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[MATH] The addition function. deriving: Used in a data declaration. It will automatically generate implementations of the type classes listed. map: A higher order function that applies another function to every element in a list. |
# Source: arxiv 0909.4956 # Title: Local Shape of Generalized Offsets to Algebraic Curves # Sections: all # Downloaded: 2026-03-03T02:20:35.471225+00:00 |
Local Shape of Generalized Offsets to Algebraic Curves Abstract In this paper we study the local behavior of an algebraic curve under a geometric construction which is a variation of the usual offsetting construction, namely the generalized offsetting process ( |
). More precisely, here we discuss when and how this geometric construction may cause local changes in the shape of an algebraic curve, and we compare our results with those obtained for the case of classical offsets ( |
). For these purposes, we use well-known notions of Differential Geometry, and also the notion of local shape introduced in Introduction |
The notion of generalized offset (see for a more formal definition of this notion and a large study of algebraic and geometric properties) arises in the literature as a generalization of usual offsets. In order to introduce this notion, one may consider the following construction over a given algebraic curve [MATH] : f... |
degrees, and consider the points [MATH] lying on [MATH] at a distance [MATH] of [MATH] . Then the generalized offset [MATH] is the Zariski closure of the set consisting of all the points [MATH] computed this way. In this context, the usual notion of offset (which corresponds to the case when [MATH] defines a rotation l... |
[MATH] of the curve (see ). For example, in Figure 1 one has, for different distances, the classical and the generalized offsets for |
[MATH] of an ellipse. Notice that this construction works over [MATH] ; nevertheless, in the following we will assume that [MATH] is real, and we will focus on the real part of its generalized offset, for a real distance and a real angle. |
Algebraic properties of generalized offsets have been considered in the literature (see ). In this sense, a nice result is that properties like the number of components, genus and therefore rationality, are invariant for the angle [MATH] ; so, they are shared by all the generalized offsets (including the classical offs... |
Questions on the shape of classical offsets have already been analyzed (see ). Moreover, in local aspects on the shape of classical offsets of possibly singular algebraic curves are studied. In that paper the notion of local shape is introduced in order to locally describe the shape of a curve. Basically, this notion d... |
) that there are four different behaviors that a real branch can exhibit, which can be found in Figure 2 (see Section ), corresponding to so-called local shapes (I), (II), (III), (IV) . Moreover, each of these possibilities has a characterization in terms of places (see also Section in this paper; for more information ... |
). Hence, given a geometric transformation like classical or generalized offsetting, in order to analyze how the transformation locally affects the curve one can take a generic place, compute the places it gives rise to in the transformed object, and compare the local shapes of the original and the final places. If all... |
for addressing not only local, but also global questions on the shape of classical offset curves. Finally, the structure of the paper is the following. In Section we provide the necessary background for developing our results; in particular, the notion of local shape is reviewed here. In Section we address the behavior... |
Acknowledgements. The author wishes to thank J. Rafael Sendra for suggesting the problem. Local Shape of an Algebraic Curve In the following we work with an algebraic curve [MATH] different from a line, a real distance [MATH] , and a real angle [MATH] . One may easily see that generalized offsets to lines are also line... |
) one says that [MATH] is the center of the place [MATH] . The functions [MATH] are called the coordinates or the components of the place, and are analytic in a neighborhood [MATH] of [MATH] . Now writing |
[EQUATION] we represent by [MATH] the order of [MATH] , i.e. the least non-zero power of [MATH] in the expression of [MATH] ; similarly we introduce [MATH] . Moreover, we speak of “real” places to denote places where the coefficients [MATH] , perhaps after a change of parameter, are real numbers. Then we consider the f... |
Definition 1 Let [MATH] be a real place of [MATH] . The signature of [MATH] is defined as the pair [MATH] where [MATH] is the first non-zero natural number such that the derivative [MATH] and [MATH] is the first natural number such that |
[MATH] are linearly independent. We denote by [MATH] the signature of [MATH] Since [MATH] by hypothesis is not a line, the numbers [MATH] in Definition always exist. Now if [MATH] then we say that [MATH] is regular , otherwise we say that it is singular . The center of a singular place is always a singular point of [MA... |
