Facial animation is a fundamental challenge that requires mathematical and computational strategies. In this paper, a novel facial animation technique using numerical traced algorithm is introduced. Homotpy-based animation methodology (HAM) uses the homotopy curve path in order to novelty generate intermediate frames for different λ values and therefore it represents the deformations from starting image to ending image. These deformations use system of equations embedded into a single homotopy equation in order to represent intermediate frames. Moreover, a hyperspherical tracking method establishes deformations with visually consistent and smooth changes. Experimental results reveal intermediate frames that can be interpreted as facial animation. Furthermore, histogram plots, homotopic trajectories, and pixel variation tables confirm that different pixel positions vary with different rates of change as the original image is transformed into the target image. Besides, these frames do not need external filters in order to correct visual interpretation errors and therefore the homotopy-based animation method can be considered as a useful alternative for animating facial images in different applications.