
Minimizing informal recycling activities is critical for the sustainable end-of-life treatment of electronics. Recent studies have started to revisit the concept of informality in recycling and reported empirical examples where informal sectors coordinate with formal sectors, jointly contributing to a greener recycling solution. This case study examines the systematic effort to transform and integrate the informal sector into the formal recycling industry for managing e-waste in Guiyu, China. This paper analyzes the policy design, implementation, technology development and market establishment of Guiyu's formal sector that enabled the evolution of the local informal recycling industry. The results show that the salient success factor is to offer advanced and centralized e-waste treatment by constructing a formal recycling sector while maintaining the competitive characteristics of the old informal businesses, including manual dismantling and private e-waste collection networks. Those characteristics ensured increased reuse value and sufficient e-waste sources. Meanwhile, the study found that many challenges and conflicts during this transition are rooted in the often-overlooked societal and historical contexts that profoundly shaped the local recycling industry. Authorities of regions facing challenges regulating informal recycling of e-waste, especially developing countries, could initiate similar systems based on local realities and the collaboration between formal and informal sectors to minimize the environmental and societal consequences of unregulated informal e-waste recycling.
Citation: Congying Wang, Fu Zhao, Carol Handwerker. Transforming and integrating informal sectors into formal e-waste management system: A case study in Guiyu, China[J]. Clean Technologies and Recycling, 2022, 2(4): 225-246. doi: 10.3934/ctr.2022012
[1] | Kemal Eren, Hidayet Huda Kosal . Evolution of space curves and the special ruled surfaces with modified orthogonal frame. AIMS Mathematics, 2020, 5(3): 2027-2039. doi: 10.3934/math.2020134 |
[2] | Kemal Eren, Soley Ersoy, Mohammad Nazrul Islam Khan . Simultaneous characterizations of alternative partner-ruled surfaces. AIMS Mathematics, 2025, 10(4): 8891-8906. doi: 10.3934/math.2025407 |
[3] | M. Khalifa Saad, Nural Yüksel, Nurdan Oğraş, Fatemah Alghamdi, A. A. Abdel-Salam . Geometry of tubular surfaces and their focal surfaces in Euclidean 3-space. AIMS Mathematics, 2024, 9(5): 12479-12493. doi: 10.3934/math.2024610 |
[4] | Cai-Yun Li, Chun-Gang Zhu . Construction of the spacelike constant angle surface family in Minkowski 3-space. AIMS Mathematics, 2020, 5(6): 6341-6354. doi: 10.3934/math.2020408 |
[5] | Emel Karaca . Non-null slant ruled surfaces and tangent bundle of pseudo-sphere. AIMS Mathematics, 2024, 9(8): 22842-22858. doi: 10.3934/math.20241111 |
[6] | Nadia Alluhaibi . Ruled surfaces with constant Disteli-axis. AIMS Mathematics, 2020, 5(6): 7678-7694. doi: 10.3934/math.2020491 |
[7] | Emad Solouma, Ibrahim Al-Dayel, Meraj Ali Khan, Youssef A. A. Lazer . Characterization of imbricate-ruled surfaces via rotation-minimizing Darboux frame in Minkowski 3-space E31. AIMS Mathematics, 2024, 9(5): 13028-13042. doi: 10.3934/math.2024635 |
[8] | Yanlin Li, H. S. Abdel-Aziz, H. M. Serry, F. M. El-Adawy, M. Khalifa Saad . Geometric visualization of evolved ruled surfaces via alternative frame in Lorentz-Minkowski 3-space. AIMS Mathematics, 2024, 9(9): 25619-25635. doi: 10.3934/math.20241251 |
[9] | Samah Gaber, Asmahan Essa Alajyan, Adel H. Sorour . Construction and analysis of the quasi-ruled surfaces based on the quasi-focal curves in R3. AIMS Mathematics, 2025, 10(3): 5583-5611. doi: 10.3934/math.2025258 |
[10] | Chang Sun, Kaixin Yao, Donghe Pei . Special non-lightlike ruled surfaces in Minkowski 3-space. AIMS Mathematics, 2023, 8(11): 26600-26613. doi: 10.3934/math.20231360 |
Minimizing informal recycling activities is critical for the sustainable end-of-life treatment of electronics. Recent studies have started to revisit the concept of informality in recycling and reported empirical examples where informal sectors coordinate with formal sectors, jointly contributing to a greener recycling solution. This case study examines the systematic effort to transform and integrate the informal sector into the formal recycling industry for managing e-waste in Guiyu, China. This paper analyzes the policy design, implementation, technology development and market establishment of Guiyu's formal sector that enabled the evolution of the local informal recycling industry. The results show that the salient success factor is to offer advanced and centralized e-waste treatment by constructing a formal recycling sector while maintaining the competitive characteristics of the old informal businesses, including manual dismantling and private e-waste collection networks. Those characteristics ensured increased reuse value and sufficient e-waste sources. Meanwhile, the study found that many challenges and conflicts during this transition are rooted in the often-overlooked societal and historical contexts that profoundly shaped the local recycling industry. Authorities of regions facing challenges regulating informal recycling of e-waste, especially developing countries, could initiate similar systems based on local realities and the collaboration between formal and informal sectors to minimize the environmental and societal consequences of unregulated informal e-waste recycling.
