The Extended Evolute as the Transition Set of Distance Functions

The Extended Evolute as the Transition Set of Distance Functions Xianming Chen Richard F. Riesenfeld Elaine Cohen Abstract This paper addresses three practical issues arising from applying singularity and unfolding theory on point/curve distance functions to any geometric applications: ) the unfolding formulas that relate the perturbation of critical distances to that of the plane point are derived; ) the theory is extended to the situation of piecewise smooth curves with at least C 0 continuity at break points; specifically, the evolute as the bifurcation set is generalized to the extended evolute as the transition set; and ) at each point of the extended evolute, which typically has self-intersections, asymptotes and line segments for a non-trivial generator curve, an inward normal or inward tangent that delineates the creation (of critical distances) direction is defined. Keywords: Distance function, critical distance, singularity, unfolding, bifurcation set, evolute, transition set, extended evolute. Introduction Singularity and unfolding theory is well known in the singularity community [,,,, 6]. J. Bruce and P. Giblin [6] especially discuss in detail the singularity of distance functions from plane points to a plane curve, showing that the bifurcation set of the unfolding of degenerate critical distances is exactly the curve evolute. It has important potential roles in many fundamental geometric computation problems, especially in providing strong guidance to the design of robust algorithms guaranteeing the correct topology. For example, offset curves are intimately related to the curve evolutes [4]. The locus of irregular points of offsets to a regular smooth curve is exactly the curve evolute; for the piecewise smooth curve case, we will show that that is exactly the extended evolute defined in this paper (see Fig. 5). e-mail: xchen@cs.utah.edu e-mail:rfr@cs.utah.edu e-mail:cohen@cs.utah.edu Minimal distances, penetration depth and collision detection (e.g., see [5, 8]) are important geometric computation problems in virtual reality, haptic rendering and simulation. Putting them in the more general context of critical distances will make the detection more robust, in the sense that no minima/maxima or collision are missed. A critical distance is usually either maximal or minimal, but occasionally it may be neither maximal nor minimal. The latter situation, being a singularity or a degeneracy, actually is the key to robust detection algorithms, because that is exactly when the topology (i.e., the number of critical distances, and their corresponding degeneracy and types) can under go change. This is especially true for a dynamic detection. Finally, the extent of the relation of singularity theory of distance functions to Blum medial axes [] and Voronoi diagrams (e.g., see [] for the closed smooth curve case) is also an interesting topic, deserving much investigation. There are three issues, however, need to be addressed before singularity theory can be possibly applied to any real world geometric applications. The first one is to derive the formulas to determine the perturbation of the new critical distance(s) relative to the original one. The second one is to extend the theory, which requires the curve to be at least G for A unfoldings (cf. Section 4), to piecewise polynomial or rational curves requiring only at least G 0 continuity at break points; included in the curve pieces are possibly line segments and circular arcs, which are very common in areas like CAD. The third issue is about the unfolding direction for the smooth curve case, or its extension as creation direction for the piecewise smooth case. Upon a perturbation of the plane point toward or along that direction, the original critical distance unfolds or transitions into one or more than one nearby regular critical distances. For the smooth curve case, the unfolding direction is usually regarded as self-evident to be the inward normal of the curve evolute. This is rightly so for simple case like the evolute of an ellipse (Fig. ), which is a simple closed There are actually two types of distance functions, namely that of point/model distances and that of model/model distances. Also the models are typically -dimensional. Currently we are investigating the extension of dynamic and continuous point/curve distances to these more practical situations. However, any result as sophisticated as the point/curve case is quite unlikely, because the singularity and unfolding theory of multivariate functions, unfortunately, is not as well-developed.

