Chapter 4: Geometry

4.20.2.3 Example: paraboloid of revolution
A Monge patch for a paraboloid of revolution is given by x f´Ù Úµ ´Ù Ú Ù ¾ ·
Ú ¾ µ, for all ´Ù Úµ ¾ Í Ê ¾ . By successive differentiation one obtains x Ù
´½ ¼ ¾Ùµ, x Ú ´¼ ½ ¾Úµ, x ÙÙ ´¼ ¼ ¾µ, x ÙÚ ´¼ ¼ ¼µ, and x ÚÚ ´¼ ¼ ¾µ.
1. Unit normal vector: n ´½·Ù ¾ ·Ú ¾ µ ½ ¾ ´ ¾Ù ¾Ú ½µ
2. First fundamental coef cients: ´Ù Úµ ½½´Ù Úµ ½·Ù ¾ , ´Ù Úµ
½¾´Ù Úµ ÙÚ, ´Ù Úµ ¾¾´Ù Úµ ½·Ú ¾
3. First fundamental form: Á ´½·Ù ¾ µ Ù ¾ ·ÙÚ Ù Ú ·´½·Ú ¾ µ Ú ¾ . Since
´Ù Úµ ¼µ Ù ¼or Ú ¼, it follows that the Ù-parameter curve Ú ¼
is orthogonal to any Ú-parameter curve, and the Ú-parameter curve Ù ¼ is
orthogonal to any Ù-parameter curve. Otherwise the Ù- and Ú-parameter curves
are not orthogonal.
4. Second fundamental coef cients: ´Ù Úµ ½½´Ù Úµ¾´½·Ù ¾ ·Ú ¾ µ ½ ¾ ,
´Ù Úµ ½¾´Ù Úµ ¼, ´Ù Úµ ¾¾´Ù Úµ ¾´½·Ù ¾ ·Ú ¾ µ
½ ¾
5. Second fundamental form: ÁÁ ¾´½·Ù ¾ ·Ú ¾ µ ½ ¾ ´Ù ¾ · Ú ¾ µ
6. Classi cation of points: ´Ù Úµ´Ù Úµ ´½·Ù ¾·Ú ¾ µ ¼ implies that all
points on Ë are elliptic points. The point ´¼ ¼ ¼µ is the only umbilical point.
7. Equation for the principal directions: ÙÚ Ù ¾ ·´Ú ¾ Ù ¾ µ Ù Ú · ÙÚ Ú ¾ ¼
factors to read ´ÙÙ· ÚÚµ´ÚÙ ÙÚµ¼.
8. Lines of curvature: Integrate the differential equations, ÙÚ · ÚÚ ¼, and
ÚÙ ÚÙ¼, to obtain, respectively, the equations of the lines of curvature,
Ù ¾ · Ú ¾ Ö ¾ , and ÙÚ ÓØ , where Ö and are constant.
9. Characteristic equation: ´½·Ù ¾ ·Ú ¾ µ ¾ ´½ · ¾Ù ¾ ·¾Ú ¾ µ´½·Ù ¾ ·
Ú ¾ µ ½ ¾ ·´½·Ù ¾ ·Ú ¾ µ ½ ¼
10. Principal curvatures: ½ ¾´½ · Ù ¾ ·Ú ¾ µ ½ ¾ , ¾ ¾´½·Ù ¾ ·Ú ¾ µ ¿ ¾ .
The paraboloid of revolution may also be represented by x f´Öµ´Ö Ó× ,
Ö ×Ò Ö ¾ µ. In this representation the Ö- and -parameter curves are lines of
curvature.
11. Gaussian curvature: Ã ´½·Ù ¾ ·Ú ¾ µ ¾ .
12. Mean curvature: À ¾´½·¾Ù ¾ ·¾Ú ¾ µ´½ · Ù ¾ ·Ú ¾ µ ¿ ¾ .
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