A novel method for manipulating polynomial curves by the Gauss–Legendre control polygon with points interpolating property

Gauss–Legendre (GL) curves have recently been introduced, offering a significant advancement over traditional Bézier curves. These curves exhibit remarkable properties, particularly in high-degree cases, where the GL control polygon closely approximates the resulting curve. This is in stark contrast to the much larger Bézier control polygon, which poses practical challenges for curve design. While GL curves retain the endpoint interpolation property, their control over intermediate curve points is limited, as the direct influence of the control polygon is confined to the hodograph at the nodes. To address the challenge of refining high-degree polynomial curves to pass through specific points, we propose a novel method that extends the functionality of GL control polygons to achieve point interpolation. This approach allows for simultaneous control of both the hodograph and the curve itself, ensuring that the curve passes through designated points. We describe the construction of suitable polynomials to serve as weights, facilitating the representation of a curve as a combination of control points. By utilizing these polynomials, we introduce a methodology for directly manipulating both the hodograph and the interpolation points at the nodes, thereby enhancing the precision and flexibility of polynomial curve design using GL polygons.