Introduction of Singularity Theory on Curves

Introduction of Singularity Theory on Curves Xianming Chen School of Computing, University of Utah, Salt Lake City, UT 84112 1 Introduction This introduction is based on the book Curves And Singularities by J.Bruce and P.Giblin [JP92]. Other useful references are [Koe90, CG00, O N97]. 2 Manifolds Defined By Submersions 2.1 Regular Points, Regular Values and Submersions A point x R m is a regular points of f : R m R q, and f is called a submersion at x, if the q m Jacobian matrix of f at x J (f) ij = f i / x j, has rank q (which is only possible if m q), or the differential Df(x) is surjective. A point v R q is a regular value of f : R m R q, if every point of its pre-image is a regular point. If the rank is m (which is only possible if m q), or the differential Df(x) is injective, f is called an immersion. Example 1 let f : R 3 R, f(x, y, z) = x 2 + y 2 + z 2 1, J (f) = ( f/ x, f/ y, f/ z) = (2x, 2y, 2z),

therefore, every point, except the origin, is a regular point of f (f is a submersion at all points except the origin), and every value other than 1 is a regular value of f. Example 2 Let f : R 2 R 3, f 0 (θ, φ) = sin(θ)cos(φ), f 1 (θ, φ) = sin(θ)sin(φ), f 2 (θ, φ) = cos(θ). The Jacobian cos(θ)cos(φ) sin(θ)sin(φ) J (f) = cos(θ)sin(φ) sin(θ)cosφ), sin(θ) 0 has three minors with determinants sin(θ)cos(θ), sin 2 (θ)cos(φ), sin 2 (θ)sin(φ); hence, f is an immersion except when all the above 3 determinants are zero, i.e., θ = kπ, k = 0, 1,. 2.2 Manifolds Defined by Submersions Given a submersion of f : R n+q, v(a, b) R q, c, defined at at some neighborhood, N(v), of v = (a R n, b R q ), where n + q = m, and c is the corresponding regular value of regular point v. Suppose the last q columns of its Jacobian matrix forms a non-singular submatrix M q q 1. Write the equation as, f(x, y) = c, (1) where x R n, y R q, and (x, y) N(v). It turns out that we can solve y from x, and therefore parametrize the level set (with c value) locally by x. That is, we get a n-dimension manifold in the domain space R n+q. 1 Because the rank of f s Jacobian matrix is q, this is always possible by some permutation of the coordinates in the (n + q)-space. 2

R q 1 f (c) y = g(x) b a n R Fig. 1. Manifold From Submersion If f : R n+q R q is a submersion at (a,b) with regular value of c, then locally f 1 (c) is a graph, and therefore a parametrized manifold. Define by F : R n+q, (a, b) R n+q, (a, c), F(x, y) = (x, f(x, y)). F is a local diffeomorphism since its Jacobian is a diagonal matrix of a n n unit matrix, followed by M q q, and consequently there is a smooth inverse, at the neighborhood of (a, c), F 1 (x, y) = (x, H(x, y)). Therefore, for each x N(a) R n, let y R q, then, by definition of F(x, y), we have y = H(x, c) g(x), f(x, g(x)) = c, Further, there cannot be another point y y, with f(x, y ) = c, otherwise we have F 1 (x, y) = (x, c), and F 1 (x, y ) = (x, c), which is impossible since F 1 is locally a diffeomorphism. In conclusion, g(x) is a graph in R n+q on R n, such that f(x, g(x)) = c, and locally f 1 (c) is a manifold parametrized by x 1,, x n. Using inverse function theorem, we have just proved what is called implicit function theorem. Some examples follow. 3

