GraphMath vs Epicycles

This graphing calculator is user friendly and powerful.  It is fully compatible with iOS 14 and 15, and mostly compatible with iOS 13 and Mac OS. Cartesian 3D Z = F (X, Y) functions:  ——————————————————— Graphs can be:  1. Drawn as surface, lines, or dots. 2. Drawn with or without perspective, and with different light and reflection settings. 3. Tilted up or down. 4. Rotated to any position. 5. Drawing is animated by either X, Y, or Z. Table of all points can be sorted by either X or Y. Cylindrical 3D Z = F (Θ, R) functions:  ——————————————————— Graphs can be:  1. Drawn as surface, lines, or dots. 2. Drawn with or without perspective, and with different light and reflection settings. 3. Tilted up or down. 4. Rotated to any position. 5. Drawing is animated by Θ, R, or Z. Table of all points can be sorted by either Θ or R. Spherical 3D R = F (Θ, Φ) functions:  ——————————————————— Graphs can be:  1. Drawn as surface, lines, or dots. 2. Drawn with or without perspective, and with different light and reflection settings. 3. Tilted up or down. 4. Rotated to any position. 5. Drawing is animated by either Θ or Φ. Table of all points can be sorted by Θ, Φ, or R. Cartesian 2D Y = F (X) functions:  ———————————————— Graph displays the following curves: 1. One or two Y = F (X) functions. 2. Derivative of the first function. 3. Integral of the first function as a shade and a curve.  Graph features:  1. Drawing is animated by X. 2. Clicking on the curves shows curve values, and finds closest point in the table. Tables: 1.  All function, integral, and derivative values. 2.  Zero intersections of first function. 3.  Intersections between first and second functions. Polar 2D R = F (Θ) functions: ———————————————  Graph displays the following curves: 1. One or two R = F (Θ) functions. 2. Integral of the first function as a shade and a curve. Graph features:  1. Drawing is animated by Θ.  2. Clicking on the curves shows curve values, and finds closest point in the table. Tables: 1.  Table of all function and integral values  2.  Table of zero intersections of first function. 3.  Table of intersections between first and second functions. Parametric 2D X = F1 (T), Y = F2 (T) functions: —————————————————————————  Graph features:  1. Drawing is animated by T.  2. Clicking on the curves shows T, X, and Y values, and finds them in the table. Table of all points. F (X, Y) = 0  equations: ———————————— Graph displays solutions as points. Clicking on the points shows X, and Y values, and finds them in the table. Table of all points. Common features: —————————— 1. List of previous entries. 2. List of preset entries. 3. Undo and redo buttons for formula entries. 4. Automatic correction of errors in function limits or steps are offered. 5. All function points can be saved to clipboard in a table format. 6. All function points can be displayed on corresponding graphs. 7. Control colors can be changed to black and white. 8. Link to Vectors and Planes 3D Geometry Guide App. 9. Link to Matrix Calculator Step by Step App. Common features of all images:  1. Animation frames can be frozen and moved back or forward. 2. Animation speed can be adjusted. 3. Zoom. 4. Scroll. 5. Save to Clipboard. 6. Save to Photos 7. Color picker. New functions and further UI improvements for our apps are coming! Developers hope to get feedback at graphmath@aol.com.

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.