Improving Frenet's Frame Using Bishop's Frame
IImproving Frenet’s Frame Using Bishop’s Frame
Daniel Carroll , Emek K¨ose & Ivan Sterling Mathematics and Computer Science Department, St Mary’s College of Maryland, St Mary’s City, MD, 20686,USACorrespondence: Ivan Sterling, Mathematics and Computer Science Department, St Mary’s College of Maryland,St Mary’s City, MD, 20686, USA. Tel: 1-240-431-8185. E-mail: [email protected]
Abstract
The main drawback of the Frenet frame is that it is undefined at those points where the curvature is zero. Further-more, in the case of planar curves, the Frenet frame does not agree with the standard framing of curves in the plane.The main drawback of the Bishop frame is that the principle normal vector N is not in it. Our new frame, whichwe call the Beta frame, combines, on a large set of curves, the best aspects of the Bishop frames and the Frenetframes. It yields a globally defined normal, a globally defined signed curvature, and a globally defined torsion. Forplanar curves it agrees with the standard framing of curves in the plane.
Keywords:
Frenet Frames, Bishop Frames.
Let γ : ( a , b ) −→ R be a curve in R . If γ is C with γ (cid:48) ( t ) (cid:44)
0, then it carries a Bishop frame { T , M , M } . If γ is C with γ (cid:48) ( t ) and γ (cid:48)(cid:48) ( t ) linearly independent, then it carries a Frenet frame { T , N , B } . The main drawback of theFrenet frame is that it is undefined when γ (cid:48)(cid:48) ( t ) =
0. This corresponds precisely to those points where the curvature κ ( t ) is zero. Also the principle normal vector N ( t ) of the Frenet frame may have a non-removable discontinuity atthese points. In the case of planar curves, the Frenet frame does not agree with the standard framing of curves inthe plane. Finally the torsion τ ( t ) is not defined when κ ( t ) =
0. The main drawback of the Bishop frame is that theprinciple normal vector N is not (except in rare cases) in the set { T , M , M } .The history of the Frenet equations for a curve in R is interesting. Discovered in 1831 by Sen ff and Bartels theyshould probably be called the Sen ff -Bartels equations. In 1847 they were rediscovered in the dissertation of Frenet,which was published in 1852. Independently they were also discovered (and published) by Serret in 1851. See(Reich, 1973) for details on this early history.Bishop frames were introduced in 1975 in the Monthly article “There is More Than One Way to Frame a Curve”(Bishop, 1975). Bishop frames are now ubiquitous in the literature on curve theory and its applications.Our new frame, which we call the Beta frame of γ , combines, on a large set of curves, the best aspects of theBishop frames and the Frenet frames. It yields a globally defined normal N β , a globally defined signed curvature κ β , and a globally defined torsion τ β . If γ is planar, it agrees with the standard framing of curves in the plane.Our approach was motivated by attempts to improve the details of our work on discrete Frenet frames (Carroll,Hankins, K¨ose & Sterling). The Beta frame introduced in this paper discretizes in a natural way consistent with ourdiscrete frame defined in (Carroll, Hankins, K¨ose & Sterling). These ideas are particularly useful in applications,such as DNA analysis and computer graphics. For example see Hanson’s technical report (Hanson, 2007), whichdiscusses several of the issues that we address here.Notation: C means continuous, C ∞ means infinitely di ff erentiable, and C ω means real analytic. For k ∈ N we saya function is C k if its derivatives up to order k are continuous. If γ (cid:48) ( s ) (cid:44) t , then γ can be reparametrizedby arclength s . If the derivative with respect to s is denoted by ˙ γ , then (cid:107) ˙ γ ( s ) (cid:107) ≡