Example 3 Some trivial manifolds defined from submersions. (1) The function in Example 1 has any c 0 as its regular value, and thus F 1 (c) is a 2D-manifold. (2) f(x 1, x 2 ) = ax 1 + bx 2 + c has Jacobian matrix of (a, b), therefore 0 is its regular value, and the kernel of it is a 1D-manifold in R 1, provided a, b are not both zero. (3) F = (f 1, f 2 ), has Jacobian matrix of, f 1 (x 1, x 2, x 3 ) = ax 1 + bx 2 + cx 3 + d, f 2 (x 1, x 2, x 3 ) = ex 1 + fx 2 + gx 3 + h, a b c. e f g 0 is a regular value of F, and F ( 1) (0) is a 1D-manifold in R 3, if the Jacobian has rank 2. Geometrically this just means that two non-parallel planes intersect into a line. 3 Envelopes Let F be a 1-parameter family of functions, with t as its parameter, F(t, x) = F t (x) : R r R, (2) Suppose 0 is a regular value of each function F t, i.e. then rank(j (F t (x))) = rank( F/ x 1,, F/ x r ) = 1, (3) C t = F 1 t (0), t R, (4) gives a 1-parameter family of parametrized hyper-surfaces in R r. On the other hand, Eq. (2) can also be explained as defining F(t, x) to be a mapping: R r+1 R. According to Equation (3), it always has a rank 1 Jacobian, ( F/ t, F/ x 1,, F/ x r ), (5) and specifically, 0 is a regular value. Therefore, F 1 (0) is a hyper-surface in R r+1. Notice that this is intuitively obvious F 1 (0) is just the lifting of the 1-family of hyper-surfaces in R r into R r+1 by its parameter value t. 4

Now we consider a special sub-manifold (silhouette generator) on this hypersurface. Let us define The Jacobian of G(t, x) is, G(t, x) = (F(t, x), F/ t(t, x)) : R r+1 R 2. (6) F/ t F/ x 1 F/ x r 2 F/ t 2 2 F/ x 1 t 2 F/ x r t. (7) For the moment, let us assume 2 F/ t 2 is not zero. In general, G(t, x) may not be regular. However, it turns out that we have a particularly nice situation for G(t, x) = 0, i.e., the zero level set of G(t, x). First, by definition (Eq. (6)), F/ t(t, x) = 0. Then, by Eq. (3), matrix 7 must have some full rank (i.e., 2) sub-matrix forming from the first column and some other column. Therefore, 0 is again a regular value of G(t, x), and G 1 (0) is a co-dimension-2 manifold in R r+1. For r = 2, this co-dimension-2 manifold is a space curve, and it has a very significant geometrical meaning. Be definition, this is a curve on the lifted surface with normals of ( F/ x 1, F/ x 2, F/ t = 0), which means the curve is the silhouette generator, or fold, of the lifted surface viewing from the vertical t-axis (the vertical normal component vanishes, i.e., F/ t = 0). For the convenience of discussion, we generalize the silhouette to any space. Finally, projecting the lifted surface, together with its silhouette generator back to R r, gives us the initial 1-parameter family of functions in R r, and in addition, its surrounding envelope(apparent silhouette of the lifted surface). Furthermore, the envelope is a hyper-surface in R r, which can be readily verified if we go back to the Jacobian of G(t, x), i.e., equation (7). Suppose F/ x r is non-zero, and the first and last columns of Eq. (7) form a non-singular submatrix (recall F/ t = 0, and our assumption that 2 F/ t 2 0). Then, by Section 2.2, t and x r can be solved as two smooth functions on x 1,, x r 1 ; specifically for x r, the smooth function defines the apparent silhouette, or the envelope of G(t, x), or, not as commonly called, the discriminant set of G(t, x). Figure 2 is adapted from [CCR06] and illustrates the envelope of normals to a plane parabola. If 2 F/ t 2 is 0, the discriminant set is not a manifold, and the locus of such points is called the regression set of G(t, x). 5

