For a curve $y\left(x\right)$ in the rectangular 2D Cartesian plane, the radius of curvature $r_{rc}\left(x\right)$ of a circle that best approximates the curve at any point can be defined by:

$r_{rc}\left(x\right)=\frac{{\left(1+y^{\prime }{\left(x\right)}^2\right)}^{3/2}}{y^{″}\left(x\right)},$(1)

where $y^{\prime }\left(x\right)$ is the first derivative (slope) and $y^{″}\left(x\right)$ is the second derivative (curvature) [6] . This formula provides an explicit relationship between the geometry of the curve and its curvature.

For other types of curves, such as those described parametrically or in multiple dimensions, the formula for $r_{rc}\left(x\right)$ can be extended to account for the general form of the curve [7] . The radius of curvature for more complex curves, such as space curves or those in multiple dimensions, is defined through the Frenet-Serret formulas, which describe the behavior of the tangent vector, normal vector, and binormal vector along the curve [8] . These formulas yield a system of differential equations that can be solved to obtain the radius of curvature in higher-dimensional settings [9] .

Moreover, the radius of curvature is a fundamental concept that arises naturally in differential geometry, where the curvature of a curve $\kappa \left(x\right)$ is defined as the reciprocal of the radius ( $1/r_{rc}\left(x\right)$) to measure the convergence/divergence of the approximate circle from the actual curve. In this research, the area under the curvature’s curve in variation with the spatial coordinate $x$ will be used and is denoted as $A_{\kappa }\left(x\right)$.

$A_{\kappa }\left(x\right)={\displaystyle \int \kappa \left(x\right)\text{d}x}$(2)

Under the assumption that the radius of curvature is known, Equation (1) is an Ordinary Differential Equation (ODE), where $y\left(x\right)$ is an unknown curve function. Due to the ODE’s structure, it is classified as nonlinear, second order, and nonhomogeneous. Therefore, a suitable set of initial conditions comprised of the initial path point $y\left(x_0\right)=y_0$ and the slope at this point $y^{\prime }\left(x_0\right)=m_0$ is employed for constructing a fully generalized closed-form analytical ODE solution $y\left(x\right)$ for any known/specified radius of curvature function.

The solution to Equation (1) is presented as follows where Ϡ $\left(x\right)$ is a quantity that determines whether the solution is real or complex, $U\left(x\right)$ is the integrand of $Є\left(x\right)$, $c_1$ and $c_2$ are constants of integration, $y\left(x\right)$ is the ODE solution, and $y^{\prime }\left(x\right)$ and $y^{″}\left(x\right)$ are successive derivatives of the ODE solution. Furthermore, the initial slope condition value $m_0$ is used in conjunction with Equation (9) to obtain the equation for $c_1$, and the initial path condition value is combined with Equation (8) to obtain the equation for $c_2$. To note, since there are multiple curve solutions that may satisfy both the initial conditions and the specified radius of curvature equation, parameters $s_1$ and $s_2$ are utilized to denote plus/minus fluctuations that account for the four possible curve solution constructs.

$$ Ϡ\left(x\right)=1-c_1^2-2c_1{A}_{\kappa }\left(x\right)-A_{\kappa }^2\left(x\right)$$ (3)

$$ U\left(x\right)=\frac{c_1+A_{\kappa }\left(x\right)}{\sqrt{ϡ\left(x\right)}}$$(4)

$Є\left(x\right)={\displaystyle \int U\left(x\right)\text{d}x}$(5)

$c_1=s_2\frac{\sqrt{m_0^2+m_0^4}}{1+m_0^2}-A_{\kappa }\left(x_0\right)$(6)

$c_2=y_0-s_1Є\left(x_0\right)$(7)

$y\left(x\right)=c_2+s_1Є\left(x\right)$(8)

$y^{\prime }\left(x\right)=s_{1}U\left(x\right)$(9)

$$ y^{\text{'}\text{'}}\left(x\right)=\frac{s_1}{r_{rc}\left(x\right)}\left(\frac{{\left(c_1+A_{\kappa }\left(x\right)\right)}^2}{ϡ{\left(x\right)}^{3/2}}+\frac{1}{\sqrt{ϡ\left(x\right)}}\right)$$(10)

For some cases of radius of curvature, antiderivative solutions to the integrals involved with the ODE solution and its successive derivatives are difficult (even for computer programs) to determine analytically. Therefore, antiderivative approximations as per Simpson’s rule (relying on left and right Riemann sums, the midpoint rule, and the trapezoid rule) are applied to the integrals seen in Equations (2) and (5) [10] . However and prior to presenting the previously mentioned approximations, the indefinite integrals are converted into definite integral forms, specifically from $x_0$ to $x$, for approximation requirements and to simplify the constants of integration.