(see Proposition 3 there) it is proven that in a suitable coordinate system, every real non-isolated point [MATH] is the center of a real place [MATH] of the type [MATH] where [MATH] is the signature of the place. If a place has this form, we say that it is in standard form; notice that when the place is in standard fo... |
it is shown that the local behavior of [MATH] around its center can be read from the signature, giving rise to the notion of local shape . We recall this notion here. |
Definition 2 Let [MATH] be a real place of signature [MATH] centered at [MATH] . Then we say that: (1) [MATH] is a thorn (or it has local shape |
(I) ) if both [MATH] are even. (2) [MATH] is an elbow (or it has local shape (II) ) if [MATH] is odd, and [MATH] is even. (3) [MATH] is a beak (or it has local shape |
(III) ) if [MATH] is even, and [MATH] is odd. (4) [MATH] is a flex (or it has local shape (IV) ) if both [MATH] are odd. In Figure 2 one can see the shape corresponding to each local shape up to rotations. In each case, the center of the place is the intersection point of the two dotted lines. Furthermore, in all cases... |
[MATH] are (II) or (IV). Moreover, if [MATH] is even we say that the place is cuspidal Behavior at regular points In the rest of the paper, we will represent the generalized offset of [MATH] , for a distance [MATH] and an angle [MATH] , as [MATH] ; in particular, if [MATH] we have the classical offset, [MATH] . Moreove... |
. So, here we focus on generalized, non-classical, offsets. Now along this section let [MATH] be a real regular place of [MATH] . Since [MATH] converges in a neighborhood [MATH] of [MATH] , we can regard [MATH] , with [MATH] , as the parametrization of a regular curve; moreover, we can assume that it has been reparamet... |
[EQUATION] Now the first result, which shows an important difference between classical and generalized offsets, is the following. |
Theorem 3 The only generalized offset which may transform a regular place into a singular offset place, is the classical offset. Therefore, the generalized, non-classical, offset, never generates a cusp from a regular point of the original curve. |
Proof. Differentiating the equality [MATH] w.r.t. the arc-length and using Frenet equations, it follows that [EQUATION] where [MATH] is the curvature of [MATH] at its center. Now [MATH] iff [MATH] . However, [MATH] . Then [MATH] iff [MATH] and simultaneously [MATH] . Since we are assuming that [MATH] and [MATH] (i.e. [... |
Remark 1 When the offset is classical, it is well-known that the tangents to the curve and its offset are parallel at corresponding points. For the generalized offset, the above expression [MATH] tells us that this no longer happens; moreover, the tangent line to the generalized offset at a point [MATH] is not even the... |
In Figure 3 one may see, for [MATH] , the classical offset to the parabola [MATH] , and a detail of this offset showing two cusps; in Figure 4 one has the generalized offset of the same curve, also for [MATH] and a very small angle, [MATH] . The reader may see in Figure 4 that in the generalized offset the cusps have b... |
Now let us address the question of checking whether the local shape of regular places is preserved or not by the generalized offsetting process (we say that the local shape of a place is preserved , if the local shapes of the places that it generates in the generalized offset coincide with the original local shape). Si... |
); however, in the generalized, non-classical case, we will see that the answer is “no”. For this purpose, we recall that the curvature at a regular flex point is [MATH] . Hence, let [MATH] denote the curvature of [MATH] at the center of the place [MATH] ; from the well-known formula of the curvature, we have that |
[EQUATION] where [MATH] is normal to the plane containing [MATH] and [MATH] . Thus, the following theorem holds. Theorem 4 The regular points of [MATH] generating flex points of [MATH] , satisfy |
[EQUATION] As a consequence, the generalized, non-classical, offset does not necessarily preserve flex points. Proof. Let us compute the numerator of the above expression for [MATH] . In order to do this, we have that [MATH] . Differentiating again, we get |
[EQUATION] Thus, [EQUATION] Notice that [MATH] (i.e. the derivative of the curvature w.r.t. the arc-length) exists because since [MATH] is regular, then [MATH] is an analytic function. Now since [MATH] represents a rotation of angle [MATH] then [MATH] (because we are assuming that [MATH] has been re-parametrized w.r.t.... |
[EQUATION] Now, expanding [MATH] and [MATH] , taking into account the formula for [MATH] in terms of [MATH] and [MATH] , and computing the dot product with [MATH] , one gets that |
[EQUATION] Now from Theorem it holds that [MATH] , and hence [MATH] iff [MATH] ; then, every point of [MATH] giving rise to a flex point of the generalized offset fulfills this equality. Finally, notice that a regular flex point of [MATH] satisfies that [MATH] , but not necessarily that [MATH] . So, such a point does n... |