curve with four cusps. However, for most practical curves, especially with G 0 break points, the curve evolute is complicated with self intersections and asymptotes, and even gaps between consecutive segments (Figs.,, 4 and 5 ), and the discrimination between the inward normal and the outward one is a subtle issue. This paper is dedicated to solve these three problems. After a brief review of curve evolutes in Section,the singularities are formulated, in Section, directly based on the given curve parametrization instead of via arc-length re-parameterization as done in [6]. In Section 4, the perturbed critical distance(s) under either an evolution event or a transition event is explicitly derived using Taylor expansion. After that in Section 5, various transitions due to curve discontinuity are investigated. Putting all these together, in Section 6, we extend the bifurcation set, which delineates the unfolding of critical distances, to the transition set, which delineates the topological changes of the critical distances. The transition set is shown to be the extended evolute; at the mean time, the unfolding direction is extended to the creation direction on the extended evolute, and it is shown to be either the same or the opposite to that of the curve tangent, with the flipping of one direction to the other occurs only at some characteristic points on the curve. Finally, the paper concludes in Section 7. The Evolute of a G > Regular Curve Given a regular generator curve γ(s) with curvature (s), its evolute is the locus of the curvature centers, E(s) = γ(s) + N(s), (.) (s) where N N N is the unit curve normal. The derivatives and curvature of the evolute are closely related to the generator curve as, E = γ + ( ) N N N + N N N = ( ) N N N, (.) E = ( ) N N N ( ) γ. E = [E E ] E = ( ) [ N N N γ ] ( ) = γ ( ). (.) By Eq. (.), the evolute has cusps at points corresponding to curve vertexes where = 0, and is regular otherwise. Moreover, the curve normal is parallel to the corresponding evolute tangent. For more details on differential properties of evolutes, refer to [0, 4]. Singularity and Classification of Critical Distances Given a plane point p and a plane curve γ(s) : R R, the squared distance function f : R R is defined as, f(s) = (γ(s) p) (γ(s) p). (.) If, for some n > 0, f () (s 0 ) = f () (s 0 ) = = f (n) (s 0 ) = 0, f (n+) (s 0 ) 0, f is said to have A n singularity at s 0, or, there is an A n distance from the plane point p to the curve point γ(s 0 ) (also called foot point). Also, an A >k distance is any A n distance with n > k. An A n distance is called a critical distance if n > 0, and it is further classified as regular if n =, or degenerate with multiplicity n if n >. Remark By Taylor expansion, the local behavior of the distance function f(s) at an A k point s 0 is determined by f (k+) (s 0 ). The A k distance is locally neither maximal nor minimal if k is even; otherwise, it is locally maximal (minimal) if f (k+) (s 0 ) < 0 (f (k+) (s 0 ) > 0). At a critical distance s 0, the first order derivative, i.e., f = (γ p) γ, (.) vanishes. We will consider singularity of the considered critical distance, i.e., f (s 0 ) = 0, f (s 0 ) = 0, etc. First, the second order derivative is, f = ( γ + (γ p) γ ). (.) Decomposing γ p with respect to the local frame { T T T = γ γ, N N N γ = r γ } (the subscript r denotes the 90 degree counter-clock-wise rotation of the considered vector ), yields ( f = γ + (γp) N N N γ r γ γ γ γ ) γ + (γp) γ γ Recall that the curvature of a plane curve γ is[9], γ r = [γ γ ] γ = γ γ r γ γ, (.4) where the notation [a a b] denotes the determinant of the matrix formed out of any two plane vectors a and b, with the obvious identity [a a b] = a r b. Thus, using also Eq. (.), The above f can be transformed into, f = γ ( + (γp) N N N ) + f γ γ γ = γ D + γ γ γ f, (.5) All the results derived in this paper are actually equally valid if the 90 degree clock-wise direction is chosen as the normal.

where we have introduced the type determinant, Let D = + (γ p) N N N. (.6) d = (p γ) N N N (.7) be the signed distance from p to γ, then D = d ρ. (.8) D is used to determined the type of the considered regular critical distance, because by Eq. (.5), at a critical distance, Λ D = Λ f. where Λ a = + if a > 0, Λ a = if a < 0 and Λ a = 0 if a = 0 for any a R. Thus, Λ D = indicates a regular minimal critical distance, Λ D = indicates a maximal regular critical distance, and Λ D = 0 indicates a neither minimal nor maximal A distance. In this paper, we simply state that the type of the critical distance is Λ D. If Λ D = 0, and the distance is A n with n >, then it is neither minimal nor maximal if n is even, otherwise the type is Λ f (n+) (cf. Remark ). With the notations of D and Λ introduced, we are ready to have, Observation f (s) is γ (s) D(s) up