$A_{\kappa }\left(x\right)={\displaystyle {\int }_{x_0}^x\kappa \left(x\right)\text{d}x}$(11)

$Є\left(x\right)={\displaystyle {\int }_{x_0}^{x}U\left(x\right)\text{d}x}$(12)

$c_1=s_2\frac{\sqrt{m_0^2+m_0^4}}{1+m_0^2}$(13)

$c_2=y_0$(14)

Since Equation (11) has only one integral, it can be approximated by Equations (16) to (20) (varying in accuracy levels) where $h_1$ is the step size equation, and parameter $n$ is the number of subdivisions used to obtain the desired accuracy level. To note, Equation (20) is the final, most accurate Simpson’s rule approximation which relies on all previous approximations.

$h_1\left(x\right)=\frac{x-x_0}{n}$(15)

$A_{\kappa L_1}\left(x\right)=LEF{T}_1\left(x\right)=\underset{i=0}{\overset{n-1}{{\displaystyle \sum }}}\kappa \left(x_0+i{h}_1\left(x\right)\right)h_1\left(x\right)$(16)

$A_{\kappa R_1}\left(x\right)=RIGH{T}_1\left(x\right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\kappa \left(x_0+i{h}_1\left(x\right)\right)h_1\left(x\right)$(17)

$A_{\kappa M_1}\left(x\right)=MI{D}_1\left(x\right)=\underset{i=0}{\overset{n-1}{{\displaystyle \sum }}}\kappa \left(x_0+i{h}_1\left(x\right)+\frac{h_1\left(x\right)}{2}\right)h_1\left(x\right)$(18)

$A_{\kappa T_1}\left(x\right)=TRA{P}_1\left(x\right)=\frac{A_{\kappa L_1}\left(x\right)+A_{\kappa R_1}\left(x\right)}{2}$(19)

$A_{\kappa S_1}\left(x\right)=SIM{P}_1\left(x\right)=\frac{2A_{\kappa M_1}\left(x\right)+A_{\kappa T_1}\left(x\right)}{3}$(20)

The approximation of Equation (12) is more complicated since it contains integrals of curvature nested within the outermost integral. To accomplish this, the approximations provided by Equations (22) to (26) are rewritten from above based on a new step size equation $h_2$ and index parameter $j$, where $m$ is the number of subdivisions used only for the integral of curvature (which is nested inside the outermost integral).

$h_2\left(x\right)=\frac{i{h}_1\left(x\right)-x_0}{m}$(21)

$A_{\kappa L_2}\left(x\right)=LEF{T}_2\left(x\right)=\underset{j=0}{\overset{m-1}{{\displaystyle \sum }}}\kappa \left(x_0+j{h}_2\left(x\right)\right)h_2\left(x\right)$(22)

$A_{\kappa R_2}\left(x\right)=RIGH{T}_2\left(x\right)=\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\kappa \left(x_0+j{h}_2\left(x\right)\right)h_2\left(x\right)$(23)

$A_{\kappa M_2}\left(x\right)=MI{D}_2\left(x\right)=\underset{j=0}{\overset{m-1}{{\displaystyle \sum }}}\kappa \left(x_0+j{h}_2\left(x\right)+\frac{h_2\left(x\right)}{2}\right)h_2\left(x\right)$(24)

$A_{\kappa T_2}\left(x\right)=TRA{P}_2\left(x\right)=\frac{A_{\kappa L_2}\left(x\right)+A_{\kappa R_2}\left(x\right)}{2}$(25)

$A_{\kappa S_2}\left(x\right)=SIM{P}_2\left(x\right)=\frac{2A_{\kappa M_2}\left(x\right)+A_{\kappa T_2}\left(x\right)}{3}$(26)

Combined with Equations (22) to (26), the complete approximation of $Є\left(x\right)$ is defined as follows, where the index parameter for the outermost integral is $i$, its number of subdivisions is $n$, and, therefore, its step size is $h_1$. Note that while the equations for $U\left(x\right)$ with subscripts L, R, and M were not defined directly, they can be obtained by using Equation (4) in conjunction with Equations (3) and (22) to (26).

