We created this animation to visualize a fascinating set of parametric equations that produce a wide range of geometric shapes, depending on the value of the variable 𝑘. As k changes, the resulting paths trace out stars, loops, spirals, and more complex patterns. These curves are often related to epicycloids and hypocycloids, which describe the motion of a point attached to a circle rolling around another circle. When k is an integer, you get sharp, symmetric star-like shapes. Non-integer value

Parametric functions describe how two or more quantities change together as they depend on a third variable, called a parameter. Instead of writing one variable directly in terms of another, like y = f(x), we express both x and y in terms of a separate variable t. For example, x = f(t) and y = g(t). As t changes, the pair (x, y) moves along a path, forming a curve. This method is useful because it can represent shapes and motions that are hard to describe with a single equation, such as circles,

Parametric functions describe a curve by expressing its coordinates as separate functions of a third variable, often called a parameter. Instead of writing one variable directly in terms of another, we define both x and y (or more variables) in terms of this parameter, usually t. This allows us to represent complex curves that cannot be described easily by a single equation, such as circles, ellipses, or spirals. As t changes, the pair of values (x(t), y(t)) traces the path of the curve, giving