Numerical diffraction can be simulated with computer-generated holography (CGH). This process is highly calculation-intensive and critical for real-time applications such as holographic video display systems. We present an analytical model for calculating the interference pattern for general line segments in 3D and adapt this model for freely drawing 3D splines. We implemented efficiently computable approximations for GPU in C++/CUDA and reported 77-fold speed gains over the reference point-wise algorithm. The results are validated on a holographic display setup.