${Є}_L\left(x\right)=LEF{T}_3\left(x\right)=\underset{i=0}{\overset{n-1}{{\displaystyle \sum }}}{U}_L\left(x_0+i{h}_1\left(x\right)\right)h_1\left(x\right)$(27)

${Є}_R\left(x\right)=RIGH{T}_3\left(x\right)=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{U}_R\left(x_0+i{h}_1\left(x\right)\right)h_1\left(x\right)$(28)

${Є}_M\left(x\right)=MI{D}_3\left(x\right)=\underset{i=0}{\overset{n-1}{{\displaystyle \sum }}}{U}_M\left(x_0+i{h}_1\left(x\right)+\frac{h_1\left(x\right)}{2}\right)h_1\left(x\right)$(29)

${Є}_T\left(x\right)=TRA{P}_3\left(x\right)=\frac{{Є}_L\left(x\right)+{Є}_R\left(x\right)}{2}$(30)

${Є}_S\left(x\right)=SIM{P}_3\left(x\right)=\frac{2{Є}_M\left(x\right)+{Є}_T\left(x\right)}{3}$(31)

Validation of the nonlinear radius of curvature differential equation theory is shown with a constant radius of curvature ( $r_{rc}\left(x\right)=2$). It is known that a circular path (which is comprised of two semicircle functions) is associated with a constant radius of curvature. The initial conditions presented below (at $x_0=0$) are expected to produce a positive semicircle ( $y\left(x\right)=\sqrt{4-x^2}$).

$y\left(x_0\right)=2$ and $y^{\prime }\left(x_0\right)=0$(32)

The exact equations of nonlinear ODE theory, with all values for $s_1$ and $s_2$, shows that the combined curve solution and successive derivatives are:

$y\left(x\right)=2\pm \left(2-\sqrt{4-x^2}\right),$(33)

$y^{\prime }\left(x\right)=\pm \frac{x}{\sqrt{4-x^2}},$(34)

$y^{″}\left(x\right)=\pm \frac{r^2}{{\left(4-x^2\right)}^{3/2}}.$(35)

The radius of curvature that results from these equations result in $r_{rc}=\pm 2$, (which is plus/minus the specified radius of curvature equation). This is not an issue due to the concept of radius of curvature, and, therefore, validates the differential equation theory. The combined graph(s) associated with the curve solution and successive derivatives are given in Figure 1 below to demonstrate the nature of the example solution.

Further validation of the nonlinear radius of curvature differential equation theory is shown with a radius of curvature that corresponds to a simple parabolic curve ( $y\left(x\right)=x^2$). The initial conditions presented below (at $x_0=0$) are expected to produce a parabolic curve with its vertex coincident with the origin of the Cartesian plane.

$r_{rc}\left(x\right)=\frac{1}{2}{\left(1+4x^2\right)}^{3/2}$(36)

$y\left(x_0\right)=0$ and $y^{\prime }\left(x_0\right)=0$(37)

The exact equations of nonlinear ODE theory, with all values for $s_1$ and $s_2$, shows that the combined curve solution and successive derivatives are:

$y\left(x\right)=\pm x^2,$(38)

$y^{\prime }\left(x\right)=\pm 2x,$(39)

$y^{″}\left(x\right)=\pm 2.$(40)

The radius of curvature that arises from these equations is:

$r_{rc}\left(x\right)=\pm \frac{1}{2}{\left(1+4x^2\right)}^{3/2},$(41)

(Which is plus/minus the specified radius of curvature equation). This is not an issue due to the concept of radius of curvature, and therefore, validates the differential equation theory. The combined graph(s) associated with the curve solution and successive derivatives is given in Figure 2 below to demonstrate the nature of the example solution.

