Surface height from surface normal function

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SUMMARY

The discussion focuses on calculating the height at a specific point (x,y) on a mountain using the known peak coordinates (Xp, Yp) and height (Zp), alongside a function G(x,y) that provides the 3D normal vector of the mountain's surface. Andrew seeks a method to derive height from the surface normal, acknowledging that the surface function F(x,y,z)=0 can yield the normal vector via ∇F(x,y,z). However, he expresses difficulty in reversing this process to find height.

PREREQUISITES
  • Understanding of surface functions, specifically F(x,y,z)=0
  • Knowledge of gradient vectors and their application in 3D space
  • Familiarity with conical shapes and their geometric properties
  • Basic concepts of vector calculus
NEXT STEPS
  • Research methods for deriving height from normal vectors in 3D geometry
  • Explore the implications of conical shapes on surface normals
  • Study the application of gradient functions in surface analysis
  • Investigate numerical methods for solving surface equations
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Mathematicians, physicists, and computer graphics developers interested in surface modeling and height calculations based on normal vectors.

AndrewD
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If you are standing on the side of a mountain at a point (x,y) and you know where the peak is (Xp, Yp) and how high it is (Zp) and you have a function G(x,y) that defines the 3D normal vector to the mountain side at all points on the surface, how do you calculate your height?

OK it's a simple way to put it but it expresses the problem neatly. I know that from the surface function F(x,y,z)=0, ∇F(x,y,z) will get you the surface normal at any point on the surface but I haven't found a simple way to express the reverse process. Can anyone help me?

Thanks,

Andrew
 
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If the mountain is conically shaped, the normal will be independent of height.
 

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