A Survey of the Differential Geometry of Discrete Curves
AA Survey of the Differential Geometry ofDiscrete Curves
Daniel Carroll, Eleanor Hankins, Emek Kose, Ivan SterlingNovember 25, 2013
Discretization of curves is an ancient topic. Even discretization of curves withan eye toward differential geometry is over a century old. However there is nogeneral theory or methodology in the literature, despite the ubiquitous use ofdiscrete curves in mathematics and science. There are conflicting definitionsof even basic concepts such as discrete curvature κ , discrete torsion τ , ordiscrete Frenet frame.Consider for example the three equally worthy definitions of the curvatureof an angle derived in Section 2 by considering the problem of approximatingan N-gon (by N-gon we mean a regular N-gon) with sides of length (cid:96) by acircle: κ = 2 (cid:96) sin θ , κ = 2 (cid:96) tan θ , κ = θ(cid:96) . (1)In the literature each of these definitions occur frequently. For example [9],[7], [6]. As we show, the source of this variety is that each author chooseswhether to normalize their curvature by using the inscribed, circumscribedor centered circle of an N-gon. Although our initial interest was in particularapplications, we realized the need for a general approach and along the waydiscovered some pleasing theorems.Using Section 2 as a guide we proceed to build three theories of discrete1 a r X i v : . [ m a t h . DG ] N ov urves all of which culminate in a discrete version of the Frenet equations: DT e = κN v ,DN e = − κT v + τ B v , (2) DB e = − τ N v . Although dozens of discrete Frenet equations can be found in the literature,all have unpleasant error terms. Our approach is new, and the resultingEquations (2) are free any error terms. We also show that our definitions ofdiscrete length (cid:96) , curvature κ and τ reproduce a unique (up to rigid motion)discrete curve with the given (cid:96) , κ and τ .In each of the three cases – inscribed, circumscribed, and centered – therecorresponds a natural differential geometric way to define the discretizationof a smooth curve. These definitions are discussed in Section 4. Converselygiven a discrete curve there is a natural differential geometric way to splinethe curve. See Section 5. Of particular interest is that discrete curves in theplane R are naturally splined by special piecewise curves: constant curvaturein the inscribed case, clothoids in the circumscribed case, and elastic curvesin the centered case. In each case we argue for our definition by showingthat these splines are the constrained minimizers of the natural variable inthat case. Section 6 contains some brief comments about applications anddiscrete surface theory. For every N ≥
3, a circle has an inscribed, circumscribed and centered N-gon which discretizes it. An N-gon is called centered about a circle if itsperimeter equals the circumference of the circle. Conversely an N-gon has aninscribed, a circumscribed and a centered circle which splines it. See Figure1 for the case N = 4. The nomenclature can be confusing, so care must betaken. For example: “An N-gon is centered about a circle” means the circleis given first, whereas “a circle is inscribed inside an N-gon” means the N-gonis given first. From trigonometry, Figure 2, we are led to three definitionsof curvature for a given N-gon. Recall the curvature of a circle is defined by κ = 1 /r and that the exterior angle for an N-gon is θ = πN . We assume allsides have length (cid:96) and N is any real N >
0. We have the curvature of the2igure 1: Discretizing a Circle, Splining an N-goncircle inscribed in an N-gon is κ = 2 (cid:96) sin θ . (3)Similarly the curvature of the circle circumscribing an N-gon is κ = 2 (cid:96) tan θ , (4)and in the centered case we have κ = θ(cid:96) . (5)We will use these basic formulas to guide us in all the definitions thatfollow. In a way that will be made more precise below we consider θ as themeasure of the angle between neighboring “tangent vectors”. θ measuresthe turning of an N-gon at a vertex. For a discrete curve in three space ifwe similarly define φ to measure the angle between neighboring “binormalvectors” then φ measures the twisting of a discrete curve along its edge.See Figures 3 and 5. We define the curvature at a vertex of a discrete curve3 Θ (cid:144) r (cid:123) Θ Θ (cid:144) r (cid:123) Θ r (cid:123) Figure 2: Trigonometry for curvature of an N-Gon Θ p qT p T p T q p q Φ B p B q B p B q