In the following app you can explore an "all-in-one" construction of the envelopes of ellipses, hyperbolas and circles.[br][br]Set the position of the focus [math]F[/math] by dragging it, then start the animation, or move point [math]A[/math] along the circle, that has center at [math]O[/math] and radius [math]r[/math].[br][br]The position of [math]F[/math] with respect to the circle determines the generated envelope as follows:[br][list][br][*] [math]F[/math] [color=#1e84cc][i]inside the circle [/i][/color]→ [color=#1e84cc][i]ellipse [/i][/color]with foci [math]O[/math], [math]F[/math] and semi-major axis [math]\frac{r}{2}[/math][br][/*][*] [math]F[/math] [color=#1e84cc][i]coincident with[/i][/color] [math]O[/math] → [color=#1e84cc][i]circle [/i][/color]with center [math]F\equiv O[/math] and radius [math]\frac{r}{2}[/math][br][/*][*] [math]F[/math] [color=#1e84cc][i]outside the circle [/i][/color]→ [color=#1e84cc][i]hyperbola [/i][/color]with foci [math]F[/math], [math]O[/math] e semi-major axis [math]\frac{r}{2}[/math][br][/*][/list]