Graphing Lemniscates

Lemniscate Graph Identification
📌 Lemniscates are the third type of polar graph discussed, characterized by being relatively straightforward to graph compared to rose curves.
📐 The general form for a lemniscate involves $r^2$ and always includes $2\theta$ (not $n\theta$), such as $r^2 = a^2 \cos(2\theta)$ or $r^2 = a^2 \sin(2\theta)$.
📏 The constant $a$ represents the distance from the pole (origin) to the tip of each leaf.

Determining Leaf Orientation
🔺 If the equation is $r^2 = \text{positive } a^2 \cos(2\theta)$, both leaves lie along the x-axis.
🔻 If the equation is $r^2 = \text{negative } a^2 \cos(2\theta)$, both leaves lie along the y-axis.
⬆️ If the equation is $r^2 = \text{positive } a^2 \sin(2\theta)$, one loop lies in Quadrant 1 and the other in Quadrant 3.
⬅️ If the equation is $r^2 = \text{negative } a^2 \sin(2\theta)$, one loop lies in Quadrant 2 and the other in Quadrant 4.

Example Applications
📐 For $r^2 = 9 \cos(2\theta)$, $a^2=9$, so $a=3$, meaning the loops lie on the x-axis, each extending a distance of 3 from the pole.
📐 For $r^2 = \sin(2\theta)$, $a^2=1$, so $a=1$, and since it is positive sine, one loop is in Quadrant 1 and the other in Quadrant 3.

Key Points & Insights
➡️ Identify a lemniscate by the presence of $r^2$ and $2\theta$ in the polar equation.
➡️ Always take the square root of the coefficient of the trigonometric term to find $a$, which defines the maximum distance of the leaves from the pole.
➡️ Lemniscates always have exactly two loops or leaves, simplifying the overall graphing process.

📸 Video summarized with SummaryTube.com on Dec 04, 2025, 06:54 UTC