SEIRAKU vs Math Surface

SEIRAKU is an interactive educational app for exploring celestial mechanics through beautiful, real-time simulations. Part of the popular science learning series including SUUGAKU (math), BUTURAKU (physics), DENRAKU (electricity), HARAKU (waves), and NETURAKU (thermodynamics). This installment focuses on celestial mechanics — ten simulations that let you experience the elegant laws governing our universe. 【10 Simulations】 ▶ Orbital Motion Simulate planets orbiting a central body under gravity. Freely adjust mass, speed, and the number of planets. Stable Verlet (leapfrog) integration produces beautiful orbital trails in real time. ▶ Universal Gravitation Visualize the gravitational field of multiple massive bodies using vector arrows. Tap anywhere to add test particles and watch how they move through the field. Experience Newton's law of gravitation. ▶ Three-Body Problem When three stars pull on one another, the motion turns wildly complex — most starting conditions diverge completely, while a few special ones trace beautiful periodic orbits. Choose between Stable Triangle, Figure-8 Orbit, Collinear, and Unstable presets to feel how gravity and velocity intertwine. ▶ Tidal Force Visualize the tidal deformation that the Moon's gravity exerts on Earth. Observe how tidal force vectors and ocean bulges (3cos²θ − 1) change in real time as the Moon orbits. Adjust distance and mass to explore the concept of the Roche limit. ▶ Kepler's Laws Experience Johannes Kepler's three laws of planetary motion in a single screen. Adjust eccentricity and semi-major axis freely, and see the equal-area law (2nd law) visualized through swept sectors. Compare the speed difference between perihelion and aphelion. ▶ Gravity Slingshot A spacecraft uses a planet's gravity to change speed and direction — the technique used by Voyager and Hayabusa. Adjust planet mass, launch speed, and approach offset to see how the trajectory, speed gain, and deflection angle change. ▶ Hohmann Transfer Visualize the classical minimum-fuel two-burn transfer between two circular orbits. The first burn (Δv₁) places the craft on a half-elliptical transfer; the second (Δv₂) circularizes onto the outer orbit. Watch Δv, the transfer half-period, and eccentricity update live — the same maneuver used in real interplanetary missions. ▶ Exoplanet Transit Simulate the dimming when a planet passes in front of its host star, along with the resulting light curve. Tune the planet/star radius ratio and impact parameter to see how transit depth and shape change, with limb-darkening producing the characteristic U-shaped curve — the technique used by NASA's Kepler and TESS missions. ▶ Gravitational Lens Visualize how light passing near a massive object (a galaxy or black hole) is bent by the curvature of spacetime. Change the lens mass and source position to watch parallel rays focus and form the Einstein ring when the source lines up directly behind it. Deflection uses the general-relativistic value (twice the Newtonian result). ▶ Axial Precession Visualize, in pseudo-3D, how the spin axis of a rapidly rotating body slowly sweeps out a cone under gravitational torque. Adjust tilt, spin rate, and torque to observe the cone, the changing pole star, and the precession period — the same phenomenon that makes Earth's axis complete one circuit in about 25,800 years. 【Features】 • Real-time parameter adjustment in all simulations • Physics formulas and live numerical readouts • Dark mode / Light mode support • Intuitive, beautiful UI • Japanese / English supported • Minimal ads — banner only at the bottom 【Pricing & Purchases】 • This app is free to use. • There are no in-app purchases. • It is supported by advertising (Google AdMob banner ads). 【Who is this for?】 • Students studying celestial mechanics in physics class • Anyone who wants to intuitively understand the laws of the universe • Teachers looking for interactive teaching materials • Everyone curious about astronomy and space

3D surfaces, 2D curves, the complex plane, probability — all in one app. Type a formula and watch it come alive. Perfect for study, exploration, or just enjoying the beauty of math. Description: MathSurface turns equations into shapes you can hold, spin, and slice. It's a math playground for iPhone and iPad — designed for students, curious minds, and anyone who learns better by seeing. ■ Features • 3D surface plots: visualize z = f(x, y) with smooth rotation and pinch-to-zoom • 2D function plots: explicit y = f(x) or implicit 0 = F(x, y) curves • Complex plane: loci such as |z − α| = r, arithmetic operations, powers, and n-th roots of unity • Probability distributions: binomial and normal, controlled by live sliders • Surface of revolution: spin any 2D curve around the x- or y-axis (rendered with SceneKit) • Cross sections: slice any 3D surface with a plane and see the resulting curve • Compare mode: overlay two functions in the same plot • Preset gallery: paraboloid, monkey saddle, Mexican hat, lemniscate, and many more • Favorites: save your own custom expressions for instant recall ■ Who it's for • High school and university students studying calculus, analysis, or probability • Teachers looking for live visual aids • Anyone curious about how a formula actually looks • People who just enjoy beautiful math A custom Liquid Glass–style formula keyboard makes entering sin, cos, √, π, and friends quick and natural. The dark neon UI keeps the math front and center.