Fig. 2. Lifting a Family of Curves (Lines) in R 2 to Form a Surface in R 3 The left image shows the normal bundle to a parametrized parabola curve. The normal lines are lifted vertically by the amount of the corresponding parameter values of the parabola curve, forming a surface in R 3, which is shown on the right. The vertical canyon lines illustrates their intersection with the surface at the neighborhood of the fold and the cusp on the fold. And one final nice result as it should have the tangent of the envelope corresponding to (t, x) is the same as that of C t, since tangent space of the silhouette generator is the kernel of the differential mapping D(G) (strictly speaking tangent mapping T (G), a translation of the differential mapping, but we make no difference in this paper), consisting of (t, ξ 1,, ξ r ) with, t F/ t + ξ 1 F/ x 1 + + ξ r F/ x r = 0, (8) t 2 F/ t 2 + ξ 1 2 F/ x 1 t + + ξ r 2 F/ x r t = 0, (9) From equation (8) and F/ t = 0, it is obvious that the projection of the tangent on the silhouette generator at (t, x), i.e., the tangent of the envelope at x, is the tangent to the curve C t. Finally, there is yet a third perspective 2 on Eq. (2). This time it is taken as a r-parameters family function on t, F(t, x) = F x (t) : R R. (10) Considering the multiplicity of the root of the parametrized function on t, it is obvious that F x has double root on any x on the envelope(a 1 -singularity), 2 the first two being a 1-parameter (t) family of function on (x 1,,x r ), and a single function on (t,x 1,,x r ) 6

and triple root if x is the regression point(a 2 -singularity), where 2 F/ t 2 = 0. 4 Contacts In this section, we formalize the concept of contact between a parametrized curve and an implicit hyper-surface by analyzing the vanishing derivatives of a composite function g : R R. Given a regular curve, γ : I R n, and a hyper-surface in R n, defined by F 1 (0) of regular value 0, F : R n R, γ and F 1 (0) are said to have k-point or k-fold contact at t = t 0, or at p = γ(t 0 ), provided the function g defined by, g : R R, g = F(γ(t)), has all its derivative up to order k 1 but not k vanished, i.e., an A k 1 singularity at t 0. This amounts to say (g(t), t 0 ) is R-equivalent(defined later in Section 5) to (f(t) = t k, 0). Example 4 (Contact of a function graphs with x-axis) Consider function y = x k, and the x-axis as the kernel of implicit function f(x, y) = y. Choosing x as the parameter of the function graph, we have g(t), g(t) = t k, and consequently, the graph has k-point contact with the x-axis. Example 5 (Contact of a curve and a line in R 2 ) Consider a regular unit speed parametrization curve γ in R 2, and a line in R 2 of normal direction u, and passing point p, x u p u = 0. We have g(t) = γ(t) u p u. Therefore, the curve and the line has 1-point contact, if g(t) is A 0 at some t 0, i.e., g(t 0 ) = γ(t 0 ) u p u. which just means the point γ(t 0 ) is on the line. If, in addition, the first derivative of g is also zero at t 0, T(t 0 ) u = 0, 7

i.e., the tangent direction (T(t)) of the curve is the same as the line direction, then the line has 2-point contact with the curve. Taking one step further, we find the line has 3-point contact with the curve if and only if κn u = 0, or κ = 0, i.e., at an ordinary inflexion point of the curve. Higher (degenerate) inflexion points give higher order contact. Example 6 (Contact of a curve and a circle in R 2 ) Consider a regular unit speed parametrization curve γ in R 2, and a circle in R 2 of radius r and center p, (x p) (x p) r 2 = 0. The function g is, g(t) = (γ(t) p) (γ(t) p) r 2, The curve and the circle has 1-point contact, if g(t) is A 0 at some t 0, i.e., g(t) = (γ(t) p) (γ(t) p) r 2 = 0 which just means the point γ(t 0 ) is on the circle. If the first derivative of g is also zero at t 0, T(t 0 ) (γ(t 0 ) p) = 0, i.e., the circle is centered at the normal direction of the curve at t 0, then the circle has 2-point contact with the curve. One step further, the vanishing of the second derivative 1 + κn(t 0 ) (γ(t 0 ) p) = 0, gives us a 3-point contact of the curve and its osculating circle (with r = 1/κ), provided κ 0. Taking one more derivative, we have κ N(t 0 ) (γ(t 0 ) p) = 0, which means the osculating circle has 4-point contact with the curve at ordinary vertex point where κ = 0. Geometrically, the vertex is the point on the curve where it is extremely round, and the curve stays on one side of the osculating circle, while it crosses it at all other non-vertex points. 8

