We develop a theoretical framework for interpolating visual contours and apply it to subjective contours. The theory is based on the idea of consistency: a curve fitting algorithm must give consistent answers when presented with more data consistent with its hypothesis, or the same data under different conditions. Using this assumption, we prove that the subjective contour through two point-tangents is a parabola. We extend the theory to include multiple point-tangents and points. Sample output of programs implementing the theory is provided.