Figure 3: θ Measures Turning, φ Measures Twisting4n three space by Equations (3), (4), and (5). We are similarly led to definethe torsion at a vertex by τ := (cid:96) sin φ , in the inscribed case, (cid:96) tan φ , in the circumscribing case, φ(cid:96) , in the centered case. A discrete map is a function with domain Z , χ : Z −→ R . Such a map iscalled a discrete function (resp. curve) if the range is R (resp. R ). Sincewe work exclusively with these special ranges, we will use without furthercomment the standard operations of R and R . If χ : Z −→ R , then weoften use the notation χ i := χ ( i ). We define discrete differentiation (resp.addition) by ( Dχ ) i := χ i +1 − χ i (resp. ( M χ ) i := χ i +1 + χ i ). We will define the lengths (cid:96) i , curvatures κ i and torsions τ i of discrete curvesin such a way that given any (cid:96) i , κ i , τ i it is possible to reconstruct a discretecurve with these lengths, curvatures and torsions. We will also require that anatural discrete version of the Frenet equations hold. As we have seen, thereare at least three reasonable definitions of the curvature of the elementaryN-gon. We will investigate these three cases using the definitions of curvatureand torsion derived from the formulas above.Let γ orig be a discrete curve γ orig : Z −→ R , which we call “the original curve.” Then we define the curve γ : Z −→ R asfollows. See Figure 4 where the larger numbers are the indices for the originalcurve and the smaller numbers are the indices for the redefined curve. Firstwe define γ ( i ) := γ orig (cid:18) i − (cid:19) if i is oddand then γ ( i ) := γ ( i + 1) + γ ( i − i is even . (cid:45) (cid:45) (cid:45) (cid:45) Figure 4: Original and Redefined Discrete CurveNote that we recover the original curve from the odd indices of γ and that theeven indices are mapped to the midpoints of the original curve. We definethe discrete length by (cid:96) i := (cid:107) ( Dγ ) i (cid:107) .γ is parametrized by arc length if (cid:96) ≡ (cid:96) is constant. Note that (cid:96) γ ≡ (cid:96) = constant if (cid:96) γ orig ≡ (cid:96) . Forclarity of presentation we will assume from now on that γ is parametrizedproportional to arc length, (cid:107) Dγ (cid:107) ≡ (cid:96) = constant . The theory goes throughwithout this restriction. In each version (Inscribed, Circumscribed and Centered) we will produce twodiscrete Frenet frames { T e , N e , B e } and { T v , N v , B v } . First for { T e , N e , B e } : T e := Dγ (cid:107) Dγ (cid:107) = Dγ(cid:96) . T ei = T ei − if i is even. Then B ei := T ei × T ei +1 (cid:107) T ei × T ei +1 (cid:107) if i is evenand B ei := B ei − if i is odd . and finally for all i : N ei := B ei × T ei . For { T v , N v , B v } we have for all i : T vi := ( M T e ) i (cid:107) ( M T e ) i (cid:107) ,B vi := ( M B e ) i (cid:107) ( M B e ) i (cid:107) ,N vi := ( M N e ) i (cid:107) ( M N e ) i (cid:107) . Note for all i , N vi = B vi × T vi .As shown again in Figure 5 the frame is turning, about the axis deter-mined by the binormal, at the “vertices”. The frame is twisting, about theaxis determined by the tangent, at the “edges”. It was precisely this alternat-ing approach which lead to the elegant form of the discrete Frenet equations(6) given below; which do not appear in the literature.7 e T e T e B e B e N e N e N e N e B e Θ Φ
Figure 5: Bird’s Eye View
The positively oriented frames { T ei , N ei , B ei } determine orientations of { T ei , N ei } and { N ei , B ei } . We define θ i as the angle between T ei and T ei +1 and note that θ i = 0 if i is odd. We define φ i as the angle between B ei and B ei +1 with φ i = 0if i is even. To avoid technical details we will assume θ i , φ i ∈ [0 , π ].The curvature κ is defined by κ := (cid:107) DT e (cid:107) = (cid:96) sin θ , in the inscribed case, (cid:107) DT e (cid:107)(cid:107) MT e (cid:107) = (cid:96) tan θ , in the circumscribing case,2 sin − ( (cid:107) DT e (cid:107) ) = θ(cid:96) , in the centered case.Note that κ i = 0 if i is odd. Similarly the torsion τ is defined by τ := φ (cid:107) DB e (cid:107) = (cid:96) sin φ , in the inscribed case, (cid:107) DB e (cid:107)(cid:107) MB e (cid:107) = (cid:96) tan φ , in the circumscribing case,2 sin − (cid:107) DB e (cid:107) = φ(cid:96) , in the centered case.With τ i = 0 if i is even. 8 .4 Discrete Frenet Equations In each version (Inscribed, Circumscribed and Centered) a direct calculationshows that the discrete Frenet equations hold.
