Plot out a graph of the Kappa curve as it pertains to your particular problem or situation. Since you will be converting slope to Kappa, you will need to plot out the slope of the tangent so that it falls within the parameters of your equation. The slope of the tangent or the tangential angle is defined by the equation in which the radius of curvature is equal to the the sine and cosine of the radius (theta) is multiplied by its negative tangent to the minus one power.
Solve for theta using the above tangential radius equation.
Plug theta into the equation for the curvature of the Kappa curve. This equation is a fairly complex one that requires you to take three times the cosecant of theta and square it. Subtract "1" from that number.
Calculate the cotangent of theta squared plus the cosecant of theta to the fourth power. Take this number to the 3/2 power and divide it into the original calculated value for theta to obtain the curvature of the Kappa curve.