Example 7 (Contact of a curve and a plane in R 3 ) Consider a regular unit speed parametrization curve γ in R 3, and a plane in R 3 of normal direction u, and passing point p, The function g is, x u p u = 0. g(t) = γ(t) u p u. The curve and the plane has 1-point contact, if g(t) is A 0 at some t 0, i.e., g(t 0 ) = γ(t 0 ) u p u. which just means the point γ(t 0 ) is on the plane. If, in addition, the first derivative of g is also zero at t 0, T(t 0 ) u = 0, i.e., the normal direction of the plane is in the normal plane of the curve, then the plane has 2-point contact with the curve. Taking one step further, we find the plane has 3-point contact with the curve if and only if κn u = 0 i.e., if it is the osculating plane of the space curve (i.e., u = B, the binormal of the space curve), provided κ 0. A 4-point contact occurs at, κ( κt + τb) B = 0, i.e. τ = 0, flat points of the space curve. The geometrical meaning here is, a general space curve crosses its ordinary the osculating plane with 3-point contact, while stays at one side of the extraordinary osculating plane with 4-point contact. Example 8 (Contact of A Line With A Surface) Consider an implicit surface in R 3 defined as some level set of F : R 3 R, and a line in R 3 of direction u, and passing point p, The function g is, g(t) = F(p + ut). As usual, the 1-point contact is trivial. There is a 2-point contact if 3 dg(t) dt = D u F = u F = 0, 3 D u F is the u-directional derivative of F, and F is the gradient of F 9

i.e. provided u is on the tangent plane of the implicit surface. Similarly, The tangent line has 3-point contact with surface, if the second order directional derivative of the function F also vanishes, i.e., and so on. d 2 g(t) dt 2 = D uu F = 0, It is nice if we could show that this 3-point contact tangent is actually an asymptotic direction on the implicit surface. This is truly so as proved below. First we have, D uu F = D u (D u F) = D u (u F). Observing D u (u) = u and u F = 0, u D u F = 0, where F is the unnormalized normal on the surface. Let it be mn, where N is the unit normal of the implicit surface. Substituting this into the above equation, we have 4, u D u (mn) = mu D u N = mu S(u) = m II(u, u) = 0, which says exactly that the second fundamental form at u is zero, and thus the tangent is an asymptotic direction. 5 A k Singularities, R-equivalent and Unfoldings t 0 is called an A k -singular point of function f : R R, if f (i) (t 0 ) = 0 holds for all i = 0, 1,, k, but not for k + 1. We make a difference between two types of functions. There are functions like the gravity potential, when the main concern is the critical points (local minimal, maximal, or inflexion points). Adding a constant term to a potentiallike function does not make any difference, and therefore A 0 is usually skipped in this case. On the other hand, the exact manifold of F 1 (c) of course depends on the specific c value. There is a simple relation between these two types, in the sense that if G is of potential-like function with A k -singularity, then F = G is non-potential-like function, and with A k 1 -singularity. 4 S is the shape operator, and II is the second fundamental form. 10