Theorem 3.1 DT e = κN v ,DN e = − κT v + τ B v , (6) DB e = − τ N v . On the other hand we can reconstruct the curve by the relations: T ei +1 = cos θ i T ei + sin θ i N ei ,N ei +1 = − sin θ i T ei +sin( θ i + φ i ) N ei − sin φ i B ei ,B ei +1 = cos φ i N ei + sin φ i B ei . and γ i +1 = γ i + T ei +1 .To summarize we have Theorem 3.2
Given θ i , φ i with θ i = 0 for i odd and φ i = 0 for i even.Then for arbitrary initial conditions γ , T e , N e , B e there exists a unique dis-crete curve γ with θ γi = θ i , φ γi = φ i satisfying γ (0) = γ , T γe = T e , N γe = N e , B γe = B e . Moreover, γ orig ( i ) := γ (2 i ) satisfies (cid:107) Dγ orig (cid:107) = 2 (cid:96) . Given a curve in R we would now like to discretize it. There is a canonicalgeometric discretization in each of our three cases. The only distinguishing feature in this case is that each vertex of the dis-cretization be on the curve itself. Thus any increasing map ι : Z −→ R willproduce an acceptable discrete curve δ := γ ◦ ι . See Figure 6.9 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 6: Inscribed Discretization
If there are no inflection points then we again take any increasing map ι : Z −→ R such that consecutive tangents are not parallel. We require that theedges of our discrete curve δ intersect tangentially with the given curve atthe points ( γ ◦ ι ) i . We define δ i to be the unique intersect point of tangentlines at ( γ ◦ ι ) i and ( γ ◦ ι ) i +1 as in Figure 7. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 7: Circumscribed DiscretizationIf there are isolated inflection points then they, as well as at least one pointin-between them, need to be included in the set of tangent points. If a curvehas infinitely many inflection points on a finite interval, then our algorithmfails. 10 .3 Centered 2D-Discretization
The natural centered discretization of a curve requires a bit more finesse.First, without loss of generality, we assume γ is parametrized by arc lengthand require our discretization to be parametrized proportional to arc length.Secondly, with loss of generality, we assume γ has no inflection points, say κ > κ i > M “large enough.” We take the specific ι : Z −→ R defined by ι ( i ) := iM andlet δ start = γ ◦ ι . For each i we offset δ starti along the (outward) normal to γ at δ starti by the amount offset i := k i M − sin k i M k i sin k i M , where k i is the curvature of γ at δ starti and we assume k i >
0. Note that thisformula is derived from the case of centered N-gons discretizing a circle. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 8: Offset DiscretizationTo see the offset more clearly we zoom into the center of the curve.11 (cid:45) (cid:45)
Figure 9: Offset Discretization ZoomThese offset points will be the even vertices, δ j , of our final discrete curve δ . Now we consider the condition that our discrete curve δ is to have thesame length as our original curve γ . With the additional conditions that (cid:107) δ j +1 − δ j (cid:107) = i M , (cid:107) δ j +2 − δ j +1 (cid:107) and κ j +1 >
0; we see there is one andonly one way to achieve this. See Figure 10. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 10: Centered DiscretizationAgain, we see more detail by zooming in, Figure 11.12 (cid:45) (cid:45)
Figure 11: Centered Discretization ZoomTo include inflection points requires more general parametrizations and wewill leave it as an exercise for the reader.
Figure 12: Discrete Curve to be Splined
In non-geometric splining, the best spline is usually related to the degreeof the polynomial γ ( t ) = ( x ( t ) , y ( t )) used to approximate the curve. Forexample a cubic spline is constructed using piece-wise cubic polynomials.Typically a cubic spline passes through the points of a discrete curve withcertain boundary conditions. Geometric splinings on the other hand are13ound by considering curves whose curvature function κ ( t ) is a low degreepolynomial. Alternatively a best geometric spline minimizes (cid:82) κ . An inscribing spline is one which tangentially goes through the midpointsof the edges of the given discrete curve. We seek a curve whose curvaturehas the lowest degree possible. Because we are assuming our discrete curvesare parametized proportional to arc length there is a trivial differentiableinscribed splining by pieces of curves of constant curvature. That is pieces ofcircles. See Figure 13. If our discrete curve is not parametrized proportionalto arc length, then the inscribed splining would require clothoids, which aredescribed in the next subsection. (cid:45)
Figure 13: Inscribed SpliningNotice the curvature jumps at the midpoints so our splining is not twicedifferentiable.
Curves with linear curvature are called first order clothoids. Given κ ( s ) = as + b (if a = 0, we get a piece of a circle, a “zeroth order clothoid”) then theturning angle θ is given by θ ( s ) = (cid:90) s κ ( t ) dt + θ . γ ( s ) = (cid:18)(cid:90) s cos θ ( t ) dt + x , (cid:90) s sin θ ( t ) dt + y (cid:19) . Similarly curves with quadratic curvature are second order clothoids, and soon.
A circumscribing spline is one which differentiable goes through the points ofthe given discrete curve. Unlike the case of inscribed splinings, it will rarelybe the case that a circumscribed splining will consist of piece of circles. Onthe other hand there will always be circumscribing splining, as in Figure 14,using first order clothoids. If there is more than one, we take the shortestone. This is called the fitting problem. See for example [3]. (cid:45)
Figure 14: Circumscribed SpliningNotice again the curvature jumps at the midpoints.