In fact, in the next section we will see that generalized, non-classical, offsets never preserve flex points (see Corollary ). Also, observe that the condition in Theorem is not sufficient because the fact that [MATH] does not necessarily imply that the point in [MATH] is a flex (it depends on the order of the first no... |
Finally, we address the turning points (i.e. points of either horizontal or vertical tangent) of the generalized, non-classical offset. In the classical case, it is well-known that the tangents to [MATH] and [MATH] at corresponding points, are parallel; hence, turning points of the offset are |
Theorem 5 Let [MATH] denote a generalized, non-classical offset of [MATH] . The following statements are true: (1) The points of [MATH] with vertical tangent, [MATH] , correspond to: (i) points of [MATH] with vertical tangent, where [MATH] ; (ii) points of [MATH] , with [MATH] , where the slope of the tangent equals [M... |
(2) The points of [MATH] with horizontal tangent, [MATH] , correspond to: (i) points of [MATH] with horizontal tangent, where [MATH] ; (ii) points of [MATH] , with [MATH] , and horizontal tangent; (iii) points of [MATH] , with [MATH] , where the slope of the tangent equals [MATH] |
Proof. From the proof of Theorem it holds that the relationship between the tangents of [MATH] and [MATH] at corresponding points is [MATH] . In order to prove (1), one considers the first component of [MATH] , namely [MATH] , and one imposes that it is [MATH] . Hence, either [MATH] and [MATH] , or [MATH] and [MATH] (n... |
Local Shape of the Generalized, Non-classical, Offset Along this section we consider a real place [MATH] , non-necessarily regular, a distance [MATH] , and an angle [MATH] (i.e. we work with a non-classical generalized offset; see |
for a study of the classical case). Moreover, we write [MATH] [MATH] , and we represent the coordinates of a place [MATH] in [MATH] as [MATH] . In order to analyze how the generalized offsetting process affects the local shape of [MATH] , the idea is to compare the local shape of [MATH] with the original local shape. F... |
[EQUATION] We recall from that, performing computations with formal power series, [EQUATION] Plugging this expression into the first equality and making computations, one gets that, whenever [MATH] |
[EQUATION] and [EQUATION] Moreover, in the special case when [MATH] (i.e. if [MATH] ) one has that [EQUATION] and [EQUATION] One may observe that the first order terms of [MATH] coincide in both cases, [MATH] and [MATH] . Furthermore, |
[MATH] . However, [MATH] and therefore it depends on the sign of [MATH] ; moreover, when [MATH] we also have to distinguish whether the coefficient of [MATH] in [MATH] , namely [MATH] , is equal to [MATH] or not. All these cases ( [MATH] [MATH] [MATH] ) and subcases will be present in our analysis. Furthermore, from Th... |
one may see that the case [MATH] happens iff the curvature vanishes at the center of the place, while the case [MATH] occurs iff the curvature tends to infinity as the center of the place is approached. |
Also, in the following we separately address results that can be reached by considering only the first order terms of [MATH] (see Subsection 4.1 ), and results which require to consider also second order terms in [MATH] (see Subsection 4.2 ). For the second type of results we will need to distinguish the cases [MATH] o... |
4.1 Results using a First-order Approximation We start with the following result; this proposition shows that in some cases, generalized offsetting processes smooth singularities, i.e. they transform singular places into regular ones. This phenomenon happens also for classical offsets (see |
). Proposition 6 Let [MATH] be a place of [MATH] with signature [MATH] . If [MATH] , then [MATH] generates regular offset places; as a consequence, if [MATH] is cuspidal (i.e. [MATH] is even) and [MATH] , then its local shape is not preserved. Conversely, if [MATH] is singular and it is smoothed by the generalized, non... |
Proof. Since [MATH] , if [MATH] we have that the places [MATH] are regular. In particular, in that case these places cannot be cuspidal; so, if [MATH] is cuspidal and [MATH] its local shape is not preserved. Conversely, if [MATH] is singular then [MATH] . Now if it generates regular places then either [MATH] or [MATH] ... |
Using the results of Section 4 of , one may check that classical offsets also smooth singular places iff [MATH] . Now we consider the case [MATH] . In this case, [MATH] and therefore [MATH] . Hence, the following theorem holds. |
Theorem 7 Let [MATH] be a place of [MATH] with signature [MATH] , where [MATH] . Then, the following statements are true: (i) If [MATH] is singular, then it generates singular offset places. |
(ii) The local shape of the offset places [MATH] behaves according to the following table: [MATH] even [MATH] is odd [MATH] even |