1. Whenever necessary to simplifynotation we assume 0 ∈ ( a , b ). Finally, by an abuse of language, we use the term “planar curve” to mean a curve in R ⊂ R .The authors would like to thank the referees for helpful comments. a r X i v : . [ m a t h . DG ] N ov Before Bishop R Let γ : ( a , b ) −→ R be a C curve in two-space with (cid:107) ˙ γ ( s ) (cid:107) ≡ T ( s ) : = ˙ γ ( s ) is called the unit tangent vector to γ at s . Let the normal N ( s ) be the unique unit vector orthogonal to T ( s ) such that { T ( s ) , N ( s ) } is positively oriented.(We think of this N as the good normal, as opposed to the bad normal below.) The signed curvature κ signed isdefined by ˙ T ( s ) = : κ signed ( s ) N ( s ) . The “unsigned curvature”, the curvature of the osculating circle, is κ ( s ) : = | κ signed ( s ) | . Alternatively, not wisely, one could first define curvature κ bad by κ bad ( s ) : = (cid:107) ˙ T ( s ) (cid:107) and then define, when κ bad ( s ) (cid:44)
0, the normal N bad ( s ) by N bad ( s ) : = ˙ T ( s ) (cid:107) ˙ T ( s ) (cid:107) = ˙ T ( s ) κ bad ( s ) . There seems to be little advantage to this “bad” alternative and at least two drawbacks. The first drawback is that N bad ( s ) is not defined when κ bad ( s ) =
0, hence N bad ( s ) has at best a removable discontinuity when κ bad ( s ) =
0. Thesecond drawback is that if γ ( s ) changes concavity then N bad ( s ) has a jump discontinuity as for example in Figure1. N (cid:72) (cid:76) Not DefinedN Jumps
Frenet Frame N (cid:72) (cid:76) Is DefinedNo Jump
Beta Frame
Figure 1: The Di ff erence Between the Frenet Frame and the Beta Frame R Even though the alternative is bad, it is this bad alternative which is used in the standard Frenet framing for C curves γ : ( a , b ) −→ R with (cid:107) ˙ γ ( s ) (cid:107) ≡
1. We first define Frenet’s curvature κ f by κ f ( s ) : = (cid:107) ˙ T ( s ) (cid:107) and then define, when κ f ( s ) (cid:44)
0, the Frenet (principal) normal N f by N f ( s ) : = ˙ T ( s ) (cid:107) ˙ T ( s ) (cid:107) = ˙ T ( s ) κ f ( s ) . Note that for planar curves κ f = κ bad and N f = N bad . As mentioned above, for planar curves the Frenet framemay not agree with the positively oriented standard frame for curves in R . Roughly speaking the purpose of thisaper is to do away with “bad” (or non-existent) normals whenever possible. The Frenet (principal) binormal B f is defined by B f ( s ) = T ( s ) × N f ( s ).If γ is C with (cid:107) ˙ γ ( s ) (cid:107) ≡ κ f ( s ) (cid:44)
0, then τ f ( s ) is defined by τ f = (cid:104) ˙ γ × ¨ γ, ... γ (cid:105) κ f . The torsion τ f ( s ) measures the rate of change of the osculating plane, the plane spanned by T ( s ) and N f ( s ). Onehas the Frenet equations: ˙ T = κ f N f , ˙ N f = − κ f T + τ f B f , ˙ B f = − τ N f . The set, { T , N f , B f } , is called the Frenet frame of γ . In Chapter 1 of (Spivak, 1990) Spivak discusses why we cannot obtain a signed curvature κ signed for curves in threespace and why we cannot, in general, hope to define torsion τ ( s ) at points where κ f ( s ) = T ( s ) in R . Furthermore the Frenet frame, in particular the principle normal N f ( s ), is only defined onintervals where κ f ( s ) (cid:44)
0. There may be no consistent way to choose the normal after passing through a point s with κ f ( s ) =
0. If κ f ( s ) = N f ( s ) may have a non-removable discontinuity at s . However, we are able, fora large set of curves, to define a new frame, the Beta frame, { T , N β , B β } , which is at least C and is defined evenwhen κ f ( s ) =
0. Furthermore N β ( s ) = ± N f ( s ), whenever N f ( s ) is defined. Once we have a global definition of N β ,we define the signed curvature κ β “the good way” by˙ T = : κ β N β . For all s we will have κ β ( s ) = ± κ f ( s ). If γ is planar, then N β = N (the good normal) and κ β = κ signed .With respect to torsion, Spivak argues that one cannot define τ f ( s ) when κ f ( s ) =
0, because there exist exampleswhere no reasonable definition would make sense.
