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Subject: How to project a point onto a surface?



This tricky problem is part of search of geodesic line. Suppose I have
a surface and I want to find out the geodesic line between two points
in the surface. This surface has curvature and is composed of many
triangular elements.

I choose two end points in the surface then I can generate a straight
line between these points. I divide this straight line into 5 or 6
segments. Now I want to project generated segment nodes onto the
surface. Firstly, the x and y co-ordinates of each of the segment nodes
are used to locate specific surface trangular element in which it lies.
The z co-ordinate of the segment node is determined using the known
nodal co-ordinates of the surface element by a standard shape function.
>From these initial positions, the projected segment nodes are allowed
to slide over the surface elements in search of the equilibrium
configuration. Once the final state of equilibrium is attained, the
path of a geodesic will be defined by these projected segment nodes.

My questions are
1) how to locate specific surface trangular element using the x and y
co-ordinates of each of the segment nodes? I know the node coordinates
of each surface triangular element but I don't know which element I
need locate.
2) how to work out the z co-ordinate of the segment node using the
known nodal co-ordinates of the surface element by a standard shape
function?

Any suggestion would be appreciated. Thanks.


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