aa r X i v : . [ m a t h . DG ] S e p A Novel Solution to the Frenet-Serret Equations
Anthony A. RuffaNaval Undersea Warfare Center Division1176 Howell StreetNewport, RI 02841October 19, 2018
Abstract
A set of equations is developed to describe a curve in space given thecurvature κ and the angle of rotation θ of the osculating plane. The setof equations has a solution (in terms of κ and θ ) that indirectly solves theFrenet-Serret equations, with a unique value of θ for each specified valueof τ . Explicit solutions can be generated for constant θ . The equationsbreak down when the tangent vector aligns to one of the unit coordinatevectors, requiring a reorientation of the local coordinate system. Given the curvature κ and torsion τ , the Frenet-Serret equations describe acurve in space parameterized by the arc length s : d T ds = κ N ; (1) d N ds = − κ T + τ B ; (2) d B ds = − τ N ; (3) d R ds = T . (4)Here R , T , N , and B are the position, tangent, normal, and binormalvectors, respectively. These equations have no explicit solution (in terms of κ and τ ) for the general case, although solutions for special cases exist .It is shown here that a set of equations can be developed to describe a curvein space given the curvature κ and the angle of rotation θ of the osculatingplane. The set of equations has a solution (in terms of κ and θ ) that indirectlysolves the Frenet-Serret equations, and has a unique θ for every value of τ .1 , i’ q Nj’ k’
Figure 1: Local coordinate system; T is normal to the plane containing j ′ & k ′ .Many problems − involve the use of the Frenet-Serret equations, requiringnumerical approximations or the use of helical arc segments (each having con-stant τ and κ ). Specifying κ and θ to generate a solution may be useful if τ isnot initially known. The torsion τ can then be determined from θ . A local coordinate system having the property T = i ′ (figure 1) supports thedefinition of N : N = j ′ cos θ + k ′ sin θ. (5)The curvature κ and the angle of rotation θ of the osculating plane (con-taining N and T ) characterize the curve. When the plane containing T andthe global coordinate j is normal to k ′ (figure 2) then2 , i’jj’ ik’k Figure 2: Angular orientation of the local coordinate system with respect to theglobal coordinate system. k ′ = T × j | T × j | = − i T k + k T i q − T j . (6)Equation (6) breaks down when T = ± j , requiring an alternate expressionfor k ′ (developed in section 4). However, when T = ± j , j ′ = k ′ × T = − i T i T j + j (cid:0) − T j (cid:1) − k T j T k q − T j . (7)Substituting (7) and (6) into (5): N i = 1 κ dT i ds = − T k sin θ − T i T j cos θ q − T j ; (8) N j = 1 κ dT j ds = cos θ q − T j ; (9) N k = 1 κ dT k ds = T i sin θ − T j T k cos θ q − T j . (10)3quation (9) can be integrated directly: Z T j T j dT j q − T j = Z ss κ cos θdσ ; (11)leading tosin − T j = sin − T j + Z ss κ cos θdσ ; T j = sin (cid:20) sin − T j + Z ss κ cos θdσ (cid:21) = sin δ ; T j = T j cos Z ss κ cos θdσ + q − T j sin Z ss κ cos θdσ. (12)Equation (8) is solved by noting that T k = q − T j − T i = p cos δ − T i and introducing the variable β so that T i = cos δ cos β ; (13) T k = cos δ sin β. (14)Substituting into (8): dT i ds = − κ cos θ sin δ cos β − cos δ sin β dβds = − κ sin θ cos δ sin β − κ cos θ cos δ cos β sin δ cos δ = − κ sin θ sin β − κ cos θ cos β sin δ. (15)Equation (15) simplifies to dβds = κ sin θ cos δ ; (16)or β = β + Z ss κ sin θ cos δ dσ = cos − (cid:18) T i cos δ (cid:19) ; (17)so that T i = T i cos δ cos δ cos Z ss κ sin θ cos δ dσ − T k cos δ cos δ sin Z ss κ sin θ cos δ dσ ; (18)where 4os δ = q − T j cos Z ss κ cos θdσ − T j sin Z ss κ cos θdσ. (19)The solution for T k follows from (14) and (16): T k = T k cos δ cos δ cos Z ss κ sin θ cos δ dσ + T i cos δ cos δ sin Z ss κ sin θ cos δ dσ. (20)It can be easily verified that (12), (18), and (20) meet the requirement: κ = (cid:12)(cid:12)(cid:12)(cid:12) d T ds (cid:12)(cid:12)(cid:12)(cid:12) . (21)Generating an expression for the torsion τ requires first computing N bysubstituting (12)-(14) into (8)-(10): N i = − cos θ sin δ cos β − sin β sin θ ; (22) N j = cos θ cos δ ; (23) N k = − cos θ sin δ sin β + cos β sin θ. (24)Next, B = T × N : B i = sin δ sin θ cos β − cos θ sin β ; (25) B j = − cos δ sin θ ; (26) B k = sin δ sin θ sin β + cos θ cos β. (27)Equation (28) expresses the torsion as a function of θ : τ = (cid:12)(cid:12)(cid:12)(cid:12) d B ds (cid:12)(cid:12)(cid:12)(cid:12) = dθds − κ tan δ sin θ. (28)Equation (29) expresses τ in terms of components of T and B : τ = dθds + κT j B j − T j . (29)Finally, (2) serves as a check on the solutions for T , N , B , and τ . Integrating (29) leads to the following expression for θ : θ = θ + Z ss τ − κT j B j − T j ! dσ. (30)5quation (30) indicates a unique value of θ for each specified value of τ when T j = ±