Further validation of the nonlinear radius of curvature differential equation theory is shown with a radius of curvature that obeys the transcendental function defined by Equation (42). The initial conditions presented below (at $x_0=0$) are arbitrarily prescribed to provide insight into what kind of curve function solutions are obtained.

$r_{rc}\left(x\right)=\frac{\sin 2x}{\sin 4x}{\left(1+\sin 2x\right)}^{3/2}$ (42)

$y\left(x_0\right)=1$ and $y^{\prime }\left(x_0\right)=0$(43)

Exact equations of nonlinear ODE theory regarding $r_{rc}\left(x\right)$, with all values for $s_1$ and $s_2$, show that the combined curve solution and successive derivatives are:

$y\left(x\right)=1\pm \frac{1}{2}\left(1-\cos 2x\right),$ (44)

$y^{\prime }\left(x\right)=\pm \sin 2x,$(45)

$y^{″}\left(x\right)=\pm 2\cos 2x.$(46)

The radius of curvature that arises from these equations result in:

$r_{rc}\left(x\right)=\pm \frac{{\left(1+\sin 2x\right)}^{3/2}}{\cos 2x}\equiv \pm \frac{\sin 2x}{\sin 4x}{\left(1+\sin 2x\right)}^{3/2}$ (47)

(which is plus/minus the specified radius of curvature equation). This is not an issue due to the concept of radius of curvature, and therefore, validates the differential equation theory. The combined graph(s) associated with the curve solution and successive derivatives is given in Figure 3 below to demonstrate the nature of the example solution.

The application process involving the nonlinear radius of curvature differential equation theory shown through this example occurs when the radius of curvature and initial conditions are determined in context of a three-point self-centering regular wedge cam mechanism design type. To note, the radius of curvature is determined parametrically in conjunction with an exact backward kinematic cam rotation solution $\theta \left(x_{cr}\right)$ determined from a developed trigonometric substitution & transformation (TS&T) method [11] .

To accomplish this, various equations are used to define the parametric regular wedge cam contour transformation equations, $x_{cr}\left(\theta \right)$ and $y_{cr}\left(\theta \right)$ (in accordance with Figure 4 ), as well as the path’s end point—as per the maximum cam angle ${\theta }_{max}$. Note that the specified/driving analysis variables necessary for all related equations regarding the regular cam design are $R_{wmax}$, $R_{wmin}$, $r$, $C_{lrx}$, ${\phi }_o$, $x_1$, $y_1$, $x_2$, and $y_2$ [11] .

The associated driven/fixed analysis variables are defined as:

$R_{twmax}=\frac{R_{wmax}+C_{lrx}+r}{\cos {\phi }_o}-r,$(48)

$L_1=\sqrt{{\left(x_1-\left(R_{twmax}+r\right)\cos {\phi }_o\right)}^2+{\left(y_1+\left(R_{twmax}+r\right)\sin {\phi }_o\right)}^2},$(49)

$L_3=\sqrt{x_1^2+y_1^2},$(50)

$\eta =\frac{\pi }{2}-{\tan }^{-1}\frac{x_1-\left(R_{twmax}+r\right)\cos {\phi }_o}{y_1+\left(R_{twmax}+r\right)\sin {\phi }_o},$(51)

$\beta ={\tan }^{-1}\frac{y_1}{x_1}.$(52)

Several variable coordinate point equations involved with the regular wedge cam equations are described as follows.

$\begin{array}{c}{x}_3\left(\theta \right)=x_1-\left(x_1-\left(R_{twmax}+r\right)\cos {\phi }_o\right)\cos \theta \\ -\left(y_1+\left(R_{twmax}+r\right)\sin {\phi }_o\right)\sin \theta \end{array}$(53)

$\begin{array}{c}{y}_3\left(\theta \right)=-y_1-\left(x_1-\left(R_{twmax}+r\right)\cos {\phi }_o\right)\sin \theta \\ +\left(y_1+\left(R_{twmax}+r\right)\sin {\phi }_o\right)\cos \theta \end{array}$ (54)

$R_{tw}\left(\theta \right)=\sqrt{x_3^2\left(\theta \right)+y_3^2\left(\theta \right)}-r$(55)

$x_5=x_2$(56)

$y_5\left(\theta \right)=y_2-\left(R_{twmax}-R_{tw}\left(\theta \right)\right)$(57)