Example 9 f(t) = t k is A k 1 at t = 0, where its graph has k-point contact with x-axis. f 1 at the neigbhorhood of t 1, denoted by (f 1, t 1 ), and f 2 at the neighborhood of t 2, denoted by (f 2, t 2 ), are R-equivalent, if they are the same up to a constant term after a parameter diffeomorphism transformation, i.e. where h(t 1 ) = t 2. f 1 (t) = f 2 (h(t)) + c, Example 10 A re-parameterization of a regular curve will change a function defined on the curve into its R-equivalent function at all the corresponding parameter values. (t 2, 0) and ( t 2, 0) are not R-equivalent, while (t 3, 0) and ( t 3, 0) are with h(t) = t. (t m, 0) and (t n, 0) are not R-equivalent if m n. It can be proved that an A k function is R-equivalent to ±t k+1, where ± has the same sign as f k+1 (t). A family of function is called an unfolding of the any function in the family. The unfolding of a function is versal folding if it includes every function close to the unfolded function. If the consider function is potential-like function, it is called a (p)versal unfolding. It can be proved that the normal form unfoldings are, (1) The (p)versal unfolding of a A k -singularity F(t, x 1,, x k 1 ) = ±t k+1 + x 1 t + x 2 t 2 + + x k 1 t k 1. (11) (2) The versal unfolding of a A k 1 -singularity F(t, x 1,, x k 1 ) = ±t k + x 1 + x 2 t + + x k 1 t k 2. (12) Example 11 Suppose f(t) = t 4 is a potential-like function, of A 3 -singularity, then it has a (p)versal unfolding of 2-parameter family of functions F(t, x 1, x 2 ) = t 4 + x 1 t + x 2 t 2. Now define g(t) = df(t)/dt = 4t 3, then it has an A 2 signularity with a versal unfolding of 2-parameter family of functions F(t, x 1, x 2 ) = t 3 + x 1 + x 2 t. Given a (p)versal r-unfolding F(t, x) of f(t) = F 0 (t), the singular set of F is the manifold of co-dimension 1 in R r+1 defined by F/ (t) = 0. The singular set has a fold, which is a co-dimension 2 manifold in R r+1 defined by F/ (t) = 11

0 and 2 F/ (t 2 ) = 0. Projecting this fold onto R r, gives us the bifurcation set, which is a co-dimension 1 manifold in R r. It can be proved that the bifurcation set is a smooth hyper-surface except at points where 3 F/ (t 3 ) = 0, the locus of which are called the regression set. If we define, G(t, x) = F/ (t). then G is r versal unfolding of g(t) = df(t)/dt = G 0 (t). The bifurcation set of F is now called the discriminant or envelope of G. 6 Unfoldings: Cusps, Swallowtails, and Butterflies There are two types of unfolding, depending on the functions we are talking about. In this section, we assume the functions are not potential-like, and thus we work on versal unfolding. 6.1 Versal Unfoldings of A 2 : Ordinary Cusps Given f : R R, f(t) = t 3, with f(t) A 2 at t = 0, the versal folding is a 2-parameter family of functions, where x = (x 1, x 2 ) is a point in R 2. F(t, x 1, x 2 ) = t 3 + x 1 t + x 2 = 0, (13) Function g(t), or F (0,0)(t) is A 2 (at t = 0). By unfoling, we are trying to find out those x such that F x is A 1 or A 0 at some t. The locus of A 1 consists all x of pair (t, x) with but not, F/ t = 3t 2 + x 1 = 0, (14) 2 F/ 2 t = 6t = 0, (15) which is exactly the envelope we were talking about in Section 3. Equation (13) and (14) actually parametrize the envelope as, x 1 = 3t 2, x 2 = 2t 3. (16) 12

