Two-Body Orbital Trajectories via GeoGebra Parameter Curves

This example demonstrates, in a 2D GeoGebra proof-of-concept visualization, how two-body (Kepler-style) orbital trajectories can be generated from simplified geometric/kinematic relations. It highlights how different initial parameters determine the orbit shape and the key points along the path, such as periapsis and apoapsis, reflecting the role of eccentricity and orbital parameters. The construction centers on parametric-curve graphing and coordinate geometry concepts: vectors, segment-based geometry, angle/length computations, and the classification of conic sections (ellipse/parabola/hyperbola) as orbit families. In GeoGebra terms, the build uses commands including Point, Vector, Segment to define mass/velocity and geometric elements; Length and Angle to compute necessary magnitudes and directions; If to switch formulas or cases depending on parameter conditions; and CurveCartesian to draw the resulting Cartesian-coordinate orbit curve (e.g., the orbit object), along with labeled quantities such as μ, v0, γ0, r0, θ0, eccentricity e, and the major/minor axis parameters (a_maj, a_min) and key points (periapsis/apoapsis). Suitable for advanced high school and early undergraduate learners for visualization of orbital geometry, conic sections, and vector-based modeling.