thorn flex [MATH] odd beak elbow As a consequence, when [MATH] the only places whose local shape is preserved are the cuspidal ones. |
Proof. Since [MATH] , then [MATH] and [MATH] ; moreover, since [MATH] then [MATH] . Hence, the signature of an offset place [MATH] is [MATH] . Now if [MATH] is singular then [MATH] ; therefore [MATH] and the offset place is singular. Moreover, the above table is also derived from the fact that [MATH] |
. From this table one may deduce that the local shape is preserved iff [MATH] is even. Remark 2 Notice that when [MATH] [MATH] and since [MATH] , it holds that [MATH] ; hence, the case [MATH] cannot occur and we find no contradiction between the first statement of Theorem and Proposition |
So, we see that the case [MATH] is completely described just by using first order terms. When [MATH] , the orders of [MATH] and [MATH] are in general both equal to [MATH] ; so, denoting as [MATH] the signature of a place [MATH] , we have that while [MATH] , in order to compute [MATH] we need to consider higher order te... |
). Nevertheless, using just the relationship [MATH] , the following result concerning the case [MATH] can be derived. Proposition 8 |
Let [MATH] be a place of [MATH] with signature [MATH] , where [MATH] . If [MATH] is odd, then the local shape of [MATH] is not preserved. |
Proof. Since [MATH] then if [MATH] is even and [MATH] is odd, [MATH] is odd and the local shape is not preserved. On the other hand, if [MATH] are both odd then [MATH] is even and the local shape is not preserved, either. |
Theorem and Proposition provide the following corollary on the non-preservation of the flex points of [MATH] Corollary 9 The generalized, non-classical, offset never preserves flex points. |
Proof. Let [MATH] be a real place with signature [MATH] , whose center is a flex point. Then, from Definition [MATH] are both odd. Hence [MATH] cannot be equal to [MATH] , i.e. either [MATH] or [MATH] hold. In the first case, the result follows from Theorem ; in the second case, the result follows from Proposition |
In order to give a more complete description of the cases [MATH] and [MATH] , we need to take into account higher order terms in [MATH] . This is considered in the next subsection. |
Example 1 Consider the curve [MATH] , and the place [MATH] centered at the origin. Here we have [MATH] , and therefore [MATH] . Since [MATH] is odd, from Proposition we deduce that the local shape of [MATH] is not preserved by any generalized offset. In fact, since this place is cuspidal and [MATH] , from Proposition i... |
4.2 Results using a Second Order Approximation In this section we provide a more complete description of the phenomenon when [MATH] . For this purpose, we consider a second order approximation of [MATH] . Furthermore, in the following we analyze in detail the case [MATH] . The analysis of the case [MATH] is similar; so... |
4.2.1 The case [MATH] We start assuming that [MATH] ; the case [MATH] will be addressed at the end of the subsection. Now in this case we have that |
[EQUATION] and therefore we have to distinguish whether [MATH] , or not; in the first case [MATH] , while in the second case [MATH] . Furthermore, the following lemma, concerning the curvature at the center of the considered place, will be useful. Here we recall that a place [MATH] can be taken as a parametrized curve ... |
Lemma 10 Let [MATH] be a real place of [MATH] with [MATH] , and let [MATH] be the center of [MATH] . Then, the function curvature [MATH] of [MATH] satisfies that: |
(1) If [MATH] (i.e. the place is regular), then [MATH] (2) If [MATH] (i.e. the place is singular), then [EQUATION] As a consequence, [MATH] and the derivative [MATH] are not continuous at [MATH] ; however, [MATH] and [MATH] have a removable discontinuity at [MATH] and therefore they can be extended to functions [MATH] ... |
Proof. The above expression for [MATH] can be obtained by plugging the coordinates of [MATH] into the curvature formula and doing computations with formal power series (see |
), taking into account that [MATH] . For the second statement one studies limits at the right and at the left of [MATH] In the following we will use the notation [MATH] , and [MATH] ; since in this subsection we are working with a place [MATH] satisfying that [MATH] , from Lemma 10 these quantities correspond to the ri... |
Theorem 11 Let [MATH] be a real place of [MATH] satisfying that [MATH] . If [MATH] , then the following behavior is obtained. (1) |
[MATH] : preserved. (2) [MATH] : preserved if and only if [MATH] are both even or both odd. (3) [MATH] : if [MATH] , preserved. Proof. Since [MATH] , the coefficient of [MATH] in the [MATH] -coordinate [MATH] of one of the offset places [MATH] , vanishes. Hence, for that place it holds that [MATH] . Thus, in order to c... |
[EQUATION] From Lemma 10 , one may check that the coefficient of [MATH] in [MATH] vanishes iff [MATH] . Thus, if this does not happen then [MATH] and therefore [MATH] ; so, the local shape is preserved. |
Similarly the following theorem holds. Theorem 12 Let [MATH] be a real place of [MATH] satisfying that [MATH] . If [MATH] , then: |