to an additive term involving f (s). At an A singular point s 0, f (s 0 ) = γ (s 0 ) D(s 0 ), where D(s 0 ) = d(s0) ρ(s. The 0) type of critical distance is Λ D(s0). Remark Geometrically, the critical distance is minimal if the plane point p is opposite to the curved side or on the same side but closer to the curve than the curvature center; it is maximal if p is on the curved side and further away from the curve than the curvature center. See Fig. (a). Remark If (s 0 ) = 0 at the critical distance point s 0, then D(s 0 ) = by Eq. (.6). Hence, the critical distance is always regular and of type minimum at an inflectional point, or when the local curve is a line segment. Consequently, we will not consider this trivial situation in the rest of this paper, and (or its left and right limits and + ; see Section 5) is assumed to be non-vanishing, unless explicitly stated otherwise. Remark 4 By Eq. (.6), D(s 0 ) can also be if γ(s 0 ) = p. Mathematically, the distance from p to (the same point) γ(s 0 ) is still a critical distance. However, the perturbation of the p cannot incur any topological change because p is not on the local evolute (see Section 4). Therefore, it may be on the evolute point corresponding to a curve point other than γ(s 0 ). we will not consider this a trivial situation either. Hence, together with Remark, we will assume D(s 0 ) in the rest of this paper. Remark 5 Consider any function g(s), and assume, for some n > 0, g (n+) (s) is ψ(s)h(s) up to some additive terms involving lower order derivatives, i.e., g (n+) (s) = ψ(s)h(s) + i=n i= φ i(s)g (i) (s). At an A n singular point s 0 of g(s), g (n+) (s 0 ) = ψ(s 0 )h(s 0 ). If ψ(s) does not vanish locally at s 0, we have g (n+) (s) = ψ(s)h (s) + i=n+ i= ϕ i (s)g (i) (s) for some ϕ i s. Hence, by inductive reasoning, for k = 0,,, g (n+k+) (s) is ψ(s)h (k) (s) up to some additive terms involving lower order derivatives, and g (n+k+) (s 0 ) = ψ(s 0 )h (k) (s 0 ) g(s) is A n+k at s 0. Now, supposing the second order derivative of the distance function f vanishes at s 0 as well, i.e., f (s 0 ) = f (s 0 ) = 0. By Observation, D(s 0 ) = 0 or d(s 0 ) = ρ(s 0 ), i.e., the plane point p is exactly the corresponding curvature center, or on the curve evolute E. Further by Remark 5, f (s) is γ (s) D (s) up to some lower order derivative terms. By Eq. (.6), D (s) is, D = (γp) N N N + (γp) N N N + (γp) N N N = (γ p) γ + (γ p) N N N = f + D, where we have already used Eqs. (.) and (.6). Hence, Observation f (s) is γ (s) D(s) (s) (s), up to some lower order derivative terms. If f(s) is A at s 0, the plane point p has to be a regular evolute point, the critical distance is neither minimal nor maximal, and f (s 0 ) = γ (s 0) (s 0). See Fig. (a). By Remark 4, γ (s) D(s) (s) does not vanish locally for a regular curve. Therefore, Remark 5 again applies; especially, Observation f (4) (s) is γ (s) D(s) (s) (s), up to some lower order derivative terms. If f(s) is A at s 0, the plane point p has to be an evolute cusp, and f (4) (s 0 ) = γ (s 0) (s 0). Remark 6 Geometrically, the local curve is most (least) curved, and the corresponding evolute cusp approaches (leaves) the local curve if (s 0 ) (s 0 ) < 0 ((s 0 ) (s 0 ) > 0). if

(c) (a) A and A Distances (c) A Unfolding (b) A Distances (d) A Unfolding Figure. Classification, Degeneracy and Unfolding of Critical Distances The critical distances (all plane points on the same normal line) in (a) are minimal, multiplicity- degenerate and maximal, respectively. The types can be easily verified from the osculating circles. The in (b) are multiplicity- degenerate minimal and maximal, respectively. (c) and (d) show the unfoldings of the A singularity in (a) and the A singularity in (b), respectively. Notice that the displacement of foot points of unfolded critical distances are much larger than that of the plane point (cf. Lemma ). In all 4 images, critical distances are colored as: red is minimal, green is neither minimal nor maximal, and blue is maximal. 4 Approximating the Evolved/Unfolded Critical Distances using Taylor Expansion Let p = (x, y) in Eq. (.) be variable; f is now a trivariate function, f(s, p) = f(s, x, y) = (γ(s) p) : R R. f can also be regarded as a -parameter (x and y) family of distance functions R R, and as such, it is called the unfolding of any of its member function. The bifurcation set of the unfolding is [6], B = {(x, y) : f(s, x, y) s = f(s, x, y) s = 0, s R}, which is exactly the evolute of the curve. Intuitively, the bifurcation set is the locus of the plane points, where the corresponding critical distances are degenerate, and may transition into one or more than one nearby regular critical distances. Using Taylor expansion of multi-variable functions, in this section, we will derive explicitly the evolved or transitioned critical