$x_6\left(\theta \right)=x_1+\left(x_2-x_1\right)\cos \theta +\left(y_2-y_1\right)\sin \theta$(58)

$y_6\left(\theta \right)=y_1-\left(x_2-x_1\right)\sin \theta +\left(y_2-y_1\right)\cos \theta$(59)

The resulting parametric transformation regular wedge cam path equations and the maximum cam angle that terminates the end point of the cam path are expressed as:

$x_{c_r}\left(\theta \right)=x_6\left(\theta \right)-x_2,$(60)

$y_{c_r}\left(\theta \right)=y_6\left(\theta \right)-y_5\left(\theta \right),$(61)

${\theta }_{max}=\eta -\beta -{\cos }^{-1}\frac{L_1^2+L_3^2-{\left(r+R_{wmin}\right)}^2}{2L_1{L}_3}.$(62)

The associated backward kinematic cam rotation equations based on TS&T are:

$\text{Θ}\left(x_{c_r}\right)=\frac{y_1-y_2}{2\left(x_1-x_2\right)-x_{c_r}}\pm \frac{\sqrt{2\left(\left(x_1-x_2\right)x_{c_r}-y_1{y}_2\right)-x_{c_r}^2+y_1^2+y_2^2}}{2\left(x_1-x_2\right)-x_{c_r}},$(63)

$\theta \left(x_c{}_r\right)=2{\tan }^{-1}\text{Θ}\left(x_{c_r}\right).$(64)

Furthermore, the successive spatial derivatives of the parametric cam equations are defined below (by Equations (74) to (77)), where $a$ and $b$ are amplitudes of the wave equations $c_w\left(\theta \right)$. These wave equations are used in conjunction with the vertical coordinate point $y_4\left(\theta \right)$ of the roller’s center point $A$ as previously shown in Figure 4 .

$a_{1_r}=2y_1\left(x_1-\left(R_{twmax}+r\right)\cos {\phi }_o\right)$ (65)

$b_{1_r}=2x_1\left(x_1-\left(R_{twmax}+r\right)\cos {\phi }_o\right)$(66)

$a_{2_r}=-2x_1\left(y_1+\left(R_{twmax}+r\right)\sin {\phi }_o\right)$(67)

$b_{2_r}=2y_1\left(y_1+R_{twmax}+r\right)\sin {\phi }_o)$(68)

$c_{w_{1_r}}\left(\theta \right)=a_{1_r}\cos \theta +b_{1_r}\sin \theta$(69)

$c_{w_{2_r}}\left(\theta \right)=a_{2_r}\cos \theta +b_{2_r}\sin \theta$(70)

$c_{w_{3_r}}\left(\theta \right)=b_{1_r}\cos \theta -a_{1_r}\sin \theta$ (71)

$c_{w_{4_r}}\left(\theta \right)=b_{2_r}\cos \theta -a_{2_r}\sin \theta$(72)

$y_4\left(\theta \right)=R_{tw}\left(\theta \right)+r$ (73)

${x^{\prime }}_{c_r}\left(\theta \right)=\frac{\text{d}{x}_{c_r}\left(\theta \right)}{\text{d}\theta }=\left(y_2-y_1\right)\cos \theta -\left(x_2-x_1\right)\sin \theta$(74)

${x^{″}}_{c_r}\left(\theta \right)=\frac{{\text{d}}^2{x}_{c_r}\left(\theta \right)}{\text{d}{\theta }^2}=-\left(x_2-x_1\right)\cos \theta -\left(y_2-y_1\right)\sin \theta$(75)

$\begin{array}{l}{y^{\prime }}_{c_r}\left(\theta \right)=\frac{\text{d}{y}_{c_r}\left(\theta \right)}{\text{d}\theta }=-\left(x_2-x_1\right)\cos \theta -\left(y_2-y_1\right)\sin \theta \\ -\frac{c_{w_{1_r}}\left(\theta \right)+c_{w_2{}_r}\left(\theta \right)}{2y_4\left(\theta \right)}\end{array}$(76)