The constant-angle ruled surfaces represent an essential concept within mathematical surfaces. These surfaces possess a specific mathematical rule at each point: the tangent lines at every point maintain a constant angle with the normal vector. This property is the primary trait that distinguishes constant-angle ruled surfaces from others. Numerous disciplines, including physics, engineering, and architecture, benefit from a comprehension of constant-angle ruled surfaces. The properties of these surfaces are particularly crucial in determining the geometric features of designs and structures. By bridging the gap between mathematics and real-world applications, the understanding of constant-angle ruled surfaces facilitates the resolution of challenging issues. The potential applications of these special surfaces in both mathematics and physics have been the subject of extensive research by a number of authors in recent years. For example, Paolo and Scala used the Hamilton-Jacobi equation to examine the characteristics of surfaces with constant angles in [1]. Their research is to comprehend the behavior of surfaces with constant-angles when the direction vector becomes singular along a particular line or point. A necessary contribution to this field was given by Munteanu and Nistor by classifying surfaces where the unit normal vector maintains a constant angle with a fixed direction vector in Euclidean 3-space [2]. Subsequent research has been conducted on developable and constant-angle surfaces, shedding light on their properties and characteristics [3,4]. Additionally, A. T. Ali explored constant-angle ruled surfaces formed by Frenet frame vectors in [5]. Latterly, the theory of constant-angle surfaces has been extended to encompass other ambient spaces. For instance, in [6,7], researchers examined these surfaces in the context of Minkowski space E31. Furthermore, in [8,9,10,11,12], various alternative approaches and perspectives to the concept of constant-angle surfaces were presented within the Lorentzian frame. The Frenet frame is the most popular tool for studying curves and surfaces. However, it is insufficient for any curve in analytic space whose curvatures have distinct zero points because the principal normal and binormal vectors may be discontinuous at the curvature's zero points. Sasai [13] introduced the modified orthogonal frame (MOF) to address this issue and derived a derivative formula that is analogous to the Frenet-Serret equation. Currently, the MOF with non-zero curvature (MOFC) and with non-zero torsion (MOFT) of a space curve were presented in Minkowski 3-space by Bükcü and Karacan [14]. After this development, the modified orthogonal frames attracted a lot of attention, and various studies were devoted to searching for novelties brought by these frames. For instance, some special curves, the evolution of curves, ruled surfaces, Hasimoto surfaces, and tubular surfaces were investigated by means of the modified orthogonal frames in recent studies [15,16,17,18,19,20,21,22,23,24,25,26].
In light of recent developments encapsulated above, the constant-angle ruled surfaces have been investigated with MOFs in Euclidean 3-space. These surfaces have been classified based on the constant-angle property. In these regards, the characterizations for the developable and minimal constant-angle ruled surfaces have been presented. Also, the conditions for these surfaces to be Weingarten surfaces have been given.
Let α be a unit speed moving space curve with the arc-length parameter s in Euclidean 3-space E3. If t, n, and b denote the tangent, principal normal, and binormal unit vectors at any point α(s) of α, respectively. Then a moving frame occurs satisfying the Frenet derivative equations t′=κn, n′=−κt+τb, b′=−τn where κ and τ are the curvature and the torsion of α, respectively.
In the cases that the principal normal and binormal vectors in the Frenet frame of any space curve are discontinuous at the points where the curvature is zero, Sasai's interpretation can be put into use, i.e., his modified orthogonal frame comes onto the stage [13]. Even though the Frenet frame can display a change at the points s∈(s0−ε,s0+ε) for any ε>0 provided that κ(s0)=0, two types of orthogonal frames can be proposed. The first one is called modified orthogonal frame with non-vanishing curvature (MOFC) for κ(s)≠0 at such points, and the second one is modified orthogonal frame with non-vanishing torsion (MOFT) for τ(s)≠0.
Let the curvature κ of a general analytic curve α be non-zero everywhere; then the elements of MOFC of a curve are defined as
T=dαds,N=dTds,B=T×N, |
where "×" represents the vector product. At non-zero points of κ, there are the relations between the MOFC and the Frenet frame as
T=t,N=κn,B=κb. |
Therefore, the derivative formulas for the elements of the MOFC are
T′=N,N′=−κ2T+κ′κN+τB,B′=−τN+κ′κB, | (2.1) |
where the prime denotes differentiation with respect to the affine arc-length parameter s and τ=det(α′,α″,α‴)κ2 is the torsion of the space curve α [13]. Also, the MOFC provides
⟨T,T⟩=1, ⟨N,N⟩=⟨B,B⟩=κ2, ⟨T,N⟩=⟨T,B⟩=⟨N,B⟩=0. |
Secondly, assume that the torsion τ of a general analytic curve α is nonzero everywhere. Then the relations between MOFT and the Frenet frame are
T=t,N=τn,B=τb. |
In this case, the derivative formulas for MOFT hold:
T′=κτN,N′=−κτT+τ′τN+τB,B′=−τN+τ′τB, | (2.2) |
where ⟨T,T⟩=1, ⟨N,N⟩=⟨B,B⟩=τ2, ⟨T,N⟩=⟨T,B⟩=⟨N,B⟩=0, [14].
The following basic definitions for any surface Φ(s,v) in Euclidean 3-space are well-known.
Definition 2.1. Let Φ(s,v) be a surface in Euclidean 3-space.
i. The unit normal vector of Φ(s,v) is defined by U(s,v)=Φs×Φv‖Φs×Φv‖, where the tangent vectors of Φ(s,v) are Φs=∂Φ∂s and Φv=∂Φ∂v.
ii. The coefficients of the first fundamental form I(s,v)=Eds2+2Fdsdv+Gdv2 of Φ(s,v) are defined by
E(s,v)=⟨Φs,Φs⟩,F(s,v)=⟨Φs,Φv⟩,G(s,v)=⟨Φv,Φv⟩. |
iii. The coefficients of the second fundamental form II(s,v)=eds2+2fdsdv+gdv2 of Φ(s,v) are defined by
e(s,v)=⟨U,Φss⟩,f(s,v)=⟨U,Φsv⟩,g(s,v)=⟨U,Φvv⟩. |
iv. The Gaussian and the mean curvatures of Φ(s,v) are defined by
K(s,v)=eg−f2EG−F2andH(s,v)=Eg−2Ef+Ge2(EG−F2), |
respectively.
v. A smooth surface Φ(s,v) satisfying a functional relationship between the Gaussian curvature K and the mean curvature H is called a Weingarten surface [27].
Proposition 2.1. Let Φ(s,v) be a surface in Euclidean 3-space.
i. If Φ(s,v) has zero Gaussian curvature everywhere, it is developable.
ii. If Φ(s,v) has zero mean curvature everywhere, it is minimal.
iii. If KsHv−KvHs=0, then Φ(s,v) is a Weingarten surface [27].