(1) [MATH] : if [MATH] , preserved. (2) [MATH] : if [MATH] , preserved. (3) [MATH] : preserved if and only if [MATH] are both even or both odd. |
Proof. We may observe that in this case [MATH] . More precisely, it holds that [EQUATION] These expressions can be written as [EQUATION] |
where [MATH] [MATH] [MATH] , etc. Now like in the previous theorem, we have to discuss the value of [MATH] , which is equivalent to discussing the value of [MATH] . So, let us consider first that [MATH] , i.e. that [MATH] . Then, we can compute the local shape of the place by directly applying Definition |
. For this purpose, we write [MATH] and we represent by [MATH] its signature. Clearly [MATH] ; moreover, [MATH] is parallel to [MATH] . So, in order to determine [MATH] we have to find the least natural number, greater than [MATH] , so that [MATH] and [MATH] are linearly independent. For [MATH] it holds that [MATH] . H... |
and similar reasonings. Example 2 Consider the curve of equation [MATH] , which contains the origin. A place of this curve centered at the origin is |
[MATH] , which satisfies [MATH] and therefore [MATH] ; moreover, [MATH] . Also, one may see that the absolute value of the curvature of [MATH] at [MATH] is [MATH] . Hence, from Theorem 11 and Theorem 12 it follows that in the following cases, the local shape is preserved: (i) when [MATH] ; (ii) when [MATH] , and [MATH]... |
Finally, the above results hold whenever [MATH] . So, let us briefly address the case when [MATH] . In this case, we have that [MATH] , with [MATH] . Hence, changing the parameter we can write the place as [MATH] , and we see that it corresponds to a regular place locally describing a parabola. Hence, by Theorem in Sec... |
4.2.2 The case [MATH] In this section we provide the results without proofs; these are tedious and similar to those in the preceding subsection, and are left to the reader. Moreover, for simplicity here we use the notation [MATH] , and [MATH] , analogous to the notation introduced in the preceding section. However, unl... |
We consider first the special case when [MATH] . In this case, the following theorem holds. Theorem 13 Let [MATH] be a real place of [MATH] satisfying that [MATH] . If [MATH] or [MATH] but [MATH] [MATH] , then the local shape of [MATH] is preserved. |
In the more general case [MATH] , the following result holds. Theorem 14 Let [MATH] be a real place of [MATH] satisfying that [MATH] and [MATH] . Then, the local behavior of [MATH] verifies the following: |
(1) If [MATH] , then: a. If [MATH] the local shape is preserved iff [MATH] are both even. b. If [MATH] the local shape is preserved iff [MATH] is even. |
(2) If [MATH] , then: a. If [MATH] the local shape is preserved iff [MATH] are both even. b. If [MATH] then: b.1 If [MATH] and [MATH] , then the local shape is preserved iff [MATH] is even. |
b.2 If [MATH] , the local shape is preserved iff [MATH] is even. (c) If [MATH] then: c.1 If [MATH] and [MATH] , the local shape is preserved iff [MATH] is even. |
c.2 If [MATH] the local shape is preserved iff [MATH] is even. (3) If [MATH] , then: a. If [MATH] , then the local shape is preserved iff [MATH] are both even. |
b. If [MATH] , then the local shape is preserved iff [MATH] is even and [MATH] are both even or both odd. c. If [MATH] , then if either [MATH] , or [MATH] and [MATH] , the local shape is preserved iff [MATH] is even and [MATH] are both even or both odd. |
Conclusions and Comparison between Classical and Non-Classical Generalized Offsets In the preceding sections we have analyzed local aspects on the shape of generalized offsets, both using tools coming from Differential Geometry and using the notion of local shape. In this section, we summarize the main results we have ... |
and Now the following table summarizes the most relevant properties concerning local aspects of the classical offset shape; we refer the reader to |
for further reading on them. Classical Offsets Regular Points [MATH] Singular Points [MATH] The following table shows analogous properties for the generalized, non-classical offset; these properties are derived from the results in this paper. |
Non-Classical Generalized Offsets Regular Points Never generate singular places; cusps do not arise. Flex points are never preserved. |
Turning points are not preserved in general. Tangents not preserved. Singular Points Smoothed iff [MATH] Singular flex points never preserved. |
[MATH] preserved iff [MATH] even [MATH] distinguish [MATH] , or not; many subcases. [MATH] many subcases. Hence, we observe a great number of differences between the local behavior in the classical and the non-classical case, both at regular and singular points (where the situation is far more intricate in the non-clas... |
# Source: arxiv 0910.1146 # Title: Dynamics of Enzyme Digestion of a Single Elastic Fiber Under Tension: An Anisotropic Diffusion Model # Sections: all # Downloaded: 2026-03-03T05:14:46.357763+00:00 |
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