Geometric Visualization of Parametric Curves in 2D and 3D

This example demonstrates how to build and visualize parametric curves in both 2D and 3D using a parameter-driven construction. It highlights key ideas of parametric equations (using a parameter t to control point coordinates), representation of curves in Cartesian coordinates, and the use of vectors to define directions and relationships that help generate the geometry in space. The construction is centered on GeoGebra commands and objects: CurveCartesian to draw curves from Cartesian coordinate expressions, Vector to encode direction/offset information, and Textfield with LaTeX together with multiple InputBox inputs to control parameters and display formulas. Intermediate variables (e.g., a, b, t, f, g, h, B, c, r, d, e, p, q, eq1, etc.) are used to link the input conditions to the final curve(s), while text labels (text1, text2, text3) clarify what each parameter controls. Suitable for advanced secondary school, early undergraduate geometry/analytic geometry instruction, and for classroom or self-study demonstrations where learners adjust parameters and observe the resulting curve transformations.