Example 3.1. (Spivak’s Example. See Figure 2.) If γ is the C ∞ curve defined by γ ( t ) : = ( s , e / s , if t > , (0 , , if t = , ( s , , e / s ) if t < . then τ ( t ) = everywhere except t = . But at t = the osculating plane jumps by an angle π . Any attempt to define τ (0) would involve distributions and delta functions, which we will not pursue in this paper. We take a di ff erent approach than Spivak. Instead of focusing on the examples where no reasonable definition ispossible, we give conditions, satisfied by many curves of interest, that allow τ β ( s ) to be defined even at points s where κ f ( s ) =
0. Furthermore τ β = τ f whenever τ f is defined. Finally we will still have the Frenet equations:˙ T = κ β N β , ˙ N β = − κ β T + τ β B β , ˙ B β = − τ β N β . Let γ : ( a , b ) −→ R be a C with (cid:107) ˙ γ (cid:107) ≡
1. The construction of a Bishop frame [2] is based on the idea of relativelyparallel fields. In particular, a normal vector field M ( s ) along a curve is called relatively parallel if˙ M ( s ) = g ( s ) T ( s )igure 2: Spivak’s Examplefor some function g ( s ). A single unit normal vector M at γ ( s ) generates a unique relatively parallel unit normalvector field M ( s ) along γ with M ( s ) = M . Moreover, any orthonormal basis { T , M , M } at γ ( s ) generates aunique C orthonormal frame { T , M , M } . Bishop’s equations are similar to the Frenet equations:˙ T = κ M + κ M , ˙ M = − κ T , ˙ M = − κ T . We have κ f ( s ) = (cid:113) κ ( s ) + κ ( s ). If κ f ( s ) (cid:44)
0, then N f ( s ) = κ ( s ) κ f ( s ) M ( s ) + κ ( s ) κ f ( s ) M ( s ) . On sub-intervals of ( a , b ) where κ f ( s ) (cid:44)
0, there exists a C function θ ( s ) such that N f ( s ) = cos θ ( s ) M ( s ) + sin θ ( s ) M ( s ). If moreover γ is C , then τ f ( s ) = ˙ θ ( s ). In most applications the normal portion of the Bishop frame, span { M , M } , is usually written using this polar coordinate approach in complex form:( κ , κ ) = κ e i (cid:82) τ . We will investigate these polar coordinates in some detail, but not using the complex form.
Remark 4.1.
The function (cid:82) κ is called the turn of γ . We have θ = (cid:82) τ . θ is related to the twist and the writhe of γ which we won’t discuss here. If γ is C and (cid:107) ˙ γ ( s ) (cid:107) ≡
1, then the curve ( κ ( s ) , κ ( s )) = ( r ( s ) , θ ( s )) in { M , M } space is called Bishop’s normaldevelopment of γ . The normal development of γ is determined up to rotation by a constant angle in the { M , M } plane and a curve γ is determined up to congruence by its normal development. The parameter s is an arclengthparameter for γ , but in general is not an arclength parameter for the normal development of γ . The normal de-velopment of a line is the constant curve whose image is the origin and the normal development of a circle is theconstant curve whose image is ( κ fconst (cid:44) , θ const ).If γ is planar, then it has vanishing torsion and the normal development is given by ( r ( s ) , θ const ). For any γ ∈ R , zeros of the normal development corresponds to points of zero curvature on γ . If the normal development( r ( s ) , θ ( s )) approaches the origin along a line and leaves the origin along a di ff erent line, the corresponding γ is like Spivak’s Example 3.1 above, it jumps from lying in one plane to lying in a di ff erent plane. The normaldevelopment of a helix is a constant speed circle around the origin. A C curve is spherical if and only if its normaldevelopment lies on a line not through the origin [2].f γ is C , we have seen that when κ f ( s ) (cid:44)
