Conformal Mapping Visualization: Preserving Angles via Splines and Vectors

This example demonstrates how a conformal mapping transforms curves and geometric directions from one plane setting to another, allowing students to visually verify the key property of local angle preservation. The main mathematical ideas include conformal maps in the complex plane (as angle-preserving local transformations) and geometric verification using angle measurements between mapped objects. Construction highlights: GeoGebra’s Spline is used to create the reference curve(s) from control points; Point defines key sampling/anchor locations; Segment and OrthogonalLine support layout and perpendicular directions for geometric comparisons. Vector and Angle are employed to represent and measure directional changes before and after mapping. Translate and Intersect are used to shift configurations and compute intersection points that define the mapped structure. A set of paired objects (e.g., multiple “U/V” and “u/v” point sets and several intermediate points like Z* and X*) is prepared to contrast source-domain geometry with its image-domain counterpart. Suitable for high school advanced topics through undergraduate pre-calculus/complex-analysis visualization, especially for introducing the geometric meaning of conformal mappings.