Exact boundary definitions used in the explorer
The canonical Mathieu equation is rendered in the standard form below. All regions on this page are generated from the exact characteristic-value boundaries, not sketch approximations.
$$\frac{d^2u}{d\xi^2} + \left[a - 2q\cos(2\xi)\right]u = 0$$
$$\begin{aligned}
X\text{-axis stable regions (first 3):}\quad
R_1^X &: a_0(q) \le a \le b_1(q) \\
R_2^X &: a_1(q) \le a \le b_2(q) \\
R_3^X &: a_2(q) \le a \le b_3(q)
\end{aligned}$$
$$\begin{aligned}
\text{Map Y into the x-frame using } a_y=-a_x,\; q_y=-q_x.\quad
R_n^{Y\rightarrow x}: -U_n(q) \le a_x \le -L_n(q)
\end{aligned}$$
$$\begin{aligned}
\text{Quadrupole first-region overlap (x-frame):}\quad
0 \le a \le \min\!\left[-a_0(q),\,b_1(q)\right]
\end{aligned}$$
The explorer precomputes these exact curves over a dense q-grid and renders them interactively in the browser (Netlify-friendly, static deployment).
Interactive Diagram
X regions are shown in positive
a; Y-axis stability is mapped into the same x-frame to make the sign symmetry and first-region overlap explicit.
Interpretation notes
- The first quadrupole mass-filter operating region is the overlap of X and Y first-region stability after mapping the Y-axis equation into the x-frame.
- Higher Mathieu stable regions are plotted for completeness and intuition, but conventional quadrupole mass filters typically operate using the first-region overlap.
- The scan-line slope slider demonstrates how the line intersects exact boundaries; near the apex slope, the line approaches the familiar first-region mass-filter operating condition.