0, we have, for some function θ ( s ), τ f ( s ) = ˙ θ ( s ). In particular, τ f ( s )will change signs at the local extrema of θ ( s ). Curves of constant torsion ± θ ( s ) = ± s + θ with r ( s )arbitrary. Curves of constant curvature 1 correspond to curves with r ( s ) ≡ θ ( s ) arbitrary. As we have seen points where κ f ( s ) = γ correspond to the points on the normal development with ( r ( s ) , θ ( s )) = (0 , κ f ( s ) ≡ γ is a line segment on that interval, and the normal development remainsat (0 ,
0) on that interval. Dealing with the case of “piecewise” defined curves including line segments is a delicateproblem which we will address elsewhere. We will only consider curves that have isolated points of zero curvature.We assume that ( κ ( s ) , κ ( s )) has an isolated zero at s .Recall that ( κ ( s ) , κ ( s )) is C . Nevertheless there may not exist any pair of C functions ˜ r ( s ), ˜ θ ( s ) such that(˜ r ( s ) , ˜ θ ( s )) = ( κ ( s ) , κ ( s )). Even if ( κ ( s ) , κ ( s )) is C ∞ , there may not exist such C functions ˜ r ( s ), ˜ θ ( s ). Moreprecisely, let the Cartesian plane be defined by R x , y ) = { ( x , y ) | − ∞ < x < ∞ , −∞ < y < ∞} , and let the (extended) Polar plane be defined by R r , ˜ θ ) = { (˜ r , ˜ θ ) | − ∞ < ˜ r < ∞ , −∞ < ˜ θ < ∞} . Note that usually the definition of Polar plane restricts to r > θ ∈ [0 , π ), but we will use “Polar plane” inthe extended sense as defined in R r , ˜ θ ) . Let π : R r ,θ ) −→ R x , y ) be given by π ( r , θ ) = ( r cos θ, r sin θ ). Note that π is C ω . Curves which are C in the Polar plane project down to curves which are C in the Cartesian plane. Howevernot all C curves in the Cartesian plane are projections of C curves in the Polar plane. Examples include thoseSpivak-like curves which enter the origin from one direction θ and leave from a “non-parallel” direction θ . Figure3 shows this and two other cases. (cid:45) (cid:45) r (cid:142) (cid:45) (cid:45) Θ(cid:142) (cid:45) (cid:45) Κ (cid:45) (cid:45) Κ (a) Spivak-Like Lift (cid:45) (cid:45) r (cid:142) Θ(cid:142) (cid:45) (cid:45) Κ (cid:45) (cid:45) Κ (b) (˜ r , ˜ θ ) = ( s , π + π sin s ) r (cid:142) Θ(cid:142) (cid:45) (cid:45) Κ (cid:45) (cid:45) Κ (c) (˜ r , ˜ θ ) = ( s , √ s ) Figure 3: Some Examples Where ˜ θ Is Not Continuouspecifically we ask, given ( κ , κ ) : ( a , b ) −→ R is C , under what conditions does there exist a C lift (˜ r , ˜ θ ) :( a , b ) −→ R such that π ◦ (˜ r ( s ) , ˜ θ ( s )) = ( κ ( s ) , κ ( s ))? If ( κ , κ ) is never (0 ,
0) then finding a lift (˜ r ( s ) , ˜ θ ( s )) istrivial. If ( κ ( s ) , κ ( s )) = (0 ,