Let a ruled surface be generated by a family of straight lines along an analytical curve σ(s), called the base curve. Its parametric equation is presented by
Φ(s,v)=σ(s)+vΥ(s). | (3.1) |
Here, f, g, and h are smooth functions of s. Let the director curve Υ(s) be a linear combination of the modified Frenet vectors introduced by Sasai as Υ(s)=fT+gN+hB in the case of the Frenet frame failing where κ(s0)=0 at any point. In that regard, we refer to the frames MOFC and MOFT of the base curve for κ(s)≠0 and τ(s)≠0 at each point s∈(s0−ε,s0+ε)∖{s0}, respectively. In the following two subsections, we examine each case separately.
Let the curvature of the base curve σ be non-zero everywhere. By using differential equations formulas (2.1) for MOFC, the partial differential equations of the surface Φ(s,v) represented by (3.1) are obtained as
Φs=(1−v(gκ2−f′))T+v(f−hτ+g′+κ′gκ)N+v(gτ+h′+hκ′κ)B |
and
Φv=fT+gN+hB. |
The cross-product of the above tangent vector fields is found as
Φs×Φv=v(h(f−hτ+g′)−g(gτ+h′))T−(h−v(f(gτ+h′)+h(gκ2+f′+fκ′κ)))N+(g+v(g(f′−gκ2)−f(f−hτ+g′+gκ′κ)))B. | (3.2) |
By a straightforward computation, the normal vector of Φ(s,v) is
U=U1T+U2N+U3B, |
where
U1=U11+vU12,U2=U21+vU22,U3=U31+vU32. | (3.3) |
If Eqs (3.2) and (3.3) are compared, the following Eq (3.4) is obtained:
{U11=0,U12=h(f−hτ+g′)−g(gτ+h′),U21=−h,U22=f(gτ+h′)+h(gκ2−f′+fκ′κ),U31=g,U32=g(f′−gκ2)−f(f−hτ+g′+gκ′κ). | (3.4) |
If we assume that the normal vector U of the surface is parallel to the tangent vector T of the base curve σ(s) according to the MOFC, then we have the following conditions:
U1≠0,U2=U3=0. | (3.5) |
Considering Eq (3.4), f=g=h=0 is obtained by the common solution of Eq (3.5). This is a contradiction because of Υ(s)≠0 and U1≠0. So, we can give the following theorem.
Theorem 3.1. There is no constant-angle ruled surface parallel to the tangent vector direction satisfying the conditions of Eq (3.5).
Suppose that the normal vector U of the surface Φ(s,v) is parallel to the direction of the modified principal normal vector N of the base curve σ(s) according to the MOFC; then there are the conditions:
U2≠0,U1=U3=0. | (3.6) |
Since U31=0, g vanishes. In that case, from Eq (3.4), we get the equations:
{U11=0,U12=h(f−hτ),U21=−h,U22=fh′−h(f′−fκ′κ),U31=0,U32=−f(f−hτ). |
Then there are two cases as follows:
Case (1): f=hτ, g=0, and h≠0. From this case, it is easy to see that the conditions given by (3.6) and then the constant-angle ruled surface is rewritten in the form
Φcn1(s,v)=σ(s)+vh(τT+B), |
such that f=hτ. By using the equations given in (2.1), the partial derivatives of the equation of the surface Φcn1(s,v) are
(Φcn1)s=(1+v(hτ)′)T+v(h′+hκ′κ)B |
and
(Φc1)v=h(τT+B). |
By a straightforward computation, the normal vector of the surface Φcn1(s,v) is calculated as
U=(Φcn1)s×(Φcn1)v‖(Φcn1)s×(Φcn1)v‖=N. |
Theorem 3.2. Let Φcn1(s,v) be a constant-angle ruled surface with MOFC; then the Gaussian and mean curvatures are
Kcn1=0andHcn1=κ(1+τ2)2(κ−vh(τκ′−κτ′)), |
respectively.
Proof. Let Φcn1(s,v) be a constant-angle ruled surface. The coefficients of the first and second fundamental forms of Φcn1(s,v) are
Ecn1=⟨(Φcn1)s,(Φcn1)s⟩=v2(h′+hκ′κ)2+(1+vτh′+vhτ′)2,Fcn1=⟨(Φcn1)s,(Φcn1)v⟩=h(v(h′+hκ′κ)+τ(1+vτh′+vhτ′)),Gcn1=⟨(Φcn1)v,(Φcn1)v⟩=h2(1+τ2), |
and
ecn1=⟨(Φcn1)ss,U⟩=v2(h′+hκ′κ)2+(1+vτh′+vhτ′)2,fcn1=⟨(Φcn1)sv,U⟩=h(v(h′+hκ′κ)+τ(1+vτh′+vhτ′)),gcn1=⟨(Φcn1)vv,U⟩=h2(1+τ2), |
respectively, since
(Φcn1)s=(1+vτh′+vhτ′)T+v(h′+hκ′κ)B,(Φcn1)v=hτT+hB,(Φcn1)ss=v(2h′τ′+τh′′+hτ′′)T+(1−vhτκ′κ+vhτ′)N+v(2h′κ′κ+h′′+hκ′′κ)B,(Φcn1)sv=(τh′+hτ′)T+(h′+hκ′κ)B,(Φcn1)vv=0. |
If the above relations are substituted in the formulas
Kcn1=ecn1gcn1−fcn12Ecn1Gcn1−Fcn12=0andHcn1=12Ecn1gcn1−2Fcn1fcn1+Gcn1ecn1Ecn1Gcn1−Fcn12, |
then the Gaussian and mean curvatures are found as in the hypothesis.
Corollary 3.1. Let Φcn1(s,v) be a constant-angle ruled surface with MOFC; then the constant-angle surface is
i. developable surface,
ii. not minimal surface,
iii. Weingarten surface.