distances, assuming that the curve has continuous derivatives of the appropriate orders as necessary. In Section 5, we will investigate the discontinuity situation based on the result from this section. Notice that, hereafter in this paper, f should be understood as the partial derivative of f with respect to s, and similarly for f, f,. The critical distance occurs when f (s, p) = (γ(s) p) γ (s), vanishes. Consider f at a point (s 0 + s, x 0 + x, y 0 + y ) neighboring a local critical distance point (s 0, x 0, y 0 ), where s are the displacements of the corresponding coordinates shown as the subscripts. We also write = ( s, x, y ) and p = ( x, y ). The Taylor expansion [7] to the third order is, f (s 0 + s, p 0 + p ) = ( s s + p )f + ( s s + p ) f + 6 ( s s + p ) f + o( ) where = ( x, y). Notice that in this paper, we assume that all the RHS expressions of any Taylor expansions, are evaluated at the local point (s 0, x 0, y 0 ); specifically, the vanishing constant term is not shown in the above Taylor expansion. The derivatives are, s k f = f (k+), k =,,,, ( p )f = p γ (s), ( s k ( p ) n) f = 0, if n > (4.) ( ) s k ( p ) f = p γ (k+) (s), k = 0,,,,. Substituting into the Taylor expansion yields, f (s 0 + s, p 0 + p ) = f s ( p γ ) f s ( p γ ) s + 6 f (4) s ( p γ ) s + o( ) (4.) 4

Besides the Taylor expansion of f, which is used to determine what the neighboring critical distances are, we also need that of the type determinant D = +(γp) N N N, which is required to determine the types of those regular critical distances. Observing that s D = (γ p) N N N γ (γ p) T T T (4.) ( p )D = p N N N, ( p ) k D = 0, k =,,, (4.4) the Taylor expansion, to the first order, is, D(s 0 + s, p 0 + p ) = D + (γ p) N N N s p N N N + o( ) where the term related to γ (γ p) T T T in Eq. (4.) does not appear, because it vanishes at the local critical point (s 0, x 0, y 0 ). At an A distance, D 0, we have, D(s 0 + s, p 0 + p ) = D + o(). (4.5) And at an A > distance, D = 0 and (γ p) N N N = ρ, we have, D(s 0 + s, p 0 + p ) = s p N N N +o( ) (4.6) In the rest of this section, given a critical distance from the plane point p 0 to the curve point γ(s 0 ), we will find all the local critical distances when p 0 is perturbed by p. Under consideration are three situations: () A evolution when p 0 is on the normal line other than the curvature center; () A transition when p 0 is the regular curvature center; () A transition when p 0 is the curvature center of an ordinary vertex (i.e. where = 0, 0). 4. A Evolution p 0 is not on the curvature center, and thus by Observation, f (s 0 ) = γ (s 0 ) D(s 0 ) 0. The Taylor expansion Eq. (4.) can be truncated to the first order, and at the perturbed critical distance, or, s f ( p γ ) + o( ) = 0, s p γ γ D. (4.7) The perturbed critical distance has the same type as the original one by Eq. (4.5). Since there is no topological change in either the number of the critical distances or the corresponding degeneracy and types, we call this type of perturbation an evolution; otherwise, it is called a transition, as we have been actually doing so so far. 4. A Transition The plane point is on the curvature center corresponding to a non-vertex curve point, and thus f = f = 0, f 0. By truncating Eq. (4.) to the second order, we have, at the neighboring critical distances, p γ + f s p γ s +o( )=0 (4.8) Tangential Transition occurs upon a tangential perturbation when p γ = 0 or, the perturbation of p 0 is precisely along the evolute tangent (cf. Eq. (.)). Eq. (4.8) is now, f s ( p γ ) s + o( ) = 0 By Observation, f = γ. Thus, the original multiplicity- degenerate critical distance unfolds into two regular ones, with 4 { s = 0 s p γ γ = δ p, (4.9) where in the second equation, for tangential perturbation only, we have introduced the signed perturbation δ p such that and used the identity, By Eq. (4.6) and Eq. (4.0), γ r γ p = δ p N N N = r δp γ ; (4.0) γ r p γ γ = γ γ r γ γ δ p = δ p. (4.) D(s 0 + s, p 0 + p ) = δ p + o(δ p ), if s = 0; D(s 0 + s, p 0 + p ) = δ p + o(δ p ). if s = δ p. Therefore, the two unfolded critical distances have exactly the opposite types. Transversal Transition occurs upon a transversal perturbation when p γ 0 or, the perturbation of p 0 is transversal to the evolute tangent. Now the third term in Eq. (4.8) can be ignored 5, i.e. f s ( p γ ) = 0. 4 The first solution is exact because the remainder in the Taylor expansion vanishes by Eq. (4.). 5 Focusing only on the order, the quadratic form can be rewritten as x + ux + u = 0, where x is s to be sought, and u = O( p ). The roots are x = u± u 4u u± 4u ± u, where the approximation holds because u is an infinitesimal number. The final result simply verifies that term ux in the original equation can be safely ignored. 5

