# Distance between points on a face

Given a 2D picture of a face, how is it possible to measure the distance between two different points on the surface of the face?

Thanks

Joel

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The use of the word "face" suggests you are talking about a polyhedron yet the tag is "spherical-geometry." Are you referring to the regions of a graph on a sphere formed by great circles? – Joseph Malkevitch Jan 3 '11 at 18:38

First you want to build a 3d-mesh out of your face. To get a general idea of this, the following paper might be interesting for you: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.58.8435&rep=rep1&type=pdf . Of course you can improve this algorithm with knowledge, that you already have about the appearance of human faces.

Then you select your points in the mesh and run a path-finding algorithm to get a good approximation for the distance. When you did everything properly you will get something like this (note that the red line should mark shortest path):

I hope this is what you wanted

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Ah the literal interpretation. I laughed hard because I thought the OP meant something like faces of a polyhedron. – AnonymousCoward Jan 3 '11 at 22:10

Use a ruler. If your needs are more specific, perhaps you could explain them?

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@Jasper Loy: I was just trying to be as helpful as possible, given the quality of the question. You are welcome to post a better solution! – TonyK Jan 3 '11 at 16:35
@ TonyK: although I think you've given the correct answer to this question, I also think the OP had something else in mind. He tagged the question as "spherical-geometry", se perhaps, Joel, you intended to ask how to find out what the length of the shortest path between two points on a sphere is? – Max Muller Jan 3 '11 at 17:36

1) Determine the longitude and lattitude of the points. (Using polar co-ordinates)

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A 2D image of a face is a certain projection of a three-dimensional object into the plane. The pre-image of any point is a line, whence a lot of information gets lost in the process. It follows that it is ${\it impossible}$ to tell the true distance between two original points by measurements in the 2D image, even if you are able to identify the individual image points with their corresponding points on the face.

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