F.A.S.T vs Epicycles

F.A.S.T (Fancy Archimedean Spiral Tool) Unlock the world of mesmerizing spirals with F.A.S.T – your ultimate tool for creating, exploring, and customizing beautiful mathematical spirals. Whether you're captivated by the elegance of curves or fascinated by the geometry of logarithmic and Archimedean spirals, F.A.S.T offers you endless possibilities for creative discovery. Key Features: - Archimedean & Logarithmic Spirals: Dive into the classic forms of Archimedean spirals or the distinctive beauty of logarithmic (Bernoulli) spirals, each customizable to perfection. - Intuitive Controls: Adjust arc length, separation, rotation, segment count, and even color to create spirals that are uniquely yours. Every control slider is designed to let you shape each aspect with precision, from the spiral's curvature to the color that brings it to life. - Dynamic Colorization: Explore the full hue spectrum with seamless color transitions that allow you to bring your spirals to life in vibrant ways. From soft gradients to intense shades, your creations can be as colorful as you wish. - Expert Mode: Ready to go beyond? Activate Expert Mode and access advanced options that make every tweak feel like an art form. Experiment with rotations, mirrored patterns, and detailed configurations to find the perfect look. - Dark Mode & More: Customize your workspace with Dark Mode, make high-contrast visualizations, and adjust your settings to fit your unique style. F.A.S.T has been built to look great on both iPhone and iPad, adapting seamlessly to your device's orientation and dark/light settings. - Made for All Ages: Ideal for curious minds of all ages, F.A.S.T offers easy-to-use controls that make it engaging for young explorers and a true creative outlet for geometry enthusiasts. Whether you're creating spirals for inspiration, education, or pure enjoyment, F.A.S.T provides a delightful environment to explore the world of curves. For a lighter experience for children and babies, be sure to check out "F.A.S.T Baby" for simpler, touch-friendly spirals. Discover the beauty of mathematics and design. Let F.A.S.T inspire your creativity today!

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.