Parametrization of Curves in the Complex Plane

Draw a smooth parametric curve in the complex plane generated from several points, and add three vectors starting from the origin, each pointing to a distinct key point on the curve. The curve should appear as a continuous path, illustrating complex parametrization. The vectors should be distinguished by color or style, all sharing the origin as their start, with endpoints on different parts of the curve to show how complex numbers move as the parameter changes. The overall layout should clearly display the geometric relationship between the curve and the vectors, suitable for high school students learning about parametric equations and complex geometry.