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    <p>there are at least two approaches to this problem</p> <h2>Approach 1</h2> <p>you can use the fact that principal curvatures are the eigenvalues of a <a href="http://en.wikipedia.org/wiki/Shape_operator#Shape_operator" rel="nofollow noreferrer">shape operator</a> - a linear function on the space defined on two its tangent vectors. </p> <p>procedure:</p> <pre><code>1. compute shape operator: </code></pre> <p>find two tangent vectors and then compute</p> <p><img src="https://i.stack.imgur.com/kmzYv.png" alt="enter image description here"></p> <p>you will find a matrix</p> <p><img src="https://i.stack.imgur.com/h46Td.png" alt="enter image description here"></p> <pre><code>2. and then the eigenvalues of this matrix are principal curvatures k1, k2 </code></pre> <h2>Approach 2</h2> <p>We will use the fact that principal curvatures of the surface <code>S</code> at the given point <code>P</code> are the roots in the real domain of the equation</p> <pre><code>(EG-F^2)k^2 - (EN-2FM+GL)k + LN-M^2 = 0 (1) </code></pre> <p>where <code>k</code> is the main curvature and coefficients are taken from first &amp; second fundamental form. They are given in terms of the parametric equation. To get these roots we will use the fact that instead of calculating <code>k1</code> and <code>k2</code> from the (1) we can find eigenvalues of a matrix <code>A</code>, where <code>A</code> is defined as</p> <p><img src="https://i.stack.imgur.com/aqQYT.png" alt="enter image description here"></p> <p>and matrix <code>F1</code> contains coefficients of the first fundamental form</p> <p><img src="https://i.stack.imgur.com/jXREc.png" alt="enter image description here"></p> <p>matrix <code>F2</code> contains coefficients of the second fundamental form</p> <p><img src="https://i.stack.imgur.com/5t898.png" alt="enter image description here"></p>
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    1. POHow to calculate principal curvature over interpolated triangular surface?
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    1. COThank you for the link, I will read more about it and try to implement it.
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    2. COIt would be great though if I could get more help in the implementation :)
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    3. COyo have to choose any two tangent vectors and solve equation 1st in 1) for S matrix or obtain it by calculating 2nd in 1) from first and second fundamental form (for e,f,...,F,G coefficients)
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