Problem:
Show that the principal curvatures are given in terms of $ K $ and $ H $ by
$ k_1 = H + \sqrt{H^2 - K} $ and $ k_2 = H - \sqrt{H^2 - K} $.
Solution:
We know $ H = \frac{k_1 + k_2}{2} $ and $ K = k_1 k_2 $. Therefore we can form the quadratic equation $ x^2 - 2Hx + K $ so that the roots are $ k_1 $ and $ k_2 $
Now using the quadratic formula, we get
$ k = \frac{2H \pm \sqrt{4H^2 - 4K}}{2} = H \pm \sqrt{H^2 - K} $, this is exactly what we needed.
Show that the principal curvatures are given in terms of $ K $ and $ H $ by
$ k_1 = H + \sqrt{H^2 - K} $ and $ k_2 = H - \sqrt{H^2 - K} $.
Solution:
We know $ H = \frac{k_1 + k_2}{2} $ and $ K = k_1 k_2 $. Therefore we can form the quadratic equation $ x^2 - 2Hx + K $ so that the roots are $ k_1 $ and $ k_2 $
Now using the quadratic formula, we get
$ k = \frac{2H \pm \sqrt{4H^2 - 4K}}{2} = H \pm \sqrt{H^2 - K} $, this is exactly what we needed.
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