Hence, the original multiplicity- degenerate critical distance unfolds into two regular ones, with s = ± γ ( p γ ), (4.) if < 0, and annihilates (disappears) otherwise. The unfolding direction is consequently Λ γ γ r. Remark 7 Geometrically, the evolute curves away from the A transversal unfolding direction, at a regular point. A simple proof follows. By Eq. (.), the normal direction to the evolute is Λ N r = Λ γ. Notice that if the curvature of the evolute, E, is positive, the evolute curves toward the normal direction, and it curves away from the normal direction if E is negative. Therefore, the evolute curves away from the direction Λ E γ, which is Λ γ by Eq. (.). From the above unfolding result, ( p N N N) = O( s ); consequently, the Taylor expansion of D (cf. Eq. (4.6)) is, D(s 0 + s, p 0 + p ) = s + o( s ). (4.) Hence, the unfolded left and right critical distances have types Λ and Λ, respectively. 4. A Transition First let us notice that the type of the original A critical distance under consideration is Λ f (4), which is Λ by Observation. Later, we use Λ 0 to refer to the type of the original critical distance. Now substituting f = f = 0 into Eq. (4.) yields, f (s 0 + s, p 0 + p ) = ( p γ ) ( p γ ) s + 6 f (4) s ( p γ ) s + o( ). (4.4) Tangential Transition If ( p γ ) = 0, i.e., p 0 moves along the evolute tangent. then ( p γ ) s in Eq. (4.4) can be ignored 6, and thus at the perturbed critical distances, ( p γ ) s + 6 f (4) s 0. Therefore, the degenerate original multiplicity- degenerate critical distance tangentially unfolds into three regular 6 Because the constant term vanishes, then either s = 0, or the cubic equation is reduced to a quadratic one, and Footnote 5 applies. ones, with (cf. Eq. (4.)) 7 { s = 0, s ± 6 p γ γ = ± 6δ p, (4.5) if δ p < 0. Otherwise, it tangentially annihilates into a regular one, with s = 0. Therefore, the tangential unfolding direction is γ r. Notice that the latter is called an annihilation event because, algebraically, the original multiplicity- degenerate critical distance transitions into a regular one with multiplicity-; that is, two potential critical distances are annihilated. If s = 0, the Taylor expansion of D, i.e., Eq. (4.6), is transformed into, D(s 0 + s, p 0 + p ) = δ p + o(δ p ). Thus, { ΛD =Λ δp =Λ =Λ 0, if δ p < 0 (i.e., unfolding); Λ D =Λ δp =Λ =Λ 0, otherwise (i.e., annihilation). Now if s ± 6 p γ γ = ± γ r 6δ p, we have δ p = 6 s + o( s), and p = o( s ). Thus, the second term in the RHS of Eq. (4.6) is p N N N = δp = 6 s + o( s). (4.6) Moreover, the second order term in s is now required because the first order term, i.e. the first term in the RHS of Eq. (4.6), vanishes due to (s 0 ) = 0. By Eq. (4.), s D = s ( (γ p) N γ (γ p) T ) = (γ p) N γ (γ p) N γ = γ + γ =, (4.7) Th other two second order derivative terms contains, respectively, p s and p, both of which are o( s ). Therefore, we finally have D(s 0 + s, p 0 + p ) = and, = Λ D = Λ = Λ 0. s 6 s + o( s ) s + o( s ), Hence, the two new regular critical distance, with s = 6δ ± p, from tangential unfolding, have the same type as the original one. 7 The solution s = 0 is exact because the remainder in the Taylor expansion vanishes by Eq. (4.). 6

Transversal Transition If ( p γ ) does not vanish, i.e. p 0 does not move along the evolute tangent, both ( p γ ) s and ( p γ ) s in Eq. (4.4) can be ignored 8. Hence, at the perturbed critical distance, ( p γ ) + 6 f (4) s 0. By Observation, f (4) = γ, and thus, the original multiplicity- degenerate critical distance annihilates into a regular one, with 6 s p γ γ = 6δp (4.8) This also implies that p = o( s ), and we have similar derivation on the Taylor expansion of D as that in the tangential case, except that now the second term in the RHS of Eq. (4.6) is o( s ) (cf. Eq. (4.6); specifically, and, D(s 0 + s, p 0 + p ) = Λ D = Λ = Λ 0. s + o( s ), 4.4 The Order Relation between Plane Point Displacements and Foot Point Displacements By Eqs. (4.7), (4.9) and (4.), and (4.5) and (4.8), we have, Lemma Considering the evolved or transitioned critical distances upon a perturbation p of the corresponding plane point. The order of the perturbation of the foot point, s, is related to p as, () A Evolution: p = O( s). () A Transversal Unfolding: p = O( s ). () A Tangential Unfolding: s = 0 or p = O( s). (4) A Transversal Annihilation: p = O( s ). (5) A Tangent Annihilation: s = 0. (6) A Tangent Unfolding: s = 0 or p = O( s ). See Fig. (c) and (d) for a comparison between s and p in a real situation. 