0) is an isolated zero, then there exist a neighborhood D of s in ( a , b ) such that( κ ( s ) , κ ( s )) (cid:44) (0 ,
0) except at s . At each such isolated zero we require the existence of a corresponding C lift(˜ r , ˜ θ ) : D −→ R r , ˜ θ ) . The first thought that comes at the reader’s mind may be that the lift exists if and only if thederivatives in s + and s − coincide. However, Figure 4(b), is a simply example to show this is not true. In fact muchmore subtle examples can be constructed. Figures 4 and 5 show some simple cases where C lifts exist. A unique C global lift (˜ r , ˜ θ ) is then constructed by patching together the pieces through (0 ,
0) and pieces which avoid (0 , C , the relationship with the Frenet frame (and in particular the principalnormal) is not. To recover a C “principal normal” a detailed analysis of these zeros is required. Precisely the lackof such a global normal in the literature was the primary motivation of this paper. (cid:45) (cid:45) r (cid:142) Θ(cid:142) (cid:45) (cid:45) Κ (cid:45) (cid:45) Κ (a) (˜ r , ˜ θ ) = ( s , s + π ) (cid:45) (cid:45) r (cid:142) Θ(cid:142) (cid:45) (cid:45) Κ (cid:45) (cid:45) Κ (b) (˜ r , ˜ θ ) = ( | s | , s + π ) (cid:45) (cid:45) r (cid:142) Θ(cid:142) (cid:45) (cid:45) Κ (cid:45) (cid:45) Κ (c) (˜ r , ˜ θ ) = ( s , | s | + π ) Figure 4: Simple Examples Where ˜ θ Is Continuous
This subsection and the following Lemma are technical and unappealing, but straightforward.We define ˆ θ : R − (0 , −→ (cid:16) − π , π (cid:105) byˆ θ ( x , y ) : = tan − (cid:16) yx (cid:17) if x (cid:44) , π if x = , y (cid:44) , undefined if x = , y = . If φ = Arg( x + iy ) is defined to be the unique argument of x + iy in [0 , π ), thenˆ θ = φ if φ ∈ (cid:104) , π (cid:105) ,φ − π if φ ∈ (cid:16) π , π (cid:105) ,φ − π if φ ∈ (cid:16) π , π (cid:17) . A graph of ˆ θ as a function of φ is shown in Figure 6. Note (cid:45) r (cid:142) Θ(cid:142) (cid:45) (cid:45) Κ (cid:45) (cid:45) Κ Figure 5: (˜ r , ˜ θ ) = ( s , π + s π sin s ) Another Example Where ˜ θ Is Continuous Π Π Π Π Φ(cid:45) Π Π Θ(cid:96)
Figure 6: ˆ θ as a function of φ ˆ θ (cid:16) π (˜ r ( s ) , ˜ θ ( s )) (cid:17) = (cid:40) ˜ θ ( s ) mod π if 0 ≤ ˜ θ ( s ) mod π ≤ π , (cid:16) ˜ θ ( s ) mod π (cid:17) − π if π < ˜ θ ( s ) mod π < π. tan − and hence ˆ θ jumps at π . Thus special care must be taken to deal with the case of ( κ , κ ) oscillating acrossthe y -axis ( x ≡
0) infinitely often as it approaches the origin. This would cause ˆ θ to oscillate wildly between near π and near − π . To avoid this cosmetic problem we (in this case only) rotate ( κ , κ ) by an angle π (so the curve isnow oscillating harmlessly about − π ). Then we can check its behavior without the jumping. We will refer to thisspecial case as Case 3; Case 1 being − π < ˆ θ + < π and Case 2 being ˆ θ + = π .Let ˆ θ + : = lim s → s + ˆ θ ( κ ( s ) , κ ( s )) , and ˆ θ + π : = lim s → s + ˆ θ √ κ ( s ) + √ κ ( s ) , − √ κ ( s ) + √ κ ( s ) . ote that these limits may not exist. Finally we define θ + : = ˆ θ + if − π < ˆ θ + < π , π if ˆ θ + π = − π , undefined otherwise . Replacing s → s + with s → s − we similarly define θ − . Lemma 6.1.
