This construction creates a Cassini Oval, the set of all points P(x, y) such that the product of the distances from P to two fixed points A and B is constant.[br][br]By adjusting the slider k, students can visualize how the shape of the Cassini Oval changes — from two loops, to a figure-eight (lemniscate), and finally to a single oval as k increases[br][br][b]Drag Test[/b][list][*]Drag points A and B — the curve should update automatically, maintaining its defining property.[br][/*][*]Move the slider k — observe how the Cassini Oval morphs in real time.[br][/*][*]Verify symmetry about the x-axis.[br][/*][/list][br]What Students Should Observe[list][*]When k is small, the Cassini Oval splits into two separate loops.[br][/*][*]As k increases to a critical value (around the half distance between AAA and BBB), the shape becomes a lemniscate of Bernoulli — a figure-eight curve.[br][/*][*]For larger k, the curve merges into one connected oval, resembling an ellipse.[br][/*][/list]
What happens when the foci A and B move farther apart?
Increasing the distance between AAA and BBB makes the curve wider and can cause it to split into two distinct parts.
How does changing k affect the shape’s symmetry and connectedness?
[*] - When k < AB/2 [b][/b] → two [b]separate loops[/b] appear, one around each focus.[br][/*][*] - When k = AB/2 → the curve becomes a [b]lemniscate of Bernoulli[/b], a perfect [b]figure-eight[/b] centered at the midpoint between AAA and BBB.[br][/*][*] - When k > AB/2 [b][/b] → the two loops [b]merge into a single connected oval[/b], symmetric about both the x- and y-axes.[br][/*]