Notice that, if Eq. (15) holds as well, the envelope will have a cusp at t = 0, which is exactly the regression of the envelope discussed in Section 3. 6.2 Versal Unfoldings of A 3 : Swallowtails Consider the singular set of the versal unfolding of A 3 -singularity, which is a hyper-surface in R 4. F(t, x) = t 4 + x 1 + x 2 t + x 3 t 2 = 0, (17) Viewing from t-axis, this surface has a fold, defined by, F(t, x)/ t = 4t 3 + x 2 + 2x 3 t = 0, (18) which is a manifold of co-dimension 2 in R 4. Projecting this fold onto R 3 space, we get the bifurcation set, which is a surface called swallowtail surface. The swallowtail can be parameterized based on equation (17) and equation (18). The Jacobian is 4t3 1 t t 2. 3t 2 0 1 2t And it is easily seen that the sub-matrix of 1 t, 0 1 is non-singular, and therefore the co-dimension-2 surface in R 4 can be parametrized by t and x 3 as, t = t, x 1 = 3t 4 + x 3 t 2, x 2 = 4t 3 2x 3 t, x 3 = x 3. Projecting to R 3 gives us the bifurcation set, which is called a swallowtail surface. The swallowtail surface is smooth except at regression set, which is a curve defined by equation (17) and (18), in addition to 2 F(t, x)/ t 2 = 12t 2 + 2x 3 = 0, (19) 13

which, all together, gives the parameterization on t, x 1 = 3t 4, x 2 = 8t 3, (20) x 3 = 6t 2. 6.3 Versal Unfoldings of A 4 : Butterflies F(t, x) = t 4 + x 1 + x 2 t + x 3 t 2 The bifurcation set is called a butterfly which lives in R 4. 7 Local Structure of Silhouettes Analysis of the topological structure of silhouettes has many application in computer graphics and computer aided design (for example, [ECC05]). Koendrink [Koe90] gives detailed discussion on the various singularities of othorgraphic silhouettes. In this section, we will show that the unfolding theory we discussed in the previous section gives us an especially simple explanation. Recall that, in Section 2, we start with a t-parameter family of implicit curves defined by the zero set of F t (x, y) = F(t, x, y) : R R in the plane, and then lift them vertically to the third dimension t to form an implicit surface (called singular set) defined by the zero set of F(t, x, y). The silhouette of this surface, when viewed vertically, is projected back to the plane, yielding the bifurcation set of F(t, x, y) regarded this time as a two-parameter (x, y) family of function on t. The bifurcation set, defined by F = 0, is a regular implicit t curve in the plane except at its regression point, defined further by 2 F = 0. t 2 The bifurcation set curve has local cusp structure at the regression point. For higher singularity, the bifurcation set has a regression set of swallowtail. In what follows, we will reverse the above procedure. Regard a surface in R 3 as defined by F(t, x, y) = c where c is a regular value of F(t, x, y) : R 3 R. Now consider all the contour curves of the surface with contour height t = c. These contour curves are the zero set of F and t c. 0 is a regular value for the corresponding mapping (from R 3 to R 2 ), because F t F x 1 0 0 F y 14

has full rank of 2, provided F x and F y and F y do not vanish simultaneously. If F x do vanish simultaneously, then F t do not because 0 is a regular value of F(t, x, y), and consequently the surface has a horizontal tangent plane at the considered point. Therefore, close to the silhouette generator (viewed vertically) where F t = 0, all the contours are regular curves. Now considering the the original surface as the lifting of this family of (projected) contours, the result in previous paragraph follows naturally. Specifically, consider an asymptotic (vertical) direction that has 3-point contact with the surface. Recall that the surface is defined as level set of F(t, x, y), and observe that the vertical line has a parametrization of (t, 0, 0) in t, a 3- point contact simply means F = 2 F = 0, following the discussion in Section 4. But the condidtion also specifies an A 2 singularity of f(t) = F (x,y) (t) t t 2 and its versal unfolding has a local cusp struture. Similarly, a 4-point contact simply means F = 2 F == 3 F = 0, i.e., an A t t 2 t 3 3 singularity that versal unfolds into a swallowtail. 8 The Distance Squared Function on a Planar Curve Given a regular unit speed curve γ(t) in R 2, for a fixed point p in R 2, define, F(t, p) = (γ(t) p) (γ(t) p), (21) We are interested in finding the points on the curve which have either local minimal or local maximal distances to p. That is, for a fixed point p, we are trying to find those ts, such that F p (t) = 0 and either F p (t) > 0 or F p (t) < 0. The singular set of F, is a surface in R 3, since the Jacobian, F (t, p) = 2(γ(t) p) T = 0, (22) 2(F T x T y ), (23) has rank 1 (T x and T y, the two components of T, can not be both zero). Equation (22) simply means p = γ(t) + λn(t), for some λ. (24) and the surface is just the spread out of all the curve normals, lifted along t-axis by the parameter value t. 15

