Epicycles1.1.4

The Epicycles app is a tool for exploring the visualization of complex Fourier series. It allows users to interactively create and visualize epicycles based on sampled parametric curves or predefined Fourier series terms. The app provides a visual representation of the complex Fourier series and demonstrates the approximation of complex-valued functions using epicycles. It can be used to study the properties of Fourier series, understand the concept of epicycles, and explore the visualization of complex functions in a fun and interactive way.

. Visualizes Fourier series of 2D functions parameterized by time that are built-in, your own drawn 2D curves, or custom frequency components using the terms editor.

. Custom terms can saved by exporting them to 'epi' documents in Files, and then later restored by importing them.

. Cyclic animations can be saved to GIFs or snapshots saved to PNGs in the Photos library, with sizes 480x480, 720x720 or 1080x1080.

. The main view consists of several functional items: Graphic Menu, Time Slider (t), Number of Fourier series terms slider (N), Current Function Menu.

Graphic Menu:

In the main view use the menubar in the graphic view to select options to hide or show the following graphic elements of the Fourier series visualization:

. Circle: The blue circles that are the paths traced by the epicycles. These illustrate Euler’s formula for the complex valued terms of the Fourier series, expressing each complex exponential term as a pair (r cos(n t), r sin(n t)) that trace a circle of radius r, n times as time t varies in the period of length 2π.

. Eye: The green circle that represents the value of the Fourier series at the current time.

. Lightning: The red line segment path that consists of the joined radii of the epicycles circles, from the origin to the value of the Fourier series at the current time.

. Pencil: The orange path that traces the current 2D function.

. Star: The black path that traces the Fourier series 2D approximation of the current 2D function.

Additionally in the menubar:

. Magnify: Hide some views to make room for the expanded display the graphic.

. Share: Save the graphic as an animated GIF or snapshot PNG in Photos library.

. Play: Animate the graphic by periodically advancing the current time.

. Info: Display internet resources conveniently in the app about Fourier series, epicycles and Euler’s formula.

Time Slider (t):

Adjust the time slider to see the state of all the graphic elements at any time within the time period [-π,π] on which the current 2D function is defined.

Number of Fourier series terms slider (N):

Adjust the number of terms included in the Fourier series approximation to the current 2D function. Frequency components in a partial Fourier series range from -N to N. The maximum value is limited to 100.

As a guide tap the wand icon to set the number of terms to a value whose corresponding highest frequency can theoretically be reproduced with the given number of samples, based on the concept of Nyquist frequency with uniform sampling. For the built-in sample functions that sample count is fixed. The number of samples of your own drawn curve is variable, and displayed in the drawing view.

The custom Fourier series using the term editor has known frequency components, limited to the range -20 to 20. The number of samples generated is sufficient for any selection in that range. Therefore in this case the wand sets N to the highest absolute frequency value of the terms.

Current Function Menu:

Use the segmented control to select from a variety of built-in 2D parametric curves or select the `?` item. Then you can either draw a 2D curve in the Draw tab view, or edit custom Fourier series terms in the Term tab view. In the latter case the app will numerically generate the Fourier series of a Fourier series, by sampling the summation of the series terms.

Version History Maintenance update to prevent some views from not displaying properly when device is set to dark mode. 1.1.4 11/23/2023 Improves the appearance of curves by rounding rather than mitering the join of lines that compose them, except when drawing the fading trail curve.

Previous update:

In the OptionsView the path for the Fourier series can be drawn with an alpha gradient, creating the effect of a fading trail by specifying the length of the trail. Access the OptionsView view by tapping on the sliders button in the toolbar of the Graphics view. Then select the tab view labelled Σ, and set the Trail Length. A length of 0 turns off the effect. The effect is demonstrated in the new animated GIF that appears in the startup view. 1.1.3 07/18/2023 In the OptionsView the path for the Fourier series can be drawn with an alpha gradient, creating the effect of a fading trail by specifying the length of the trail. Access the OptionsView view by tapping on the sliders button in the toolbar of the Graphics view. Then select the tab view labelled Σ, and set the Trail Length. A length of 0 turns off the effect. The effect is demonstrated in the new animated GIF that appears in the startup view. 1.1.2 07/18/2023 • Adjust the line widths and colors of the different paths for the epicycles graphic. Apply from a new options view, accessed from the toolbar in the graphic view (sliders icon).

• Bug fix: During PNG or GIF export the selected term colors of the custom Fourier series, if any, were being applied when the terms view was not selected. The result was that the epicycle circles may have different colors than expected. 1.1.1 07/12/2023 • For functions defined using a custom Fourier series, the PNG and GIF generation epicycle circles will be drawn in their respective colors set in the Terms view.

• A graphical angle indicator was added to the custom Fourier series terms editor, as a visual aid for selecting the phase (angle) of the coefficients.

• New custom Fourier series terms are created with random properties.

• All custom Fourier series terms can be deleted at once, and a feature has been added that will create an random number of terms with random color, amplitude, phase, component frequency.

• Deleting custom Fourier series terms ask for confirm. New `Remove all` button. When there are no terms there is a new button to add random number of new terms with random properties.

• Option to set the background white or transparent is available for PNG and GIF images.

• The direction of time in animations (and GIFs) has been changed from negative to positive.

• The time (t) slider has been fixed, the right end was not quite reaching π.

• The Dial control for time in the expanded view has a 0 button, buttons to set time to -π/2 and π/2.

• Document icons were flipped in iOS

• The splash screen only displays once per launch, rather than for each new window.

• A reference to a web page describing how the app works is included in the help view (Access with the 'info' button of the Graphic Menu) 1.1 07/10/2023 1.0 06/30/2023