Case (2): f=0, g=0, h≠0, and τ=0. In this case, the conditions given by (3.6) are satisfied, and then we have obtained a constant-angle ruled surface which takes the form
Φcn2(s,v)=σ(s)+vhB, |
where the base curve σ(s) is planar. The partial derivatives of the surfaces Φcn2(s,v) using (2.1) are found as
(Φcn2)s=T+v(h′+hκ′κ)B |
and
(Φcn2)v=hB. |
By a straightforward computation, the normal vector of the surface Φcn2(s,v) is
U=−N. |
Theorem 3.3. Let Φcn2(s,v) be a constant-angle ruled surface; then the Gaussian curvature and mean curvature are
Kcn2=0andHcn2=0, |
respectively.
Proof. This theorem is proved in a manner akin to that of Theorem 3.2.
Corollary 3.2. Let Φcn2(s,v) be a constant-angle ruled surface with MOFC; then Φcn2(s,v) is
i. developable surface,
ii. minimal surface,
iii. Weingarten surface.
Let the normal vector U of the surface Φ(s,v) be parallel to the modified binormal vector B of the base curve σ(s) according to the MOFC; then we have the following conditions:
U3≠0,U1=U2=0. | (3.7) |
Since U21=0, h vanishes. In that case, from Eq (3.4), we get the following equations:
{U11=0,U12=−g2τ,U21=0,U22=fgτ,U31=g,U32=g(f′−gκ2)−f(f+g′+gκ′κ). |
Then, there are the cases below that satisfy the conditions in (3.7).
Case (1): f≠0, g=0, and h=0. From this case, we have
{U11=U12=U21=U22=0,U31=0,U32=−f2. |
These mean that the constant-angle ruled surface takes the form
Φcb1(s,v)=σ(s)+vfT. |
Also, this case is achieved whenever the base curve σ(s) is planar. The normal vector of the surface Φcb1(s,v) is obtained as
U=−B, |
since
(Φcb1)s=(1+vf′)T+vfNand(Φcb1)v=fT. |
Theorem 3.4. Let Φcb1(s,v) be a constant-angle ruled surface; then the Gaussian and mean curvatures are
Kcb1=0andHcb1=−τ2vf, |
respectively.
Proof. This theorem is proved in a similar manner to the proof of Theorem 3.2.
Corollary 3.3. Let Φcb1(s,v) be a constant-angle ruled surface; then the constant-angle surface is
i. developable surface,
ii. minimal surface if and only if the base curve is planar,
iii. Weingarten surface.
Case (2): f=0, g≠0, h=0, and τ=0. In this case,
{U11=U12=U21=U22=0,U31=g,U32=−g2κ2, |
and then it is found that the constant-angle ruled surface is presented in the form
Φcb2(s,v)=σ(s)+vgN, |
where the base curve σ(s) is planar. The normal vector of the surface Φcb2(s,v) is obtained as
U=B, |
where
(Φcb2)s=(1−vgκ2)T+v(h′+hκ′κ)Nand(Φcb2)v=gN. |
Theorem 3.5. Let Φcb2(s,v) be a constant-angle ruled surface; then the Gaussian and mean curvatures are
Kcb2=0andHcb2=0, |
respectively.
Proof. The proof of this theorem follows a similar manner to the proof of Theorem 3.2.
Corollary 3.4. Let Φcb2(s,v) be a constant-angle ruled surface; then Φcb2(s,v) is
i. developable surface,
ii. minimal surface,
iii. Weingarten surface.
Case (3): f≠0, g≠0, h=0, and τ=0. In this case,
{U11=U12=U21=U22=0,U31=g,U32=g(f′−gκ2)−f(f+g′+gκ′κ), |
and then the constant-angle ruled surface is represented by
Φcb3(s,v)=σ(s)+v(fT+gN), |
where the σ(s) is planar. The normal vector of the surface Φcb3(s,v) is calculated as
U=B, |
by the facts that
(Φcb3)s=(1−vgκ2+vf′)T+v(f+g′+gκ′κ)Nand(Φcb3)v=fT+gN. |
Theorem 3.6. Let Φcb3(s,v) be a constant-angle ruled surface; then the Gaussian and mean curvatures are
Kcb3=0andHcb3=0, |
respectively.
Proof. This is proved similar to the proof of Theorem 3.2.
Corollary 3.5. Let Φcb3(s,v) be a constant-angle ruled surface; then Φcb3(s,v) is
i. developable surface,
ii. minimal surface,
iii. Weingarten surface.
Let the torsion of the base curve be non-zero everywhere. The partial derivatives of the surface Φ(s,v) are represented by Eq (3.1) using derivative formulas (2.2). In this section, let's investigate under what conditions the surfaces are constant-angle ruled surfaces using MOFT. Considering the derivative formulas (2.2), the partial differential equations of the surfaces Φ(s,v) presented by Eq (3.1) are
Φs=(1+v(−gκτ+f′))T+v(g′−hτ+gτ′+fκτ)N+v(gτ+h′+hτ′τ)B |
and
Φv=fT+gN+hB, |
where f, g, and h are smooth functions of s. The cross-product of the above vector fields is found as
Φs×Φv=v(fhκτ−g2τ+h(−hτ+g′)−gh′)T+(−h+v(f(gτ+h′)+h(gκτ−f′+fτ′τ)))N+(g+v(−f2κτ+g(f′−gκτ)+f(hτ−g′−gτ′τ)))B. |
From here, the normal vector of the surface Φ(s,v) can be given in the form
W=(W11+vW12)T+(W21+vW22)N+(W31+vW32), |
such that
{W11=0,W12=fhκτ−g2τ+h(−hτ+g′)−gh′,W21=−h,W22=f(gτ+h′)+h(gκτ−f′+fτ′τ),W31=g,W32=−f2κτ+g(f′−gκτ)+f(hτ−g′−gτ′τ), | (3.8) |
where τ≠0.
In this subsection, let the normal vector W of the surface Φ(s,v) be parallel to the tangent vector T of the base curve σ(s) according to the MOFT; then we have the following conditions:
W1≠0,W2=W3=0. | (3.9) |
Case (1): Considering Eq (3.8), f=g=h=0 is obtained from the solution of Eq (3.9). This situation contradicts the conditions Υ(s)≠0 and W1≠0.