5 Evolution and Transition at G (<) Break Points In this section, we formulate the transition set of distance functions on a closed piecewise smooth plane curve 8 Unlike what we did in Footnote 5 for a quadratic equation, the simple way to justify this claim is to verify it by the achieved result, which is p = O( s ). with at least G 0 continuity at its break points. A plane point p is in the transition set, if there will be topological change to the local critical distances (from p to the curve) under the perturbation of p. By the result from Section and Section 4, clearly the bifurcation set, or the curve evolute, is part of the transition set. However, for a closed piecewise smooth curve, there will be some subtle issues at the curve break points with continuity less than G ; specifically there will be extra transition events, which are caused by discontinuity in addition to the unfolding of some algebraic degeneracy. Remark 8 We discuss geometric discontinuity at the break points because that determines the exact type of the topological change. However, the formulas for the perturbation of the critical distances are related to the specific parameterization, which may have lower order continuity at the considered break point. For example, later in Section 5. on G break points, we will discuss the situation that a critical distance transitions into a left and right regular ones with, s = p γ γ D and s = p γ γ D +, with an implicit assumption that the curve is actually C as well. However, the result still applies if the curve is C 0 but G ; the necessary change, though, is to replace the first and the second γ s by γ and γ +, respectively. 5. Transition at G Break Points Consider the critical distance from a plane point p 0 to a G curve break point s 0. If p 0 is not the curvature center, we have exactly an A evolution as in Section 4., because only the first and the second order derivatives are involved in the derivation of an A evolution. On the other hand, if p 0 is the curvature center, the transition is not exactly A as in Section 4.. However, the key point is, Observation 4 At a A distance with G foot point, there are actually two A transitions related, respectively, to the left and right limits at s 0. Moreover, for the unfolded critical distances from the transition at the left limit to be realized, the s s have to be non-positive, while they have to be non-negative for the right limit case. The resulting transition, which is shown shortly to even look like an A in some situation, is nevertheless called A G transition because algebraically it is really all about A transitions. The superscript G denotes the specific continuity at the considered point. Later, we will use similar notations such as A G, etc. There are two different scenarios for an A G transition (see Fig. (a)). 7

. The left and right curvature derivatives have different signs, i.e. + < 0. The considered curve point is a curvature extreme point. By Eq. (.), the evolute has a cusp and reverse its speed at s 0 just like it does at an ordinary cusp 9. By Observation, Λ f = Λ ; therefore, if > 0 and thus + < 0, we have f < 0 and f + > 0. Consequently the considered critical distance is minimal. Thus, the type of the original A distance, with a G foot point, can be denoted as Λ 0 = Λ. 4 4 5 6. Upon a tangential perturbation δ p, The left limit tangential A unfolding would have created two critical distances, with s = δ p, 0, and with Λ D = ±Λ δp, respectively. Those related to the right limit are correspondingly s = δ p, 0, and + Λ D =±Λ δp. Therefore, by Observation 4, (a) Either G or Cusp 5 (b) Evolute of a G Cubic B-spline 6 (a) If δ p > 0 and thus δ p + < 0, the original multiplicity- degenerate critical distance, unfolds into three regular ones, with s = δ p, 0, δ p +, respectively, and with types being Λ δp = Λ = Λ 0, Λ δp = Λ = Λ 0, Λ δp = Λ = Λ 0, respectively. (b) If δ p < 0 and thus δ p + > 0, the original multiplicity- degenerate critical distance unfolds into a regular one with s = 0, and with Λ D =Λ δp =Λ =Λ 0. In conclusion, the A G tangential transition, corresponding to a curvature extreme point, is algebraically two A unfoldings, while appears to be either an A tangential unfolding or an A tangential annihilation. The tangential creation direction is γ γ r. Notice that here we use creation direction rather than unfolding direction, because, algebraically, unfoldings occur in both direction γ γ r and direction γ γ r, while only the former results in a creation event, i.e., the number of critical distances increases.. Upon a transversal perturbation p toward Λ γ, the right limit A transition is an annihilation, while the left limit one is an unfolding, but only one of it, i.e cusp. ( p γ ) s = γ, is realized, with Λ D = Λ = Λ = Λ 0. Similarly, upon a transversal perturbation p toward Λ + γ, the new regular critical distance is s = γ ( p γ ), with + Λ D = Λ + = Λ = Λ 0. In conlusion, the A G 9 The difference, though, is that now the speed does not vanish at the Figure. Evolute of a Curve with G Breaks Cusps 5 and 6 in (b) correspond to G curve breaks with + < 0, rather than any real vertexes. Two asymptotes to the evolute are also shown in light