Equation of Egg Shaped Curve of the Actual Egg Author: Nobuo Yamamoto Updated: Apr. 29, 2007 (with multiple revisions up to Oct. 27, 2022) Lab: TDCC Laboratory --- Overview This page presents an equation for an egg-shaped curve closely resembling an actual chicken egg, improving upon existing oval curves such as Cassini ovals. The original work started with a 2006 newsletter publication, followed by progressive versions of egg-shaped curve equations (Egg Shaped Curve I to VII) released between 2007 and 2011. The study addresses both 2D curves and 3D egg-shaped solids, with detailed derivations, visualizations, and computational programs provided. --- Key Contributions Basic Derivation of Egg Shaped Curve The shape is derived based on a point \( P \) moving in the xy-plane relative to a moving point \( Q \) on the x-axis. \( Q \) moves cosinusoidally with respect to an angle \( \theta \), and \( PQ \) length also varies cosinusoidally. Under certain assumptions, these relations yield a fourth-order algebraic equation defining the egg-shaped curve. The final standard form (Eq. 10) is: \[ (x-b)^2 + y^2 = a x \sqrt{x} \] where \( a, b \) are constants controlling the shape. Graphical comparisons show this curve closely matches actual egg shapes, especially with careful choice of constants (\( b = 0.7a \) gives the closest shape). The gradient at \( x = 0 \) is infinite, ensuring a smooth rounded tip rather than pointed. Numerical Programs and Visualizations C++ programs are provided for: Generating coordinate data for these curves. Calculating minor axis, circumference (one round length), and area. Plotting for Excel and other graphical tools. Several visual figures display the variations across parameter ranges and comparison with real eggs, including an animation created by Mr. Akira Nagashima. Volume and Surface Area of 3D Egg Solids Extending the 2D curve into 3D by revolving around the x-axis gives: \[ \left\{(x-c)^2 + y^2 + z^2\right\}^2 = x\left[(x-c)^2 + y^2 + z^2\right]^{1/2} b - a x (x-c)^2 \] Volume \( V \) and surface area \( S \) formulas and calculations are presented, including special cases like spheres. Results tabulated for different \( b \) values with respect to constants, showing volume and surface area decrease as \( b/a \) deviates from 0. Using Ellipsoid as Base Figure Replacing circle with an ellipse as the base yields more diverse egg-like shapes. The corresponding equations and resulting curves (e.g., spreading sideways or lengthwise) are analyzed. Volume formulas for revolving such ellipsoidal shapes are provided. Extensions Beyond Defined Parameter Regions For \( b < 0 \) and \( a > 0 \), shapes resembling sea urchins appear. For \( b > a > 0 \), shapes similar to children's fish drawings appear. Introducing an additional scaling constant \( c \) can modify these shapes back towards egg-like forms. Generalization to Pear-Shaped Curves Extending the base equation by adding more terms, inspired by Dr. Hiroyasu Okuyama, allows generation of pear-shaped curves analytically. New equation has up to 6th and 8th degree terms but retains an analytic solution. Necessary conditions ensuring closed curves are discussed and visualized. Shows a continuous family from egg to pear shapes. 7 & 8. Higher Order Equations Additional extensions to 8th order polynomial equations for curve definition. Several new fascinating shapes produced (including spade-like curves). Analytical solutions remain possible, with conditions on parameters for closed shapes. Various visual examples confirm the versatility of these formulations. --- Equations and Constants The standard egg-shaped curve is controlled primarily by parameters \( a \) and \( b \). Displacement \( c \) and other constants (e.g., \( d, e, f \)) appear in generalized and higher-order forms. -