What does (x^2+y^2-1)^3 and (xy)^2 mean in equation of Astroid(not asteroid) which is (x^2+y^2-1)^3+27(xy)^2?

Question

How is the behaviour of above two curves? Does one curve define the interior and other the exterior and together they constitute ASTROID? No restictions on values of x or y?

Answer 1

The surfaces decribed by the following equations,

z = (x² +y² -1)³
z = -27(xy)²

have 4-fold symmetry. The intersection of the 2 surfaces is the astroid. If the coefficient of 27 was 1 instead, the intersection would be closer to a square. If it was 1000 instead of 27, the intersection would be a skinny 4-pointed star. Neither one of those curves is a true hypocycloid, which is formed by a point on a circle rolling inside of a larger one. For a 4-cusped hypocycloid, the rolling circle has to be exactly 1/4 the diameter of the larger one, or else the point on it will not come back to the starting point when it touches the larger circle for the 4th time. So, no, you cannot create other hypocycloids by changing the value of the coefficient 27. And there is no connection between those two terms of the equation with the circles that form the hypocycloid, separately.

The equation, in fact, satisfies the parametric equations:

x = (Cos(θ))³
y = (Sin(θ))³

so that (x² + y² -1)³ + 27(xy)² = 0, which is actually a trigonometric identity.

Answer 2

x^2 + y^2 - xy^2 = -15 differentiating implicitly 2x + 2y y' - 2xyy' - y^2 = 0 => y ' (2y - 2xy) = y^2 - 2x y ' = (y^2 - 2x) /(2y - 2xy) for horizontal tangent, y ' = 0 => y^2 - 2x = 0 => y^2 = 2x replace y^2 = 2x in the eqn of curve x^2 + 2x - 2x^2 = -15 => x^2 - 2x - 15 = 0 => (x - 5)(x + 3) = 0 x = 5 and -3 y^2 = 10 and - 6 y = ±?10 (x,y) = (5, -?10) and (5, ?10)