Let$\ R\ $be the radius of curvature of the plane curve$\ γ$,$\ α\ $be the angle between the constant vector and the current tangent vector of the curve$\ γ$. Find the parametric equation of the curve$\ γ,\ $if $R=\frac{A}{cos\ α}$
I don't know how to approach this assignment.. I haven't encountered this kind of problem at all
From Euler's integral solution for the tangent angle of plane curves we have
$$\theta =\int{\kappa \left( s \right)\,ds;\,\,\,\,\,\,\,\,{d\theta }/{ds=\kappa \left( s \right)}\;}$$
where $\theta$ is the tangent angle and $\kappa=1/R$ is the curvature.
From which the curve is solved with the parametric equations
$$x=\int{\frac{\cos \theta }{\kappa }\,d\theta };\,\,\,\,\,\,\,\,y=\int{\frac{\sin \theta }{\kappa }\,d\theta }$$
or
$$x=\int R\cos \theta \,d\theta;\,\,\,\,\,\,\,\,y=\int R \sin \theta \,d\theta $$
With $R=A/\cos\theta$ we get
$$ x=A\theta+\text{const};\ \ \ \ y=-A\ln(\cos\theta)+\text{const}$$
The problem is completed by specifying $x,y$ at some $\theta_0$.