The fold (or silhouette generator viewed from t-axis) of this surface is defined by the following equation, in addition to equation (22), F (t, p) = 2(1 + (γ(t) p) κn) = 0, (25) which is regular, since the Jacobain of the map (F, F ) : R 3 R 2, 0 2T x 2T y (26) F 2κN x 2κN y has a non-singular submatrix consisting of the last two columns if κ 0, or a non-sinuglar submatrix consisting of the first column and either the seond or the third column (T x and T y can not be both zero) if F 0. When projecting back to R 2, we have the envelope of the normals of curve, which is just the evolute of the curve, because from equation 25, we have, κ 0, and p = γ(t) + 1/κN(t). (27) The evolute is therefore smooth so long as F 0 (See Section 3). Otherwise, we have the regression point, F (t, p) = 2 κ (γ(t) p) N = 0, (28) which is the curvature center of some vertex on the curve. And the evolute has an ordinary cusp there if F 0, because (1) F p0 (t) (p 0 is the curvature center of the vertex) has A 3 singularity at the corresponding t. (2) It can be proved that F(t, x) in equation (21) is the (p)versal unfolding of F p0 (t). (3) By Section 6.1, the evolute has an ordinary cusp there. Now back to our minimal or maximal distance problem, we have the following local result, (1) For a fixed point p, the local minimal or maximal distance point has to be the curve point, say γ(t 0 ), with the normal passing through p. See equation (24). (2) γ(t 0 ) is a local minimal distance point if F > 0, i.e., κ = 0, or p is closer to it than the curvature center. (3) γ(t 0 ) is a local maximal distance point if F < 0, i.e., p is farther to it than the curvature center. (4) if γ(t 0 ) is not a vertex, and p is its curvature center, i.e. F = 0, but F 0, then γ(t 0 ) is neither a local minimal distance point nor a local maximal one. 16

Fig. 3. Evolute of the parabola γ(t) = (t,t 2 ). It has an ordinary cusp, which is also the local structure of any versal 2-unfolding of A 2 singularity (5) γ(t 0 ) is a local minimal distance point, if it is a vertex of maximal curvatur, and p is its curvature center. i.e. F = 0, and F 0, but F > 0. (6) γ(t 0 ) is a local maximal distance point, if it is a vertex of minimal curvatur, and p is its curvature center. i.e. F = 0, and F 0, but F < 0. In the neighborhood of curvature center of vertex of maximal curvature, we have the following multi-local result(figure 3), (1) For any point p, outside the folded area, there is only one minimal distance point. (A 1 -singularity) (2) For any point p, inside the folded area, there are two minimal distance points and one maximal point. (all A 1 -singularity) (3) For any point p, on the fold but not the cusp, there is one minimal distance point.(a 1 -singularity, and there is another curve point of A 2 singularity). (4) For p on the cusp, there is one minimal distance point. (A 3 singularity). (5) The evolute divides the R 2 control space into 2 regions, with 1 and 3 critical points respectively. When moving the point p from the 1 critical point (which is minimal) region into the 3 critical point region, the change of minimal point on the curve is gradual. If we keep moving point p all the way through the 3 critical point region and crossing out into the other side of the 1 critical point region, there is a sudden change of minimal point on the curve, which is called catastrophe. 17