Case (2): Considering Eq (3.8), f≠0 and g=h=κ=0 are obtained from the solution of Eq (3.9). This situation contradicts the condition W1≠0. So, we can give the following theorem:
Theorem 3.7. Let the normal vector W of a surface Φ(s,v) be parallel to the tangent vector of the MOFT; then there is no constant-angle ruled surface parallel to the tangent vector that satisfies conditions Eq (3.9).
Let the normal vector W of the surface Φ(s,v) be parallel to the modified normal vector N of the base curve σ(s) according to the MOFT; then we have the following conditions:
W2≠0,W1=W3=0. | (3.10) |
Since W31=0, g vanishes. In that case, from Eq (3.8), we get the following equations:
{W11=0,W12=h(fκτ−hτ),W21=−h,W22=fh′−h(f′−fτ′τ),W31=0,W32=−f(fκτ−hτ), |
for τ≠0. Then, considering the conditions specified in Eq (3.10), there are some situations as follows:
Case (1): g=0, fκ=hτ2, and h≠0, From this case, we have
{W11=0,W12=0,W21=−h,W22=hh′τ2κ−h((hτ2κ)′−hτ2τ′κτ),W31=0,W32=0, |
where κ≠0 in addition to τ≠0. Hence, the conditions specified in Eq (3.10) are satisfied, and then we see that the constant-angle ruled surface takes the form
Φtn1(s,v)=σ(s)+vh(τ2κT+B), |
where fκ=hτ2. The partial derivatives of the surface Φtn1(s,v) using (2.2) are
(Φtn1)s=(κ2−vhτ2κ′+vκτ(τh′+2hτ′)κ2)T+v(h′+hτ′τ)Band(Φtn1)v=h(τ2κT+B). |
The normal vector of the surface Φtn1(s,v) is found as
W=(Φtn1)s×(Φtn1)v‖(Φtn1)s×(Φtn1)v‖=−N. |
Theorem 3.8. Let Φtn1(s,v) be a constant-angle ruled surface; then the Gaussian and mean curvatures of Φtn1(s,v) are
Ktn1=0andHtn1=κ(κ2+τ4)2τ(vhτ2κ′−κ(κ+vhττ′)), |
respectively, where τ≠0.
Proof. Let Φtn1(s,v) be a constant-angle ruled surface. The coefficients of the first and second fundamental forms, respectively, are determined
Etn1=⟨(Φtn1)s,(Φtn1)s⟩=v2(h′+hτ′τ)2+(κ2−vhτ2κ′+vκτ(τh′+2hτ′))2κ4,Ftn1=⟨(Φtn1)s,(Φtn1)v⟩=h(v(h′+hτ′τ)+τ2(κ2−vhτ2κ′+vκτ(τh′+2hτ′))κ3),Gtn1=⟨(Φtn1)v,(Φtn1)v⟩=h2(1+τ4κ2), |
and
etn1=⟨(Φtn1)ss,U⟩=−κτ+vhτκ′κ−vhτ′,ftn1=⟨(Φtn1)sv,U⟩=0,gtn1=⟨(Φtn1)vv,U⟩=0, |
by the aid of the equations
(Φtn1)s=(κ2−vhτ2κ′+vκτ(τh′+2hτ′)κ2)T+v(h′+hτ′τ)B,(Φtn1)v=hτ2κT+hB,(Φtn1)ss=v(2hτ2κ′2κ3−2τ2h′κ′+hτ2κ′′+4hτκ′τ′κ2+4τh′τ′+2hτ′2+τ2h′′+2hττ′′κ)T+(κτ−vhτκ′κ+vhτ′)N+v(2h′τ′τ+h′′+hτ′′τ)B,(Φtn1)sv=(τ(−hτκ′+κ(τh′+2hτ′))κ2)T+(h′+hτ′τ)B,(Φtn1)vv=0. |
If the coefficients of the first and second fundamental forms are substituted in the formulas of Gaussian and mean curvatures, the proof is completed.
Corollary 3.6. Let Φtn1(s,v) be a constant-angle ruled surface with MOFT; then Φtn1(s,v) is
i. developable surface,
ii. minimal surface if and only if the base curve is a line,
iii. Weingarten surface.
Case (2): f=0, g=0, and h≠0. This situation contradicts the fact that W1=0. So, the ruled surface cannot be a constant-angle surface.
Case (3): f≠0, g=0, and h=0. This situation contradicts the fact that W2≠0 and W3=0. So, the ruled surface cannot be a constant-angle surface.
Let the normal vector W of the surface Φ(s,v) be parallel to the modified binormal vector B of the base curve σ(s) according to the MOFT; then we have the following conditions:
W3≠0,W1=W2=0. | (3.11) |
Since W21=0, h vanishes. In that case, from Eq (3.8), we get the following equations:
{W11=0,W12=−g2τ,W21=0,W22=fgτ,W31=g,W32=−f2κτ+g(f′−gκτ)−f(g′+gτ′τ), |
for τ≠0. Then, considering the conditions in Eq (3.9), there exist the following cases.
Case (1): f≠0, g, h, and κ≠0. From this case, we have
{W11=W12=W21=W22=0,W31=0,W32=−f2κτ, |
and we obtain the constant-angle ruled surface, which takes the form
Φtb1(s,v)=σ(s)+vfT. |
By a straightforward computation, the normal vector of the surface Φtb1(s,v) is obtained as
W=−B, |
by the partial derivatives
(Φtb1)s=(1+vf′)T+vfκτNand(Φtb1)v=fT. |
Theorem 3.9. Let Φtb1(s,v) be a constant-angle ruled surface; then the Gaussian curvature and mean curvature of Φtb1(s,v) are
Ktb1=0andHtb1=−τ22vfκ, |
respectively.
Proof. The proof of this theorem is similar to the proof of Theorem 3.2.
Corollary 3.7. Let Φtb1(s,v) be a constant-angle ruled surface with MOFT; then Φtb1(s,v) is
i. developable surface,
ii. not minimal surface,
iii. Weingarten surface.
Case (2): f≠0, g=0, h=0, and κ=0. This is a contradiction because of W3≠0, which means that in this case, the ruled surface cannot be a constant-angle surface.
