In this paper, we revisit the role of the image of the absolute conic (IAC) in recovering the camera geometry. We derive new constraints on IAC that advance our understanding of its underlying building blocks. The new constraints are shown to be intrinsic to IAC, rather than exploiting the scene geometry or the prior knowledge on the camera. We provide geometric interpretations for these new intrinsic constraints, and show their relations to the invariant properties of the IAC. This in turn provides a better insight into the role that IAC plays in determining the camera internal geometry. Since the new constraints are invariant properties of the IAC, they can be used to reparameterize its elements. We show that such reparameterization would allow to recover a more general camera geometry from a single view, compared to existing methods. We apply the new constraints to single view calibration using vanishing points, investigate the error resilience, and compare our results to Liebowitz-Zisserman (1999).