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In this post, we extend the idea of cycloidal curves to a more complex path: the parabola. Using the tangent angle, arc length, and no-slip condition, we derive the parametric description of the tracing point and show how these formulas can be translated into Python code for accurate Manim visualization.

The purpose of this post is to systematically derive the corresponding parametric equations and implement them in Python for visualization using Manim. The goal is not only to understand these curves analytically but also to see them come alive through animation.