the terms to give, Substituting these expressions in the identity cos 2 α + sin 2 α = 1 we obtain With the center and the radius. In order to recover the case where b → 0, we consider a Taylor expansion for the exponential up to second order in the exponential in Eq. (40) and up to first order in Eq. (41). The fact that this circle exists in the presence of a drag force is remarkable. We obtained an explicit form for the locus C m com- posed by the set of maxima of all the trajectories of a projectile launched at an initial velocity ⃗ V$_{0}$ , and in the presence of a linear drag force, − mb⃗ v , i.e. C m is the lo- cus of the apexes. In polar coordinates, C m is written in terms of the principal branch of the Lambert W function for negative values. This represents the parameterization of the curve by the polar angle θ$_{m}$ only and gives C m in a closed form and exhibits the deep relationship between the Lambert W function and the linear drag problem. The curvature of C m was calculated for different values of the dimensionless parameter ε ≡ bV$_{0}$/g in two param- eterizations. The first one, the polar parameterization, shows a maximum that slightly departs from the drag- free case in θ = 1 / 2 arctan(4 / 3). A wider exploration of the functional dependence respect to ε is pending due to numerical accuracy in the calculation of the Lambert W function near the limit at x = − 1 /e . In the case of a pa- rameterization using the launch angle α there is not such a restriction. In this case, the curvature was calculated for a wide range of the parameter ε yielding maximum at angle values larger than those corresponding to max- imum range. Comparison with the maximum skewness angle [12] was also done and the difference is larger than the previous one. As an addendum, we demostrate that the synchronous curve, in this case, is a circle as in the drag-free case. and that f ( α ) = sin α ( θ$_{m}$ ) cos α ( θ$_{m}$ ). The first expres- sion is obtainable from the drag-free solutions Eqs. (A1) and (A2), and the second is obtained by setting b → 0 in Eq. (11). Where we used the polar coordinates x$_{m}$ = r$_{m}$ cos θ$_{m}$ and y$_{m}$ = r$_{m}$ sin θ$_{m}$ . The solutions are r$_{m}$ = 0 and The second one is the required form for the ellipse. Taking the squares in both expressions, summing them and arranging terms, we arrive to being ρ ≡ V 2 0 g . With the help of the trigonometric re- lations sin 2 α = 2 sin α cos α and 2 sin 2 = 1 − cos 2 α we transform the upper equations into and Ellipse canonical form or the polar form with the origin considered in one of the focus are standard knowledge. In the present case, however, we require to consider the origin of coordinates located in the bottom of the ellipse, since, in the presence of a drag force, the launching origin is the only invariant point when we change the drag force value. To obtain the ellipse form, we depart from the drag-free solutions at the locus of the apexes, This work was supported by PROMEP 2115/35621. HHS thanks to M. Olivares-Becerril for useful discussions and encouragement. Acknowledgments Appendix A: Polar form of an Ellipse with origin at bottom. Appendix B: Drag-free limit for ˜ r VI. CONCLUSIONS In order to obtain the drag-free limit for the locus C m given in Equation (13), we note that tan θ = (1 / 2) tan α (B1) r$_{m}$ ( θ$_{m}$ ) = 2 ρ sin θ$_{m}$ 1 + 3 sin 2 θ$_{m}$ . (A6) 4 r$_{m}$ ρ ( r$_{m}$ ρ (1 + 3 sin 2 θ$_{m}$ $^{)}$− 2 sin θ$_{m}$ ) = 0 . (A5) sin 2 α = 2 x$_{m}$ ρ , (A3) cos 2 α $^{= 1}$− 4 y$_{m}$ ρ . (A4) x$_{m}$ = ρ sin α cos α, (A1) y$_{m}$ = ρ 2 sin 2 α, (A2) cos α = b V$_{0}$ x $^{1}$− $^{exp(}$− b ) , (37) sin α = b V$_{0}$ y − b 2 $^{(1}$− bt − $^{exp(}$− bt )) $^{1}$− $^{exp(}$− bt ) . (38) x 2 + ( y − y$_{c}$ ( t )) 2 = R $^{2}$( t ) . (39) y$_{c}$ ( t ) = g b $_{2}$(1 − bt − $^{exp(}$− bt )) , (40) R ( t ) = V$_{0}$ b (1 − $^{exp(}$− bt )) , (41) 6