(x2+y2-1)3-x2y3=0 Meaning

4 min read Jun 17, 2024

Unraveling the Equation: (x² + y² - 1)³ - x²y³ = 0

The equation (x² + y² - 1)³ - x²y³ = 0, while seemingly complex, holds a hidden beauty in its representation. Let's break it down to understand its meaning and the fascinating shapes it can create.

Unveiling the Heart: A Family of Curves

This equation doesn't define a single curve, but rather a family of curves. To understand this, we can rewrite the equation as:

(x² + y² - 1)³ = x²y³

This emphasizes that the left-hand side, which represents a cubic function in x and y, is equal to the right-hand side, a product of x² and y³. This equality holds true for a range of x and y values, leading to various curves that fulfill this relationship.

Visualizing the Equation: The Astroid's Tale

One of the most interesting curves emerging from this equation is the astroid. This star-shaped curve can be visualized by considering the special case where:

y = 1: In this case, the equation simplifies to (x² + 1 - 1)³ - x² = 0, which reduces to x⁶ = x² and finally x⁴ = 1. The solutions to this equation are x = ±1, which, along with y = 1, define two points on the astroid.
x = 1: Similarly, substituting x = 1 into the original equation leads to (1 + y² - 1)³ - y³ = 0, simplifying to y⁶ = y³ and finally y⁴ = 1. The solutions are y = ±1, defining two more points on the astroid.

These four points, along with the symmetry of the equation, hint at the shape of the astroid. It's a closed, four-pointed star-like curve, with its vertices touching the x and y axes at ±1.

Beyond the Astroid: A Universe of Curves

While the astroid is a prominent example, the equation (x² + y² - 1)³ - x²y³ = 0 encompasses a vast array of curves. The specific shapes are influenced by the relationship between x² and y³ within the equation.

The Beauty of Complexity

The equation (x² + y² - 1)³ - x²y³ = 0 is a testament to the power of mathematical equations to represent complex and beautiful geometric shapes. The astroid is just one fascinating example, and exploring its family of curves opens a window into the intricate world of algebraic geometry.