital limit if the curve cannot be defined analytically. Results of the analysis demonstrate that Frenet frame and Serret-Frenet equations are well defined at curvature-vanishing points. While the use of Frenet angles in the analysis of curve geometry is not necessary, these angles give a clear geometric interpretation for curve geometry and allow obtaining simple expressions for curve geometric invariants.

This research was supported by the National Science Foundation (Project # 1632302).

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent is not applicable. This article does not include any research in which animal participants were involved.

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. Data provided by a third party are listed in Acknowledgment. No data, models, or code were generated or used for this paper.