gray. transversal transition, corresponding to a curvature extreme point, is algebraically an A transversal unfolding together with another A transversal annihilation, while appears to be an A transversal annihilation.. The left and right curvature derivatives have the same sign, i.e. + > 0. By Eq. (.), the local evolute is really G instead of G 0 ; by Eq. (.), the left and right limits of E, though different, have the same sign, and the evolute is thus not inflectional at s 0. Hence, the evolute at s 0 appears as if the curve were G (>) at s 0, and any result dependent only on the sign of the curvature derivative remains valid. Without going into detail, by Observation 4, an A G transition is just like an A transition, except that in the unfolding formula for left perturbed critical distance has to be replaced by, and that for the right one by +. (cf. 4.). 5. Evolution and Transition at G Break Points The evolute now is discontinuous with two curvature centers corresponding to the considered break point. A G Evolution and Transition Assume that p 0 is not on either of the two curvature centers (i.e., D 0 and D + 0). Algebraically, there are two A evolutions, and we could have made similar observation as Observation 4. Considered again in two different situations (see Fig. (a)). 8

. D D + > 0 (p 0 is on the dashed vertical line segments in Fig (a)) The considered critical distance has type Λ D. Only one of the A evolution events is realized and the perturbed critical distance has the same type. Notice that the corresponding curvature center jumps when p 0 move across the normal line; the foot point, though, moves continuously. This is essentially an A evolution.. D D + < 0 (p 0 is on the solid vertical line segments in Fig (a)), and thus the considered A distance is neither maximal nor minimal. The A evolution corresponding to the left limit would have the critical distance evolved to p γ γ D, while that corresponding to the right limit would have the critical distance evolved to p γ γ D +. (a) Upon tangential perturbation of δ p, the original critical distance simply evolves to another one with the same foot point and the same type. (b) Upon transversal perturbation of p toward Λ D+ γ, both A evolution events are realized; that is, the original critical distance perturbed into two regular critical distances, with s s being p γ γ D and p γ γ D +, and with types Λ D and Λ D+. This is an creation event due to discontinuity rather than any algebraic unfolding. Consequently, Λ D+ γ is the creation direction (see the arrows in Fig (a)). (c) Upon transversal perturbation of p away from Λ D+ γ, both A evolution events are not realized; that is, the original critical distance disappears. This is an annihilation event due to discontinuity rather than any algebraic unfolding. In conclusion, this is algebraically A evolutions, while appears to be either an A tangential evolution or an A transversal transition. Remark 9 If p 0 is on either of the two curvature centers (p 0 is the point connecting a solid vertical line segment to a dashed one in Fig (a)), algebraically there would be an A evolution and an A transition event. We call the resulting effect A G / transition. Due to page limit, and also because the A G / transition has a zero measure, we will not go into more details. Similar comments apply to other transitions such as A G /, AG0 /, AG0 / (cf. Section 5.). Remark 0 In the discussion, we did not assume any properties of the G connected left and right curve segments. Therefore, if any of the two segments is an arc, with the local evolute degenerated into a point, the result remains valid. See Fig 4. (a) Evolute at Curve G Breaks; + < 0 in left images, and + > 0 in right images. Arrows indicate the creation directions. 4 5 6 6 7 6 5 (b) Evolute of a Cubic B-spline with 8 G Figure. Evolute of a Curve with G 4 7 4 8 8 8 7 Breaks Breaks 5. Evolution and Transition at G 0 Break Points There is a simple trick to turn a G 0 break point into two G points, by inserting an arc of radius 0 and of length 0. By Remark 0, if the plane point p 0 is on the left limit normal line, there is either an evolution or transition event due to the G discontinuity between the left segment and the imaginary arc. On the other hand, if p 0 is on the right limit normal line, the event is due to the G discontinuity between the imaginary arc and the right segment. The specific type of the event is determined, by D and D for the first case, and by D and D + for the second case, where 9