1. Thus, (12), (18), and (20) indirectly solve (1)-(3).The angle θ can also be expressed in terms of components of T , N , B : θ = − sin − B j q − T j = cos − N j q − T j = − tan − B j N j . (31) θ An explicit solution often results when θ is constant. Setting T i = 1, so that β = δ = 0 (and setting s = 0) leads to δ = Z s κ ( σ ) cos θ dσ ; (32) β = 2 tan θ tanh − (tan δ/
2) ; (33)so that T i = cos (cid:2) θ tanh − (tan δ/ (cid:3) cos Z s κ ( σ ) cos θ dσ ; (34) T j = sin Z s κ ( σ ) cos θ dσ ; (35) T k = sin (cid:2) θ tanh − (tan δ/ (cid:3) cos Z s κ ( σ ) cos θ dσ. (36)The torsion becomes τ ( s ) = − κ ( s ) sin θ tan Z s κ ( σ ) cos θ dσ . (37)As an example, when κ = κ e − s , (38) T i = cos (cid:20) κ √ π s ) cos θ (cid:21) cos (cid:20) θ tanh − (cid:18) tan (cid:20) κ √ π s ) cos θ (cid:21)(cid:19)(cid:21) ;(39) T j = sin (cid:20) κ √ π s ) cos θ (cid:21) ; (40) T k = cos (cid:20) κ √ π s ) cos θ (cid:21) sin (cid:20) θ tanh − (cid:18) tan (cid:20) κ √ π s ) cos θ (cid:21)(cid:19)(cid:21) ;(41) τ ( s ) = − κ e − s sin θ tan (cid:20) κ √ π s ) cos θ (cid:21) . (42)6 .2 Constant κ When κ = κ but θ = θ , the solution will typically involve undeterminedintegrals. For example, when κ = κ and θ = κ s , T i = cos (sin κ s ) cos Z s κ sin κ σ cos (sin κ σ ) dσ ; (43) T j = sin (sin κ s ) ; (44) T k = cos (sin κ s ) sin Z s κ sin κ σ cos (sin κ σ ) dσ ; (45)and τ ( s ) = κ − κ tan (sin κ s ) sin κ s. (46) κ and θ When κ = κ and θ = θ , (34)-(37) become: T i = cos ( κ s cos θ ) cos h θ tanh − (cid:16) tan h κ s θ i(cid:17)i ; (47) T j = sin ( κ s cos θ ) ; (48) T k = cos ( κ s cos θ ) sin h θ tanh − (cid:16) tan h κ s θ i(cid:17)i ; (49) τ ( s ) = − κ sin θ tan ( κ s cos θ ) . (50)When θ = π/ τ ( s ) = 0, confining T and N to a plane. When T alignswith j , τ → ∞ in (50), and the equations break down. The equations break down when T j → ±
1, requiring a different orientation forthe local coordinate system. The angle of rotation of the osculating plane isdesignated φ here. In general, φ = θ , reflecting differences in angular orientationbetween the local and global coordinate systems for the two cases. Defining k ′ as the normal to the plane containing T and i , i.e., k ′ = i × T | i × T | = − j T k + k T j p − T i . (51)The j ′ unit vector becomes: j ′ = k ′ × T = − i (cid:0) − T i (cid:1) + j T i T j + k T i T k p − T i . (52)7ubstituting into the expression for N : N i = 1 κ dT i ds = − cos φ q − T i ; (53) N j = 1 κ dT j ds = − T k sin φ + T i T j cos φ p − T i ; (54) N k = 1 κ dT k ds = T j sin φ + T i T k cos φ p − T i . (55)Equations (53)-(55) have the following solution: T i = sin γ ; (56) T j = cos γ cos α ; (57) T k = cos γ sin α ; (58) N i = − cos γ cos φ ; (59) N j = sin γ cos α cos φ − sin α sin φ ; (60) N k = sin γ sin α cos φ + cos α sin φ ; (61) B i = cos γ sin φ ; (62) B j = sin γ cos α sin φ + sin α cos φ ; (63) B k = − sin γ sin α sin φ + cos α cos φ ; (64) τ = dφds − κ tan γ sin φ = dφds − κB i T i − T i . (65)Here γ = sin − T i − Z ss κ cos φdσ ; (66) α = cos − (cid:18) T j cos γ (cid:19) = α + Z ss κ sin φ cos γ dσ ; (67) T j = T j cos γ cos γ cos Z ss κ sin φ cos γ dσ − T k cos γ cos γ sin Z ss κ sin φ cos γ dσ ; (68) φ = − tan − B i N i . (69)Even though θ and φ both represent the angle of rotation of the osculatingplane, (31) and (69) differ because of differences in angular orientation of thelocal coordinate system.When T j → ± T i → ±
1, switching from one set of equations to anotheravoids numerical difficulties. 8
Concluding Remarks
Unlike the Frenet-Serret equations, (8)-(10) are nonlinear, and do not involve N , B , or τ . The solution (in terms of κ and θ ) indirectly solves the Frenet-Serret equations, and leads to a precise definition of τ as a function of κ and θ .A unique value of θ can be obtained for each specified value of τ through a firstorder ordinary differential equation. The equations break down when T → ± j ,requiring an alternative set of equations that break down when T → ± i . Theexpressions for the angle of the osculating plane in the two approaches differbecause of differences in the angular orientation of the local coordinate system. Acknowledgement
This work was funded by the Office of Naval Research,Code 321US (M. Vaccaro).
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