Epicycles 1.1.4

The Epicycles application serves as a sophisticated tool for the visualization and exploration of complex Fourier series. It enables users to interactively generate and examine epicycles derived from sampled parametric curves or predefined Fourier series components. The app offers a visual representation of complex Fourier series, illustrating how such series can approximate intricate functions through the use of epicycles. This functionality facilitates a deeper understanding of Fourier series properties, the concept of epicycles, and the visualization of complex functions in an engaging and interactive manner.

• Supports visualization of Fourier series corresponding to 2D functions parameterized over time, including built-in examples, user-drawn curves, and custom frequency components editable via the terms editor.
• Custom Fourier terms can be saved by exporting them into 'epi' files within the device's file system and restored later through import functionality.
• Cyclic animations are exportable as GIF files or static snapshots in PNG format, with selectable resolutions of 480x480, 720x720, or 1080x1080 pixels.
• The primary interface features several interactive controls: a Graphic Menu, a Time Slider (t), a Fourier series term count slider (N), and a Current Function Menu.

Graphic Menu:

Utilize the menu bar within the graphic view to toggle visibility of specific visual elements that contribute to the Fourier series visualization:

• Circle: Blue circles representing the paths traced by epicycles, illustrating Euler’s formula for complex exponential terms as circles with radius r that trace (r cos(n t), r sin(n t)) — reflecting their dependence on frequency n over the period 2π.
• Eye: Green circle indicating the current value of the Fourier series at a specific point in time.
• Lightning: Red line segments depicting the radii connecting successive epicycle centers, culminating at the current Fourier series value.
• Pencil: Orange path tracing the actual 2D function over time.
• Star: Black path illustrating the Fourier series approximation of the current 2D function.

The menu also includes additional options:

• Magnify: Collapses ancillary views to maximize display space for detailed visualization.
• Share: Exports current graphics as animated GIFs or static PNG snapshots to the Photos library.
• Play: Animates the visualization by advancing the current time parameter periodically.
• Info: Provides access to online resources related to Fourier series, epicycles, and Euler’s formula for further study.

Time Slider (t):

This control allows users to adjust the temporal parameter to observe the dynamic behavior of all visual elements within the interval [-π, π], thus examining how the functions evolve over their period.

Number of Fourier Series Terms Slider (N):

This slider modulates how many terms are included in the Fourier approximation. The partial sum encompasses frequency components from -N to N. The maximum allowable value is 100. Users are advised to utilize the wand icon to select an N that aligns with Nyquist sampling principles, ensuring reproducibility of highest frequencies based on available sample points. For built-in functions with fixed sampling, this limit is implicit. When drawing custom curves, sampling count varies and is displayed accordingly. When editing terms in the custom Fourier series editor, frequency components are restricted between -20 and 20, with sample counts sufficient for accurate reconstruction within this range. In such cases, N automatically adjusts to match the highest absolute frequency present in the terms list.

Current Function Menu:

This segmented control allows selection among various pre-defined 2D parametric curves. Selecting '?' prompts users to either draw a new curve in the Draw tab or edit custom Fourier terms in the Term tab. In editing mode, the app numerically computes the Fourier series by sampling the summation of selected terms, enabling detailed analysis of custom configurations.