D is the type determinant of the critical distance from p to the imaginary arc. Therefore, the only remained problem is to decide D. Assuming the signed exterior angle at the G 0 break point is e (π, π), the curvature of the arc is = Λ e ; that is, the curvature is positive infinite if e > 0 and is negative infinite if e < 0. Therefore, Λ D = Λ d = Λ e Λ d. 6 The Extended Evolute of a Piecewise Smooth Curve Finally we put the result from Section 4 and Section 5 together, as the following lemma on evolution and transition of critical distances to a piecewise smooth curve. Lemma The A evolution, A and A transitions, corresponding to G (>) curve points, and the A G0, AG, and A G transitions, corresponding to curve G 0, G, and G break points respectively, are shown in Table 6. Notice that gray cells indicate the direction for tangential perturbation. Also, by the result from Section 5, the following lemma generalizes the evolute to identify with the transition set. The inward normal direction, at a regular point, is the transversal unfolding or creation direction. The inward tangent direction, at a cusp, is the tangential unfolding or creation direction. Also notice that the notation Ia b denotes an oriented line segment from a to b in R. Lemma The traditional evolute of a smooth curve is extended for a piecewise smooth curve γ(s), together with its inward normal N in (s) at regular points and its inward tangent T in (s) at cusp points, is defined as,. For each smooth piece, the evolute is E = γ + N N N. At a regular point of the evolute, N in = Λ γ. Corresponding to an ordinary curve vertex, the evolute has an ordinary cusp, where T in = Λ γ γ r.. At a G (>) break point or a G break point with + > 0, two adjacent evolute pieces join together with at least G, and the junction is not inflectional.. At a G break point with + < 0, there is a cusp arising from joining two adjacent evolute pieces together, with T in = Λ + γ r. 4. At a G break point, the evolute is extended by a line segment L = {γ + N N N : I + } with N in = Λ D+ γ. Moreover, (a) if ( + ) < 0, the left end of L is a cusp with T in = Λ γ r ; (b) if + ( + ) < 0, the right end of L is a cusp with T in = Λ + γ r. 5. At a G 0 break point, the evolution is extended by two line segments joined at the curve point; one is L = {γ + N N N : I Λe } with N in = Λ D γ, and the other is L = {γ + N N N + : I + Λ } with e N in = Λ D+ γ. Moreover, (a) if Λ e < 0, the left end of L is a cusp with T in = Λ γ r ; (b) if +Λ e > 0, the right end of L is a cusp with T in = Λ + γ r. Remark Intuitively, the way to extend the evolute at a G break point p, where is not continuous, is essentially to insert an imaginary segment with G connection with the original curve. The imaginary segment collapses to the point p, and consequently it has only one normal line. However, it still has a curvature range of ( + ) and thus unfolds into a straight line segment (in projective space P ) on the extended evolute. On the other hand, the extension at a G 0 break point p takes two stages. First, an arc is inserted to span the exterior angle at the break point. The insertion of arc turns the original G 0 break point into two G points, and the earlier insertion procedure follows at these two G points. Corresponding to p, there are two end points on the traditional evolute. The evolute is extended from one of the two ends to p by a line segment in P, and from p, is extended again to the other end on the traditional evolute by another line segment in P with different direction. Notice that the break point p is the collapsed evolute of the inserted arc. Remark Geometrically, the extended evolute is still the envelope or boundary of the normal bundle to the curve. For a closed curve in R, its extended evolute can be regarded as closed in P. See Figs.,, 4 and 5 for some examples of extended evolutes. Notice that, all the extended line segments that end at the image boundary actually extend to infinity. The curve break points, as well as their corresponding curvature centers, are numbered in order. 7 Conclusion In this paper, we have derived explicitly the various unfolding formulas expressed in the given curve parameterization. We have investigated A G evolution and transition, A G0 evolution and transition, and A G transition; correspondingly, we have also generalized the traditional 0

Table. Evolution and Transition of Critical Distances to Piecewise Smooth Curves p 0 original type p direction new critical distances # s s types A Λ 0 = Λ D any p γ γ D Λ 0 If D D + > 0 : Λ 0 = Λ D any p γ γ D, if D ( p γ ) 0 p γ γ D +, if D ( p γ ) > 0 Λ 0 A G If D D + < 0 : 0 D + γ p > 0 D + γ p < 0 0 p γ γ D, p γ γ D + Λ D, Λ D+ ±γ γ r 0 0 A G0 Convert to A G points, joined by an arc, with = Λ e, Λ D = Λ d = Λ d e. γ p > 0 ± γ ( p γ ) ±Λ A 0 γ p < 0 0 ±γ γ r δ p, 0 Λ δp, Λ δp If + > 0 : just like an A, with replaced by and + appropriately. A G γ p > 0 γ ( p γ ) Λ 0 If + < 0 : Λ 0 = Λ γ p 0 γ ( p γ ) + Λ 0 γ γ r δ p, 0, δ p + Λ 0, Λ 0, Λ 0 γ r ± 6δ p, 0 Λ 0, Λ 0 A Λ 0 = Λ 6 p γ all else γ Λ 0

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