Epicycles vs TalkPro

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.

TalkPro helps you practice putting your thoughts into words. Pick a word, set a timer, speak, then listen back. EXPRESSION PRACTICE Open Expression Practice in the Special tab. Roll through five visible words, tap to pause on the centered word, then tap Next. Select 60 or 120 seconds and review the data-sharing explanation. After consent, use up to five minutes to organize your thoughts or gather material. Start recording when ready and follow the circular timer. Each recording requires its own upload consent. After it is saved on your device, AI transcribes your speech and offers text-based feedback on clarity and flow, with references to what you said. Replay recordings, read transcripts and reports in History, and practice again using the suggestion. AI recognition and feedback can be inaccurate; they do not assess your accent or personal ability. AI expression practice currently supports Chinese and English. MORE WAYS TO PRACTICE Use cue words, guided questions, pyramid exercises and scenarios for interviews, meetings and conversations. A simulated live room offers a camera preview and generated audience activity for private practice, without broadcasting. Browse History and progress trends, set a daily goal, and allow an optional daily practice reminder that you can turn off in Settings. Recordings and long-term history stay on your device and can be exported or deleted. FREE PRACTICE AND TALKPRO PLUS Free users can complete one basic cue-word session each day. Expression Practice and other member features require an active TalkPro Plus subscription. Eligible previous paid purchasers retain their original access. Monthly and annual plans offer the same benefits. Online AI analysis has a separate allowance, explained in the app. Prices and billing periods are shown before purchase. Subscriptions renew automatically and can be managed or canceled in your Apple Account settings. Restore Purchases recovers eligible access. PRIVACY AND CHOICE Local practice needs no TalkPro account or personal AI API key. Audio is uploaded for transcription and analysis only after you consent to that recording; video is never uploaded. AI services connect to Volcano Engine through Cloudflare. See the privacy policy for short-term encrypted caching and allowance records. Optional AI-generated prompts also require consent. English, Simplified Chinese, Japanese, Korean and Thai interfaces and local practice content are available for iPhone and iPad. AI expression practice currently appears in the Chinese and English interfaces. Privacy: https://talkpro-app.pages.dev/privacy/ Terms: https://talkpro-app.pages.dev/terms/ Apple EULA: https://www.apple.com/legal/internet-services/itunes/dev/stdeula/