CONSTRUCTING PERIODIC CURVES WITH AN EXPONENTIAL-ALGEBRAIC HYBRID INTERPOLATING POLYNOMIAL

By “periodic” we mean a parametric curve that replicates itself on some predefined period up to a translation. A good example of a parametric curve is a helix. The domain of the function that sets the parametric curve is real numbers, the codomain is then points in n-dimensional space. The period can be an arbitrary number, but from now on in this work we’ll always presume that the period is 2π.In this work, we propose a way to construct a periodic curve that goes through a set of predefined points and, while doing so, provides additional control over its shape. To build such a periodic curve, we use an exponential-algebraic hybrid polynomial that is partially similar to the Fourier polynomial in complex exponential notation. The difference is that the hybrid polynomial we propose also has algebraic terms, moreover while exponential terms inherently require imaginary numbers as their input, the algebraic terms do not. This polynomial interpolates a set of complex numbers, each assigned to a single real parameter. The real parts of the complex data points constitute the points in n-dimensional space the target curve passes through, while the imaginary parts of the same data points affect the shape of the curve without compromising the interpolating properties of the polynomial in real numbers.In this way, we effectively split our input in two. The “curve goes through the points” conditions are set by the real parts of the complex input data points, and we can then use the imaginary parts to control the curve’s shape. An example of such control is minimizing the curve’s length on a given parameter range.