For the centered spline we first offset the vertices using the centered circlesof N-gons and take the directions of the desired spline at these offset pointto be the average of the incoming and outgoing directions of the edges at thevertices. See Figure 15. 15 .5 1.0 1.5 2.0 2.50.51.01.52.0
Figure 15: Centered Splining OffsetsWe then seek differentiable splines passing through these offset pointswhose length agrees with that of the given discrete curve. These curves arefound using (cid:82) κ and are called elastica. These are solutions to a variationalproblem proposed by Bernoulli to Euler in 1744; that of minimizing thebending energy of a thin inextensible wire. Among all curves of the samelength that not only pass through points A and B but are also tangent togiven straight lines at these points, it is defined as the one minimizing thevalue of the expression (cid:82) κ .The one parameter family of elastic curves introduced by Euler [10] is wellknown. They are all given by explicit formulas involving elliptic integrals.These formulas arise by solving the one-dimensional sine-Gordon differentialequation. Which is alternatively written as θ (cid:48)(cid:48) = sin θ or [8] θ (cid:48)(cid:48)(cid:48) + ( θ (cid:48) ) + Cθ (cid:48) =0. In applied problems, such as finding the elastic curve with the boundaryconditions care must be taken for several reasons. One issue is that thereare several types of “elastic intervals” (inflectional, non-inflectional, critical,circular, and linear). Another issue is that in some cases there are multiplesolutions. An excellent survey of the subject is in Andentov [1]. As discussedin [1] these problems persist when attempting to numerically approximateelastic curves.Sogo [11] shows how, at least in some cases, “integrable discretization”theory can be used to construct a discretized one-dimensional sine-Gordonequation satisfied by discretized elliptic integrals. For example the inflec-tional type elastic curve has a turning angle which is given by a formula,involving the Jacobi sn function, of the formsin θ θ KL ( L − s ) , k )16nd Sogo shows that sin θ j θ KN ( N − j ) , k )is the turning angle of an approximating discrete elastic curve.Figure 16 shows (one of) the differentiable elastic splines with minimalbending energy and length nine. (cid:45) Figure 16: Centered Splining • The discretization of smooth curves and the splining of discrete curvesin three space is also well studied. Similar, though at times more in-volved, case by case constructions can be carried out in all six casesdiscussed in detail for the curves in the plane. A complete understand-ing of circles and N-gons guides the discretization and splining methodsin the plane. Similarly by first carrying out the most basic case of thehelix one is able to succeed with curves in three space as well. • The three settings (Inscribed, Circumscribed, and Centered) and onlythese three settings, are used extensively in the literature. This istrue in both pure and applied differential geometry. Other settings weconsidered, although formally feasible, are not as natural. For examplethe reader can consider circles whose enclosed areas agree with theenclosed areas of a regular N -gons.17 We feel there is no absolute “right” definition of discrete curvatureor torsion. A particular application may inform the researcher as towhich definitions to use. For example clothoids arise in the buildingof highway off ramps. So in that case the circumscribed setting mightbe more natural. In a more abstract context the circumscribed settingis also used by T. Hoffman in his dissertation on discrete curves andsurfaces [7]. It seems clear that Gauss would have used the centeredsetting, as it agrees most closely with his definition of the curvatureof an angle between two intersecting curves and with his definition ofcurvature given by the normal Gauss map. This setting is used, forexample, by Doliwa and Santini [6] in their work on the integrabledynamics of discrete curves. • There is also a vast literature on discrete surface theory which goes backover one hundred years. See [4] and references there. Not surprisinglythere is an even wider variety of definitions for the standard conceptssuch as discrete Gauss curvature, discrete mean curvature, discreteumbilics, etc. Again it seems clear that there is no absolute “right”definition. How one chooses to define “the discretization” of a smoothsurface will again depend on which properties one wishes to preserve.The theory of “integrable discretizations” in particular has been appliedto soap bubbles, minimal surfaces, Hasimoto surfaces (i.e. the surfacesswept out by smoke-rings) and surfaces of constant Gauss curvature.Similar comments apply to the theory of splining discrete surfaces. • We have highlighted the Frenet frame because it is the most well knowncurve framing. Discrete versions of the Bishop frame [2],[5] can also bederived using the ideas of this paper. The Bishop frame is particularlyuseful for curves that have points of zero curvature. • We have considered only the simplest discretizations and the simplestsplinings. One which are as local as possible, taking into account onlythe “nearest neighbors.” We feel the diversity and elegance of the casescovered give a nice survey. Third order versions, either taking intoaccount more points for each calculation or by including curvature intoboundary conditions, can be found in the literature.18 eferences [1] Ardentov, A. A. and Sachkov, Yu. L.
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