1: Converting intrinsic form [s=f(w)] to cartesian form [y=g(x)]. Differentiate to give (ds/dw)=f'(w) so ds=f'(w) dw
(dy/ds)=sinw ∫1 dy = ∫sinw ds y=∫sinw.f'(w) dw Integrate to give a relationship between y and w. Remember not to neglect the arbitrary constant.
(dx/ds)=cosw ∫1 dx = ∫cosw ds x=∫cosw.f'(w) dw Integrate to give a relationship between x and w. Remember not to neglect the arbitrary constant.
Eliminate the parameter 'w' to give the cartesian relationship between y and x. Typically you'll use a trigonometric identity or make w the subject in the two equations to eliminate w.
2: Converting cartesian form [y=f(x)] to intrinsic form [s=g(w)] y=f(x) so (dy/dx)=f'(x)=tanw s=∫[1+(dy/dx)^2]^0.5 dx with lower limit a(where arc length is measured from) and upper limit x. s=∫[1+(f(x))^2]^0.5 dx with lower limit a and upper limit x. Perform the integration to find the relation s=h(x) and eliminate x using (dy/dx)=tanw, to form a relation involving only s and w. Tou can convert from x=f(y) or parametric equations x=f(t), y=g(t) in a similar manner using the appropriate arc length formula.