In a recent study of laser eye surgery by Gatinel, Hoang-Xuan, and Azar, a vertical

Chapter 13, Problem 68

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In a recent study of laser eye surgery by Gatinel, Hoang-Xuan, and Azar, a vertical cross section of the cornea is modeled by the halfellipse of Exercise 67. Show that the half-ellipse can be written in the form x = f (y), where f (y) = p1 r r2 py2 . During surgery, tissue is removed to a depth t(y) at height y for S y S, where t(y) is given by Munnerlyns equation (for some R>r): t(y) = R2 S2 R2 y2 r2 S2 + r2 y2 After surgery, the cross section of the cornea has the shape x = f (y) + t(y) (Figure 20). Show that after surgery, the radius of curvature at the point P (where y = 0) is R. Segment of length t(y) y S S Eye shape before surgery x = f(y) Eye shape after surgery x = f(y) + t(y) P x