Specular surfaces, like water surfaces, create caustics by focusing the light being refracted or reflected. These caustics are very important for scene realism, but also challenging to render: to compute them, we need to find the exact path connecting two points through a specular reflective or refractive surface. This requires finding the roots of a complicated function on the surface. Manifold-Exploration methods find these roots using the Newton-Raphson method, but this involves computing path derivatives at each step, which can be challenging. We show that these roots lie on a curve on the surface, which reduces the dimensionality of the search. This dimension reduction greatly improves the search, allowing for interactive rendering of caustics. It also makes implementation easier, as we do not need to compute path derivatives.