# On the Measurement of Particle Contact Curvature and Young's Modulus Using X-ray \(\mu\)CT
## 3 Contact Curvature Theories
### Curvature Formula  
Curvature is the amount by which a geometric object deviates from being flat with the equivalent definition as the reciprocal of the radius of the osculating circle. For a planar curve given explicitly as \(\text{y}=f(x)\), and using primes for derivatives with respect to the coordinate \(x\), the curvature is:  
\[k=\frac{\left|y^{\prime}\right|}{\left(1+y^{\prime 2}\right)^{3/2}} \tag{13}\]  
For an implicit surface \(F(x,y,z)=0\), three notations including the gradient \(\
abla F\), the Hessian \(H(F)\), and the adjoint of the Hessian for surface are needed to derive curvature formula [28]:  
\[\
abla F=\left(\frac{\partial F}{\partial x}\frac{\partial F}{\partial y} \frac{\partial F}{\partial z}\right)=\left(F_{x}\ F_{y}\ F_{z}\right) \tag{14}\]  
\[H(F)=\left(\begin{array}{cc}\frac{\partial^{2}F}{\partial x^{2}}&\frac{ \partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial x\partial z }\\ \frac{\partial^{2}F}{\partial y\partial x}&\frac{\partial^{2}F}{\partial y \partial y}&\frac{\partial^{2}F}{\partial z^{2}}\\ \frac{\partial^{2}F}{\partial z\partial x}&\frac{\partial^{2}F}{\partial z \partial y}&\frac{\partial^{2}F}{\partial z^{2}}\\ \end{array}\right)=\left(\begin{array}{cc}F_{xx}&F_{xy}&F_{xz}\\ F_{yx}&F_{yy}&F_{yz}\\ F_{zx}&F_{zy}&F_{yz}&F_{yz}\\ F_{zx}&F_{zy}&F_{zz}\\ \end{array}\right)=\
abla(\
abla F) \tag{15}\]  
\[H*(F)=\left(\begin{array}{cc}Cofactor(F_{xx})&Cofactor(F_{yy})&Cofactor(F_{ xz})\\ Cofactor(F_{yx})&Cofactor(F_{yy})&Cofactor(F_{yz})\\ Cofactor(F_{xx})&Cofactor(F_{zy})&Cofactor(F_{zz})\\ Cofactor(F_{xx})&Cofactor(F_{zy})&Cofactor(F_{zz})\\ \end{array}\right)=\left(\begin{array}{cc}F_{yy}F_{zz}-F_{yz}F_{zy}&F_{yz} F_{xx}-F_{yx}F_{zz}&F_{yx}F_{zy}-F_{yy}F_{zx}\\ F_{xz}F_{yy}-F_{xz}F_{zy}&F_{xx}F_{zz}-F_{xx}F_{xx}F_{xx}F_{xx}F_{xx}-F_{xx}F_{xy} \\ F_{xy}F_{yz}-F_{xz}F_{yy}&F_{yx}F_{zz}-F_{xx}F_{yz}&F_{xx}F_{yy}-F_{xy}F_{yz} \\ \end{array}\right) \tag{16}\]  
Here \(\
abla\) applied to a row vector refers to taking the gradient of each component and storing these component gradients in a matrix as consecutive column vectors. The gradient \(\
abla F\) is parallel to the normal of the surface \(F(x,y,z)=0\).  
Figure 5: Sketch of principal curvature on a surface (**a**) in 3D Euclidean space and (**b**) by top view.  
The following curvature formulas for implicit surfaces appear in [29]:  
Gaussian curvature \(K_{G}\) gives:  
\[K_{G}=\frac{\
abla F*H(F)*\
abla F^{T}}{\left|\
abla F\right|^{4}}=-\frac{\left| \begin{array}{cc}H(F)&\
abla F^{T}\\ \
abla F&0\end{array}\right|}{\left|\
abla F\right|^{4}} \tag{17}\]  
Mean curvature \(K_{M}\) gives:  
\[K_{M}=\frac{\
abla F*H(F)*\
abla F^{T}-\left|\
abla F\right|^{2}\text{{Trace} }(H)}{2\left|\
abla F\right|^{3}}=\frac{-coeff(\lambda)in}{\left|\begin{array} []{cc}H(F)-\lambda I&\
abla F^{T}\\ \
abla F&0\end{array}\right|}{2\left|\
abla F\right|^{3}} \tag{18}\]  
The principal curvatures \(k_{1}\) and \(k_{2}\) can be computed from the mean and Gaussian curvature \(K_{M}\) and \(K_{G}\), respectively, from  
\[k_{1},\ k_{2}=K_{M}\pm\sqrt{K^{2}_{M}-K_{G}} \tag{19}\]  
The detailed derivation for Gaussian and mean curvatures of implicit surfaces can be found in [28].