Case (3): f=0, g=0, h=0, and κ≠0. This situation contradicts the fact that Υ(s)≠0 and W3≠0. Thus, we say that the ruled surface cannot be a constant-angle surface in this case.
Example 3.1. Let us consider a curve given by the parametric equation
σ(s)=(1√2s∫0cos(πt22)dt,1√2s∫0sin(πt22)dt,s√2), |
which is known as the Cornu spiral or Euler spiral [16]. Also, the components s∫0cos(πt22)dt and s∫0sin(πt22)dt of the curve are called Fresnel integrals. The elements of the Frenet trihedron of the curve σ(s) are found as
t=(1√2cos(πs22),1√2sin(πs22),1√2),n=(−sin(πs22),cos(πs22),0),b=(−1√2cos(πs22),−1√2sin(πs22),1√2),κ=πs√2,τ=πs√2. |
Here, we refer to the modified Frenet vectors presented by Sasai because the principal normal and binormal vectors are discontinuous at the neighborhood of the point s0=0. In that regard, we find the modified Frenet vectors of σ as follows:
T=(1√2cos(πs22),1√2sin(πs22),1√2),N=(−πs√2sin(πs22),πs√2cos(πs22),0),B=(−πs2cos(πs22),−πs2sin(πs22),πs2). |
If we assume that f=πssin(s)√2, g=0, and h=sin(s), the equation of the constant-angle ruled surface parallel to the modified normal vector of σ for Case 1 with the MOFC is represented as
Φcn1(s,v)=(1√2s∫0cos(πt22)dt,1√2s∫0sin(πt22)dt,s√2+vπssin(s)), |
see Figure 1.
Let us take that f=sin(s) and h=g=0. Then the equation of the constant-angle ruled surface parallel to the modified binormal vector of σ for Case 1 with MOFC is
Φcb1(s,v)=1√2(s∫0cos(πt22)dt+vcos(πs22)sin(s),s∫0sin(πt22)dt+vsin(πs22)sin(s),s+vsin(s)), |
see Figure 2.
For f=cos(s)πs√2 and h=cos(s), the equation of the constant-angle ruled surface parallel to the modified principal normal vector of σ for Case 1 with the MOFT is represented by
Φtn1(s,v)=1√2(s∫0cos(πt22)dt,s∫0sin(πt22)dt,s(√2πvcos(s)+1)), |
see Figure 3.
In this study, the modified orthogonal frames in Euclidean 3-space have been employed to investigate the constant-angle ruled surfaces. The investigation involves determining the necessary and sufficient conditions for any ruled surface to stand the angles between each modified Frenet vector of the base curve and the unit normal vector of the surface to be constant. Within this context, the conditions for such surfaces to be minimal, developable, and Weingarten surfaces have been derived. Notably, this study presents novel insights, as constant-angle ruled surfaces have not been previously examined in the context of MOFs. Finally, the study provides examples of some constant-angle surfaces, accompanied by their graphics, and offers a new perspective for future research in the field of surface theory.
Kemal Eren, Soley Ersoy and Mohammad N. I. Khan: Conceptualization, methodology, investigation, writing - original draft, writing - review and editing. All authors have read and approved the final version of the manuscript for publication.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2025).
The authors declare no conflicts of interest.
[1] |
Cao J, Lu B, Chen Y, et al. (2016) Extended producer responsibility system in China improves e-waste recycling: Government policies, enterprise, and public awareness. Renew Sust Energ Rev 62: 882–894. https://doi.org/10.1016/j.rser.2016.04.078 doi: 10.1016/j.rser.2016.04.078
![]() |
[2] |
Chi X, Streicher-Porte M, Wang MY, et al. (2011) Informal electronic waste recycling: A sector review with special focus on China. Waste Manage 31: 731–742. https://doi.org/10.1016/j.wasman.2010.11.006 doi: 10.1016/j.wasman.2010.11.006
![]() |
[3] |
Guibrunet L (2019) What is "informal" in informal waste management? Insights from the case of waste collection in the Tepito neighbourhood, Mexico City. Waste Manage 86: 13–22. https://doi.org/10.1016/j.wasman.2019.01.021 doi: 10.1016/j.wasman.2019.01.021
![]() |
[4] |
Miranda ITP, Fidelis R, de Souza Fidelis, et al. (2020) The integration of recycling cooperatives in the formal management of municipal solid waste as a strategy for the circular economy—The case of Londrina, Brazil. Sustainability 12: 10513. https://doi.org/10.3390/su122410513 doi: 10.3390/su122410513
![]() |
[5] |
Oduro-Appiah K, Afful A, Kotey VN, et al. (2019) Working with the informal service chain as a locally appropriate strategy for sustainable modernization of municipal solid waste management systems in lower-middle income cities: lessons from Accra, Ghana. Resources 8: 12. https://doi.org/10.3390/resources8010012 doi: 10.3390/resources8010012
![]() |
[6] | Balde CP, Forti V, Gray V, et al. (2017) The Global E-waste Monitor 2017: Quantities, Flows and Resources E-book, Tokyo: United Nations University. |
[7] | Bournay E (2006) Vital Waste Graphics 2, 2 Eds., Nairobi: United Nations Environment Programme. |
[8] | World Economic Forum, A new circular vision for electronics: Time for a global reboot. PACE, 2019. Available from: http://www3.weforum.org/docs/WEF_A_New_Circular_Vision_for_Electronics.pdf. |
[9] |
Yang C, Tan Q, Liu L, et al. (2017) Recycling tin from electronic waste: A problem that needs more attention. ACS Sustain Chem Eng 5: 9586–9598. https://doi.org/10.1021/acssuschemeng.7b02903 doi: 10.1021/acssuschemeng.7b02903
![]() |
[10] |
Deng W, Zheng J, Bi X, et al. (2007) Distribution of PBDEs in air particles from an electronic waste recycling site compared with Guangzhou and Hong Kong, South China. Environ Int 33: 1063–1069. https://doi.org/10.1016/j.envint.2007.06.007 doi: 10.1016/j.envint.2007.06.007
