Knot Surfaces: Constructing Complex 3D Surfaces by Rotation

This example demonstrates how to build “knot surfaces” by starting from a parameterized Cartesian curve and then generating a full 3D surface through spatial rotation. The visualization highlights the emergence of tangled, intricate surface geometry that looks knotted and structured in space. Key mathematical ideas include parameterized curve geometry (via CurveCartesian), rotation transformations (Rotate) in 3D, and surface creation by sweeping/rotating curves using the Surface tool. From the construction summary, it is built by drawing base curve objects (e.g., Knot1, Knot2 and related parameters/points) with CurveCartesian, defining the rotation center/axis using elements such as O, R, and r, and then applying Rotate to obtain rotated profiles; finally, Surface combines the rotated geometry into the complete knot surface, with supporting objects (f, g, h, a, b, c, d, e, i, j, k) and styling such as Color. Suitable for advanced high-school students and early university learners studying 3D geometry, coordinate/parametric modeling, and surface concepts in a calculus-aware context, as well as for interactive visualization and project-based exploration.