Let ( κ , κ ) be C with an isolated zero at s . Then there exists a C lift (˜ r , ˜ θ ) near s such that π ◦ (˜ r , ˜ θ ) = ( κ , κ ) if and only if both θ + and θ − exist and θ + = θ − .Proof. First assume there does exist a C lift (˜ r , ˜ θ ) near s such that π ◦ (˜ r , ˜ θ ) = ( κ , κ ). In other words for all s near s we have (cid:16) ˜ r ( s ) cos ˜ θ ( s ) , ˜ r ( s ) sin ˜ θ ( s ) (cid:17) = ( κ ( s ) , κ ( s )) . Since the zero is isolated at s we may assume ˜ r ( s ) (cid:44) s .We next prove θ + exists. By definition this means that either ˆ θ + exists with − π < ˆ θ + < π and / or ˆ θ + π exists withˆ θ + π = − π . By continuity lim s → s + ˜ r ( s ) = = lim s → s − ˜ r ( s ) and lim s → s + ˜ θ ( s ) = ˜ θ ( s ) = lim s → s − ˜ θ ( s ) . (1)When s (cid:44) s we have ˆ θ ( κ ( s ) , κ ( s )) = tan − (cid:16) ˜ r ( s ) sin ˜ θ ( s )˜ r ( s ) cos ˜ θ ( s ) (cid:17) if cos ˜ θ ( s ) (cid:44) , π if cos ˜ θ ( s ) = . = ⇒ ˆ θ ( s ) = (cid:40) tan − (cid:16) tan(˜ θ ( s )) (cid:17) if ˜ θ ( s ) mod π (cid:44) π , π if ˜ θ ( s ) mod π = π . By conditions (1) eventually ˜ θ ( s ) is near ˜ θ ( s ). If ˜ θ ( s ) mod π (cid:44) π , then eventually ˜ θ ( s ) mod π (cid:44) π and θ + = ˆ θ + = tan − (cid:16) tan(˜ θ ( s )) (cid:17) . If ˜ θ ( s ) mod π = π , then we rotate by π and by the same argument we haveˆ θ + π = tan − (cid:18) tan(˜ θ ( s ) + π (cid:19) = tan − (cid:32) tan( 3 π (cid:33) = − π . So θ + = π . Thus in either case θ + exists.Similarly θ − exists and by condition (1) θ + = θ − .Conversely assume θ + and θ − exist and θ + = θ − . Let (˜ r + ( s ) , ˜ θ + ( s )) (resp. (˜ r − ( s ) , ˜ θ − ( s )) be any C lift for s > s (resp. s < s ). Still assuming r ( s ) (cid:44) s (cid:44) s , without loss of generality assume both ˜ r + ( s ) > r − ( s ) > s (cid:44) s .First we consider the Case 1 where − π < ˆ θ + < π (and hence − π < ˆ θ − < π ). We want to show there is a C lift(˜ r , ˜ θ ). We claim lim s → s + ˜ θ + ( s ) and lim s → s − ˜ θ − ( s ) exist. More precisely we have − π < lim s → s + tan − (cid:32) κ ( s ) κ ( s ) (cid:33) < π . Or − π < lim s → s + tan − (cid:32) ˜ r + ( s ) sin ˜ θ + ( s )˜ r + ( s ) cos ˜ θ + ( s ) (cid:33) < π . Or − π < lim s → s + tan − (cid:16) tan ˜ θ + ( s ) (cid:17) < π . Eventually ˜ θ + ( s ) mod π avoids π . Thus eventuallytan − (cid:16) tan ˜ θ + ( s ) (cid:17) = (cid:40) ˜ θ + ( s ) mod π or (cid:16) ˜ θ + ( s ) mod π (cid:17) − π. (2)n either case ˜ θ + ( s ) = lim s → s + ˜ θ + ( s ) exists. By the same argument ˜ θ − ( s ) = lim s → s − ˜ θ − ( s ) exists. Since θ + = θ − weknow by Equation (2) that ˜ θ + ( s ) − ˜ θ − ( s ) is a multiple of π . Since we have assumed both ˜ r + ( s ) > r − ( s ) > r ( s ) , ˜ θ ( s )) depending on whether ( κ , κ ) approaches the origin from the same or opposite directionsas s approaches s from the left and right. In the first case we have j ( s ) = ˜ θ + ( s ) − ˜ θ − ( s ) = π and(˜ r ( s ) , ˜ θ ( s )) : = (˜ r − ( s ) , ˜ θ − ( s ) + j ( s )) if s < s , (0 , ˜ θ + ( s )) if s = s , (˜ r + ( s ) , ˜ θ + ( s )) if s > s . If j ( s ) = ˜ θ + ( s ) − ˜ θ − ( s ) = π mod 2 π then(˜ r ( s ) , ˜ θ ( s )) : = ( − ˜ r − ( s ) , ˜ θ − ( s ) + j ( s )) if s < s , (0 , ˜ θ + ( s )) if s = s , (˜ r + ( s ) , ˜ θ + ( s )) if s > s . In Case 2 we assume ˆ θ + = π . Since this implies ˆ θ + π = − π we see that Case 2 is included in Case 3.Finally we consider Case 3: ˆ θ + π = − π . As discussed above ( κ , κ ) is oscillating across the y -axis, but otherwise converges nicely. We can C lift therotated ( κ , κ ) and then shift that (˜ r , ˜ θ ) by π . (cid:3) As mentioned above, if the conditions of Lemma 6.1 are valid at all points of zero curvature, then we have a global C lift (˜ r , ˜ θ ) of ( κ , κ ). Without loss of generality we will assume ˜ θ (0) = Without loss of generality we assume κ f (0) (cid:44) M ( s ) , M ( s ) by the following initial conditions:1. If γ is planar, then M (0) = N (0) , M (0) = T (0) × N (0),2. If γ is not-planar, then M (0) = N f (0) , M (0) = B f (0). Assume that the normal development of γ has a continuous lift (˜ r , ˜ θ ) with ˜ θ (0) = κ , κ ) = (˜ r cos ˜ θ, ˜ r sin ˜ θ ) . Then we can globally define N β by N β ( s ) : = cos ˜ θ ( s ) M ( s ) + sin ˜ θ ( s ) M ( s )and note that N β = ± N f whenever N f is defined. We define our signed curvature κ β by˙ T ( s ) = : κ β ( s ) N β ( s ) . Note κ β = ˜ r and κ β = ± κ f henever κ f is defined. We define B β by B β ( s ) : = T ( s ) × N β ( s )and note that B β = − sin ˜ θ M + cos ˜ θ M . The globally defined frame { T , N β , B β } is called the Beta frame. The Beta frame (when defined) is unique and isinvariant under regular, orientation preserving, base point fixing reparametrizations. If the base point changes, itmay happen that N β and B β globally switch signs. Finally we assume γ is C , (cid:107) ˙ γ (cid:107) ≡ r , ˜ θ ) of ( κ , κ ) as in the last section.In this case we have that κ , κ are C and after a bit of checking ˜ r is C . ˜ θ is once again more di ffi cult. Evenif both κ , κ are C , there is no guarantee that ˜ θ is C if κ β ( s ) = (cid:107) ( ˙ κ ( s ) , ˙ κ ( s )) (cid:107) (cid:44)
0. For example(˜ r , ˜ θ ) = ( s , s ). See Figure 7. (cid:45) (cid:45) r (cid:142) Θ(cid:142) (cid:45) (cid:45) Κ (cid:45) (cid:45) Κ Figure 7: (˜ r , ˜ θ ) = ( s , s ) An Example Where ˜ θ is C but not C Assuming ˜ θ ( s ) is C at all curvature zero points, then the lift (˜ r , ˜ θ ) is globally C and we define τ β by τ β ( s ) : = ˙˜ θ ( s )and note that τ β = τ f whenever τ f is defined.As promised we will still have the Frenet equations:˙ T = κ β N β , ˙ N β = − κ β T + τ β B β , ˙ B β = − τ β N β . eferences Carroll, D., Hankins, E., K¨ose, E. & Sterling, I.
A Survey of the Di ff erential Geometry of Discrete Curves , inpreparation.Bishop, R. (1975). There is more than one way to frame a curve , Amer. Math. Monthly 82, 246-251.http: // dx.doi.org / / Quaternion Frenet Frames: Making Optimal Tubes and Ribbons from Curves , Indiana Uni-versity Technical Report TR407.Reich, K. (1973).
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