An instrument and method for creating a precise capsulorhexis as a step in cataract surgery is described. The instrument has a curvilinear cutting band attached to a longitudinal support component and cutting band extending on two sides of the support. In an embodiment, the cutting band has a sinusoidal shape. The support is coupled to a supporting structure, and then to a handle, the supporting structure is shaped to permit it to pass through an incision at a side of a cornea of the eye while positioning the cutting band on an anterior surface of an anterior capsule of a lens of the eye. In an embodiment of the method, the incision in the anterior capsule is formed by pressing the cutting band into the anterior capsule, and capsulorhexis is extended by tearing away the two tabs formed and defined by curvature of the cutting band.