![]() |
[11] |
Guo Y, Huang C, Zhang H, et al. (2009) Heavy metal contamination from electronic waste recycling at Guiyu, Southeastern China. J Environ Qual 38: 1617–1626. https://doi.org/10.2134/jeq2008.0398 doi: 10.2134/jeq2008.0398
![]() |
[12] |
Leung AOW, Duzgoren-Aydin NS, Cheung KC, et al. (2008) Heavy metals concentrations of surface dust from e-waste recycling and its human health implications in Southeast China. Environ Sci Technol 42: 2674–2680. https://doi.org/10.1021/es071873x doi: 10.1021/es071873x
![]() |
[13] |
Lin X, Xu X, Zeng X, et al. (2017) Decreased vaccine antibody titers following exposure to multiple metals and metalloids in e-waste-exposed preschool children. Environ Pollut 220: 354–363. https://doi.org/10.1016/j.envpol.2016.09.071 doi: 10.1016/j.envpol.2016.09.071
![]() |
[14] |
Luo Q, Cai ZW, Wong MH (2007) Polybrominated diphenyl ethers in fish and sediment from river polluted by electronic waste. Sci Total Environ 383: 115–127. https://doi.org/10.1016/j.scitotenv.2007.05.009 doi: 10.1016/j.scitotenv.2007.05.009
![]() |
[15] |
Quan SX, Yan B, Lei C, et al. (2014) Distribution of heavy metal pollution in sediments from an acid leaching site of e-waste. Sci Total Environ 499: 349–355. https://doi.org/10.1016/j.scitotenv.2014.08.084 doi: 10.1016/j.scitotenv.2014.08.084
![]() |
[16] |
Song Q, Li J (2014) Environmental effects of heavy metals derived from the e-waste recycling activities in China: A systematic review. Waste Manage 34: 2587–2594. https://doi.org/10.1016/j.wasman.2014.08.012 doi: 10.1016/j.wasman.2014.08.012
![]() |
[17] |
Huo X, Peng L, Xu X, et al. (2007) Elevated blood lead levels of children in Guiyu, an electronic waste recycling town in China. Environ Health Persp 115: 1113–1117. https://doi.org/10.1289/ehp.9697 doi: 10.1289/ehp.9697
![]() |
[18] |
Zhang B, Huo X, Xu L, et al. (2017) Elevated lead levels from e-waste exposure are linked to decreased olfactory memory in children. Environ Pollut 231: 1112–1121. https://doi.org/10.1016/j.envpol.2017.07.015 doi: 10.1016/j.envpol.2017.07.015
![]() |
[19] |
Xu X, Liu J, Huang C, et al. (2015) Association of polycyclic aromatic hydrocarbons (PAHs) and lead co-exposure with child physical growth and development in an e-waste recycling town. Chemosphere 139: 295–302. https://doi.org/10.1016/j.chemosphere.2015.05.080 doi: 10.1016/j.chemosphere.2015.05.080
![]() |
[20] |
Yang H, Huo X, Yekeen TA, et al. (2012) Effects of lead and cadmium exposure from electronic waste on child physical growth. Environ Sci Pollut R 20: 4441–4447. https://doi.org/10.1007/s11356-012-1366-2 doi: 10.1007/s11356-012-1366-2
![]() |
[21] |
Zeng X, Xu X, Boezen HM, et al. (2017) Decreased lung function with mediation of blood parameters linked to e-waste lead and cadmium exposure in preschool children. Environ Pollut 230: 838–848. https://doi.org/10.1016/j.envpol.2017.07.014 doi: 10.1016/j.envpol.2017.07.014
![]() |
[22] |
Zhang L (2009) From Guiyu to a nationwide policy: e-waste management in China. Env Polit 18: 981–987. https://doi.org/10.1080/09644010903345736 doi: 10.1080/09644010903345736
![]() |
[23] |
Murray A, Skene K, Haynes K (2015) The circular economy: An interdisciplinary exploration of the concept and application in a global context. J Bus Ethics 140: 369–380. https://doi.org/10.1007/s10551-015-2693-2 doi: 10.1007/s10551-015-2693-2
![]() |
[24] |
Shinkuma T, Huong N (2009) The flow of e-waste material in the Asian region and a reconsideration of international trade policies on e-waste. Environ Impact Assess Rev 29: 25–31. https://doi.org/10.1016/j.eiar.2008.04.004 doi: 10.1016/j.eiar.2008.04.004
![]() |
[25] | Zhang L (2009) Guiyu, when it can achieve the real lndustrial transition (in Chinese)? Resour Recy 5: 16–18. Available from: https://d.wanfangdata.com.cn/periodical/ysjszsyly200905006. |
[26] |
Zhou L, Xu Z (2012) Response to waste electrical and electronic equipments in China: Legislation, recycling system, and advanced integrated process. Environ Sci Technol 46: 4713–4724. https://doi.org/10.1021/es203771m doi: 10.1021/es203771m
![]() |
[27] |
Hu Y, Poustie M (2018) Urban mining demonstration bases in China: A new approach to the reclamation of resources. Waste Manage 79: 689–699. https://doi.org/10.1016/j.wasman.2018.08.032 doi: 10.1016/j.wasman.2018.08.032
![]() |
[28] | Li Y (2010) E-waste recycling of on the Guiyu case study[Master's thesis] (in Chinese). University of Lanzhou, China. Available from: http://d.wanfangdata.com.cn/thesis/Y1703836. |
[29] | Secretariat of the Basel Convention, Basel convention on the control of transboundary movements of hazardous wastes and their disposal. Secretariat of the Basel Convention, 2018. Available from: https://www.basel.int/Portals/4/Basel%20Convention/docs/text/BaselConventionText-e.pdf. |
[30] |
Wong N (2018) Electronic waste governance under "One Country, Two Systems": Hong Kong and Mainland China. Int J Environ Heal R 15: 2347. https://doi.org/10.3390/ijerph15112347 doi: 10.3390/ijerph15112347
![]() |
[31] |
Yu L, He W, Li G, et al. (2014) The development of WEEE management and effects of the fund policy for subsidizing WEEE treating in China. Waste Manage 34: 1705–1714. https://doi.org/10.1016/j.wasman.2014.05.012 doi: 10.1016/j.wasman.2014.05.012