9 The Gravitational Catastrophe Machine Imagine a 2-dimension shape with some regular curve, of unit speed parametrization γ(t), as its boundary. For a given gravity center at point p(x, y), if the shape is rested on the horizontal x-axis with the contact point of γ(t), the gravity potential 5 is, V (t, x, y) = (p(x, y) γ(t)) N(t) = (p γ) N. (29) We are trying to find the positions of stable equilibrium, i.e. the minimal value of the potential function, for a fixed point p, and most importantly to find out how the answer depends on the specific position p. Obviously V (t, x, y) is a potential-like function, the singular set of which is, U(t, x, y) = V/ t(t, x, y) = T N kt (p γ) = kt (γ p) = 0, (30) which simply says the point p has to be on the normal line, provided k 0 6. The bifurcation set is, U/ t(t, x, y) = 2 V/ t 2 (t, x, y) = k T (γ p) + k 2 N (γ p) + k = k + k 2 N (γ p) = k (1 kn (p γ)) = 0, (31) which says that the point p has to be on the curvature center, provided k 0. It is also obvious from this equation that the equilibrium is stable if and only if the point p is on the normal line and closer to the curve than the curvature center is. And the third order derivative is, 2 U/ t 2 (t, x, y) = 3 V/ t 3 (t, x, y) = k kn (γ p) + k 2kk N (γ p) + k = k + k + 2k + k = 3k = 0, (32) 5 we leave out the inessential term related to the mass, i.e., mg 6 Notice that the singular set of this potential function is essentially the same as that of the squared distance function (cf. Eq.(22)). Therefore, all the rest of this section is, in essence, a repetion of Section 8 18

which says the regression point is the curvature center of a vertex, provided k 0, and locally it is the cusp point of the evolute. At such point, say p 0, function U p0 (t) has A 2 singularity, and equation U p0 (t) = 0 has triple root. By versal A 2 unfolding, we know that locally U p (t) = 0 has 3 roots if p is inside the cusp and 1 root outside it. If the vertex is a type of minimal curvature radius, then V p (t) has 2 and 1 stable equilibrium positions respectively. On the other hand, if the vertex is a type of maximal curvature radius, then V p (t) has 1 and 0 stable equilibrium positions respectively. For the discussion below, we assume the former case. Now, let us regard the gravity center point p as a control variable. Suppose initially we are in a stable equilibrium at curve point γ(t 0 ). If we move p along the normal line at γ(t 0 ) from outside into the cusp area, i.e., increase the distance of p to the curve, we get two extra equilibrium positions and anyway the old one, γ(t 0 ), is still stable. And if we move p gradually in any direction, we have new stable point γ(t 0 ) which is very close to γ(t 0). However, if we keep moving until that the distance of p to the curve is larger than the radius of curvature, i.e., moving out of the cusp area, then γ(t 0 ) or any point in its neighbor is not a stable position anymore, and the shape is under a sudden flip to another stable position - this is the so called a catastrophe. References [JP92] [CCR06] Xianming Chen, Elaine Cohen, and Richard Riesenfeld. Dynamic curve-point critical distances. Springer-Verlag Lecture Notes in Computer Science 4077 (GMP 2006): 87-100, 2006. [CG00] Roberto Cipolla and Peter Giblin. Visual Motion of Curves and Surfaces. Canbrudge University press, 2000. [ECC05] Gershon Elber, Xianming Chen, and Elaine Cohen. Mold accessibility via gauss map analysis. ASME Transactions, Journal of Computing & Information Science in Engineering, June 2005:79-85, 2005. J.W.Bruce and P.J.Giblin. Curves And Singularities. Cambridge University Press, 2 edition, 1992. [Koe90] Jan. J. Koenderink. Solid Shape. MIT press, 1990. [O N97] B. O Neill. Elementary Differential Geometry. Academic Press, 2 edition, 1997. 19