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

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The butterfly curve is a stunning example of how mathematics can produce natural, almost lifelike beauty through simple equations. It’s defined by parametric functions involving exponential and trigonometric components that combine to create a shape resembling a butterfly with delicate wings and a slender body. As the parameter moves through its range, the curve traces elegant, symmetrical loops that expand and contract smoothly, giving it the visual impression of fluttering motion. Despite its

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