we all know about circles from our 5th std or so.. and i'm sure it was a really great pleasure meeting the ELLIPSE in 10th std. Circle was all we knew... we even wrote logo programs for drawing circles (both filled and non filled) and now suddenly this new thing seems more complicated and more general than the circle. Just when i thought it was all.. i meet SUPER ELLIPSE. The one ellipse which can transform into a square or a circle depending on the values of its' parameters.
Circle Equation: X^2 + Y^2 = 1
Ellipse Equation: (X/a)^2 + (Y/b)^2 = 1
Super Ellipse Equation: (X/a)^n + (Y/b)^n = 1
Since we already know about circles and ellipses lets' just look at super ellipses. Here are the images (from answers.com) that best describe the equation.
The super ellipse's equation can be further generalized
as the following: (X/a)^m + (Y/b)^n = 1; m,n>0
Hope to find out more such beautiful geometric equations. :)