Epicycles vs Tinalp

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.

TINALP is a revolutionary solution for the training and education 4.0 that takes advantage of Mixed Reality and latest cloud and network technologies to create "Augmented Classroom". An Augmented Classroom is an advanced hybrid learning spaces where students and professors can participate from everywhere and share traditional 2D slides and innovative 3D contents such as 3D models and volumetric videos, all in real time and perfectly synchronized. Thanks to a simple but powerful user interface based on gesture control, voice recognition and full hand tracking, the interaction between trainers and trainees is seamless and as natural as being in a real classroom. Travel costs and safety risks can be minimized by leveraging TINALP ability to teleport people and data anywhere, at any time. The main features are: - Professors/trainers can create structured lectures using a web portal like Keynote/PowerPoint (with images, videos, 3d models, 3d videos, ...) - Professors/trainers can create quizzes, evaluation tests and other activities that can be done in a shared way by students collecting data in reports. - Professors/trainers can create live lectures with augmented classes at any time, with students in the same physical space or remotely - Students can participate at live lectures and, raising their hand, ask to intervene. - Students can r download the training materials and review it offline (if the professor enables it).