Algebraic curves are mathematical objects defined by polynomial equations in two variables, typically represented as f(x, y) = 0 . These curves can take various shapes, such as lines, circles, and more complex forms like ellipses and hyperbolas. The degree of the polynomial determines the curve's complexity and properties. These curves are studied in algebraic geometry, a branch of mathematics that explores the relationships between algebraic equations and geometric shapes. Algebraic curves can also be classified into different types, such as smooth curves and singular curves, based on their characteristics and behavior at specific points.