$\begin{array}{l}{y^{″}}_{c_r}\left(\theta \right)=\frac{{\text{d}}^2{y}_{c_r}\left(\theta \right)}{\text{d}{\theta }^2}=\left(x_2-x_1\right)\sin \theta -\left(y_2-y_1\right)\cos \theta \\ +\frac{{\left(c_{w_{1_r}}\left(\theta \right)+c_{w_{2_r}}\left(\theta \right)\right)}^2}{4y_4{\left(\theta \right)}^3}-\frac{c_{w_{3_r}}\left(\theta \right)+c_{w_{4_r}}\left(\theta \right)}{2y_4\left(\theta \right)}\end{array}$(77)

The first and second derivatives of the cam contour itself are:

${y^{\prime }}_{c_r}\left(x_{c_r}\right)=\frac{{y^{\prime }}_{c_r}\left(\theta \left(x_{c_r}\right)\right)}{{x^{\prime }}_{c_r}\left(\theta \left(x_{c_r}\right)\right)},$(78)

${y^{″}}_{c_r}\left(x_{c_r}\right)=\frac{{y^{″}}_{c_r}\left(\theta \left(x_{c_r}\right)\right){x^{\prime }}_{c_r}\left(\theta \left(x_{c_r}\right)\right)-{y^{\prime }}_{c_r}\left(\theta \left(x_{c_r}\right)\right){x^{″}}_{c_r}\left(\theta \left(x_{c_r}\right)\right)}{{x^{\prime }}_{c_r}{\left(\theta \left(x_{c_r}\right)\right)}^3}.$(79)

The radius of curvature equation for this specific regular wedge cam application can now be written as:

$r_{rc}\left(x_c\right)=\frac{{\left(1+{y^{\prime }}_c{\left(x_c\right)}^2\right)}^{3/2}}{{y^{″}}_c\left(x_c\right)},$(80)

and the initial conditions (at $x_{c{0}_r}=0$ and ${\theta }_0=0$) are:

$y_{c_r}\left(x_{c{0}_r}\right)=y_{c{0}_r}\text{and}{y^{\prime }}_{c_r}\left(x_{c{0}_r}\right)=\frac{{y^{\prime }}_{c_r}\left({\theta }_0\right)}{{x^{\prime }}_{c_r}\left({\theta }_0\right)}$(81)

The nonlinear ODE application and associated validation procedures for the design configuration shown in Figure 4 are constructed from several specified/driving analysis variables: $R_{wmax}$ = 4 in. (0.1016 m), $R_{wmin}$ = 0.5 in. (0.0127 m), $r$ = 2 in. (0.0508 m), $C_{lrx}$ = 0.5 in. (0.0127 m), ${\phi }_o$ = 30˚, $x_1$ = 12.21 in. (0.3101 m) and $y_1$ = 11.14 in. (0.2829 m), and $x_2$ = 13.21 in. (0.3355 m) and $y_2$ = 18.29 in. (0.4646 m). Due to the complexity of this specific regular wedge cam theory, Simpson’s rule (where $n=100$ and $m=100$) is utilized for approximating all integrals involved within this applied nonlinear ODE theory.

With experimentation, the plus/minus values for $s_1$ and $s_2$ were determined in alignment with Figure 5 below—as they affect the solution output due to the possibility of having more than one curve satisfying both the given radius of curvature function and the initial conditions (as seen within the three previous mathematical validation examples). The specific application to regular wedge cam design coupled with the chosen analysis variable parameters requires $s_1=+1$ and $s_2=+1$. This was determined by testing all solution numbers and comparing the ODE solution and its derivatives with the exact path equation and its derivatives. Due to the use of Simpson’s rule, the maximum error $E_1$ (between exact parametric and approximate ODE paths) is a 0.186% overestimate. The errors $E_2$ and $E_3$ (between exact parametric form and approximate ODE successive cam path derivatives) are all 0.000% underestimates.

Consequently, the nonlinear differential equation theory and Simpson’s equations for both singular and nested integrals are validated. For other validation aspects regarding the parametric cam path equations, consult with research on the detailed derivation of generalized robust systems-based three-point self-centering motion theory in the design of regular and inverse wedge cams [11] . ( Table 1 )

In closing, the rectangular form of the regular wedge cam contour utilized the derived nonlinear radius of curvature ODE solution which was performed as an exploratory effort as well as for its use in the optimization of related curvature and cam dynamic characteristics [12] .