• E. Torricelli, who was the last Galileo's secretary, suggested a purely geometrical method to find the envelope in his De motu Proiectorum. He also coined the term parabola of safety'. Apparently it was the first example of computation of an envelope. The method is briefly described in this note.

• Another approach is to launch identical missiles with the same velocity at all possible angles simultaneously. At time $t$, their positions describe a circle $$x^2+\left(y-\frac{t^2}{2}\right)^2=(vt)^2.$$ The latter equation has a unique solution in $t$ provided $(x,y)$ belongs to the parabola $$y=\frac{v^2}{2}-\frac{x^2}{2v^2}.$$

• In the case of missiles moving in a Kepler field (with the attractive potential $\sim -1/r$), the envelope of elliptic trajectories is indeed an ellipse. A web search gave the nice short article which contains several elementary geometric proofs of this and related results.

Edit. A free version of J.-M. Richard article can be found here.

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• E. Torricelli, who was the last Galileo's secretary, suggested a purely geometrical method to find the envelope in his De motu Proiectorum. He also coined the term parabola of safety'. Apparently it was the first example of computation of an envelope. The method is briefly described in this note.

• Another approach is to shoot launch identical missiles with the same velocity at all possible angles simultaneously. At time $t$, their positions describe a circle $$x^2+\left(y-\frac{t^2}{2}\right)^2=(vt)^2.$$ The latter equation has a unique solution in $t$ provided $(x,y)$ belongs to the parabola $$y=\frac{v^2}{2}-\frac{x^2}{2v^2}.$$

• In the case of missiles moving in a Kepler field (with the attractive potential $\sim -1/r$), the envelope of elliptic trajectories is indeed an ellipse. A web search gave the nice short article which contains several elementary geometric proofs of this and related results.

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• E. Torricelli, who was the last Galileo's secretary, suggested a purely geometrical method to find the envelope in his De motu Proiectorum. He also coined the term `parabola of safety'. Apparently it was the first example of computation of an envelope. The method is briefly described in this note.

• Another approach is to shoot identical missiles with the same velocity at all possible angles simultaneously. At time $t$, their positions describe a circle $$x^2+\left(y-\frac{t^2}{2}\right)^2=(vt)^2.$$ The latter equation has a unique solution in $t$ provided $(x,y)$ belongs to the parabola $$y=\frac{v^2}{2}-\frac{x^2}{2v^2}.$$

• In the case of missiles moving in a Kepler field (with the attractive potential $\sim -1/r$), the envelope of elliptic trajectories is indeed an ellipse. A web search gave the nice short article which contains several elementary geometric proofs of this and related results.

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