Designing a Lens Surface

The design problem in one line

Chasing the angles through the surface

Eliminating to a slope condition

Reading the slope condition

Suppose a lens surface were shaped with a constant slope, a flat slanted plane rather than a curve whose slope grows with height. What would happen to rays leaving one object point?

In the angle-chasing derivation, the two unknown angles inside the glass are $\theta_2$ and $\theta_3$. Which pair of relations removes them and connects the outer angles to the surface slope?

A lens designer needs $1/a + 1/b = 0.2$ per centimeter with glass of index $n = 1.5$. Using $\theta_S = \frac{c}{2(n-1)}(1/a + 1/b)$, compute the surface slope at height $c = 0.1$ centimeters and write it as a decimal.