![]() |
[32] | Gmünder S, Recycling-from waste to resource: Assessment of optimal manual dismantling depth of a desktop PC in china based on eco-efficiency calculations. Swiss Federal Institute of Technology (ETH) and Swiss Federal Laboratories for Materials Testing and Research (EMPA), 2007. Available from: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=ba730e16957ed2d0edcc35effe3477059de23d2b. |
[33] | Tong L, Hai R, Xie T, et al. (2005) Study on the programming of guangdong Guiyu waste machinery and electronic products disassembly and using demonstration district. China Resour Compr Util 24: 4–8. |
[34] |
Kirby PW (2019) Materialities meet the mangle: Electronic waste scavenging in Japan and China. Geoforum 102: 48–56. https://doi.org/10.1016/j.geoforum.2019.03.011 doi: 10.1016/j.geoforum.2019.03.011
![]() |
[35] | Lines K, Garside B, Sinha S, et al. (2016) Clean and Inclusive? Recycling E-Waste in China and India, London: International Institute for Environment and Development. |
[36] |
Tuncuk A, Stazi V, Akcil A, et al. (2012) Aqueous metal recovery techniques from e-scrap: Hydrometallurgy in recycling. Miner Eng 25: 28–37. https://doi.org/10.1016/j.mineng.2011.09.019 doi: 10.1016/j.mineng.2011.09.019
![]() |
[37] | Zhong W (2013) A treatment method for using waste plastic-encapsulated IC card boards to extract gold (in Chinese). China National Intellectual Property Administration, No. ZL201210215343.0. Available form: https://patents.google.com/patent/CN102703710B/en. |
[38] |
Huang K, Guo J, Xu Z (2009) Recycling of waste printed circuit boards: A review of current technologies and treatment status in China. J Hazard Mater 164: 399–408. https://doi.org/10.1016/j.jhazmat.2008.08.051 doi: 10.1016/j.jhazmat.2008.08.051
![]() |
[39] | Zeng L, Liu F, Zhang P (2016) Plant practice on treating waste printed circuit boards in top-blown furnace. NMEMS 12: 20–22. |
[40] | Ministry of Ecology and Environment, "E-Waste Capital" transformation and leap-over—Comprehensive remediation practice of Guiyu Town, Shantou City, Guangdong Province (in Chinese). Ministry of Ecology and Environment of the People's Republic of China, 2019. Available from: https://www.mee.gov.cn/xxgk2018/xxgk/xxgk15/201908/t20190828_730337.html. |
[41] | Management Committee of Guiyu Circular Economy Industrial Park, Departmental budget of the administrative committee of Guiyu circular economy industrial park (in Chinese). Management Committee of Guiyu Circular Economy Industrial Park, 2020. Available from: http://www.gdcy.gov.cn/attachment/0/9/9240/1770384.pdf. |
[42] |
Oliveira CRD, Bernardes AM, Gerbase AE (2012) Collection and recycling of electronic scrap: A worldwide overview and comparison with the Brazilian situation. Waste Manage 32: 1592–1610. https://doi.org/10.1016/j.wasman.2012.04.003 doi: 10.1016/j.wasman.2012.04.003
![]() |
[43] | Greenpeace, The potentiality of the circular economy of waste electronic products in China (in Chinese). Greenpeace, 2019. Available from: https://www.greenpeace.org.cn/the-potentiality-of-the-circular-economy-of-waste-electronic-products-in-china-report/. |
[44] |
Zeng X, Duan H, Wang F, et al. (2017) Examining environmental management of e-waste: China's experience and lessons. Renew Sust Energ Rev 72: 1076–1082. https://doi.org/10.1016/j.rser.2016.10.015 doi: 10.1016/j.rser.2016.10.015
![]() |
[45] |
Tong X, Lifset R, Lindhqvist T (2004) Extended producer responsibility in China: Where is "best practice"? J Ind Ecol 8: 6–9. https://doi.org/10.1162/1088198043630423 doi: 10.1162/1088198043630423
![]() |
[46] |
Hicks C, Dietmar R, Eugster M (2005) The recycling and disposal of electrical and electronic waste in China—legislative and market responses. Environ Impact Assess Rev 25: 459–471. https://doi.org/10.1016/j.eiar.2005.04.007 doi: 10.1016/j.eiar.2005.04.007
![]() |
[47] |
Kojima M, Yoshida A, Sasaki S (2009) Difficulties in applying extended producer responsibility policies in developing countries: case studies in e-waste recycling in China and Thailand. J Mater Cycles Waste Manage 11: 263–269. https://doi.org/10.1007/s10163-009-0240-x doi: 10.1007/s10163-009-0240-x
![]() |
[48] |
Davis JM, Garb Y (2020) Toward active community environmental policing: Potentials and limits of a catalytic model. Environ Manage 65: 385–398. https://doi.org/10.1007/s00267-020-01252-1 doi: 10.1007/s00267-020-01252-1
![]() |
[49] |
Sasaki S (2020) The effects on Thailand of China's import restrictions on waste: measures and challenges related to the international recycling of waste plastic and e-waste. J Mater Cycles Waste Manage 23: 77–83. https://doi.org/10.1007/s10163-020-01113-3 doi: 10.1007/s10163-020-01113-3
![]() |
[50] |
Wang F, Huisman J, Meskers CE, et al. (2012) The Best-of-2-Worlds philosophy: Developing local dismantling and global infrastructure network for sustainable e-waste treatment in emerging economies. Waste Manage 32: 2134–2146. https://doi.org/10.1016/j.wasman.2012.03.029 doi: 10.1016/j.wasman.2012.03.029
![]() |
[51] |
Schulz Y, Lora-Wainwright A (2019) In the name of circularity: Environmental improvement and business slowdown in a Chinese recycling hub. Worldw Waste J Interdiscip Stud 2: 1–13. https://doi.org/10.5334/wwwj.28 doi: 10.5334/wwwj.28
![]() |
![]() |
![]() |