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POChecking if two cubic Bézier curves intersect
 

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  1. POChecking if two cubic Bézier curves intersect
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    <p>For a personal project, I'd need to find out if two cubic Bézier curves intersect. I don't need to know where: I just need to know if they do. However, I'd need to do it fast.</p> <p>I've been scavenging the place and I found several resources. Mostly, there's <a href="https://stackoverflow.com/questions/109364/bezier-clipping">this question here</a> that had a promising answer.</p> <p>So after I figured what is a <a href="http://en.wikipedia.org/wiki/Sylvester_matrix" rel="nofollow noreferrer">Sylvester matrix</a>, what is a <a href="http://mathworld.wolfram.com/Determinant.html" rel="nofollow noreferrer">determinant</a>, what is a <a href="http://mathworld.wolfram.com/Resultant.html" rel="nofollow noreferrer">resultant</a> and <a href="https://stackoverflow.com/questions/109364/bezier-clipping#comment-4332265">why it's useful</a>, I thought I figured how the solution works. However, reality begs to differ, and it doesn't work so well.</p> <hr> <h1>Messing Around</h1> <p>I've used my graphing calculator to draw two Bézier splines (that we'll call B<sub>0</sub> and B<sub>1</sub>) that intersect. Their coordinates are as follow (P<sub>0</sub>, P<sub>1</sub>, P<sub>2</sub>, P<sub>3</sub>):</p> <pre><code>(1, 1) (2, 4) (3, 4) (4, 3) (3, 5) (3, 6) (0, 1) (3, 1) </code></pre> <p>The result is the following, B<sub>0</sub> being the "horizontal" curve and B<sub>1</sub> the other one:</p> <p><img src="https://i.stack.imgur.com/ZYVaW.png" alt="Two cubic Bézier curves that intersect"></p> <p>Following directions from the aforementioned question's top-voted answer, I've subtracted B<sub>0</sub> to B<sub>1</sub>. It left me with two equations (the X and the Y axes) that, according to my calculator, are:</p> <pre><code>x = 9t^3 - 9t^2 - 3t + 2 y = 9t^3 - 9t^2 - 6t + 4 </code></pre> <hr> <h1>The Sylvester Matrix</h1> <p>And from that I've built the following Sylvester matrix:</p> <pre><code>9 -9 -3 2 0 0 0 9 -9 -3 2 0 0 0 9 -9 -3 2 9 -9 -6 4 0 0 0 9 -9 -6 4 0 0 0 9 -9 -6 4 </code></pre> <p>After that, I've made a C++ function to calculate determinants of matrices using <a href="http://en.wikipedia.org/wiki/Laplace_expansion" rel="nofollow noreferrer">Laplace expansion</a>:</p> <pre><code>template&lt;int size&gt; float determinant(float* matrix) { float total = 0; float sign = 1; float temporaryMatrix[(size - 1) * (size - 1)]; for (int i = 0; i &lt; size; i++) { if (matrix[i] != 0) { for (int j = 1; j &lt; size; j++) { float* targetOffset = temporaryMatrix + (j - 1) * (size - 1); float* sourceOffset = matrix + j * size; int firstCopySize = i * sizeof *matrix; int secondCopySize = (size - i - 1) * sizeof *matrix; memcpy(targetOffset, sourceOffset, firstCopySize); memcpy(targetOffset + i, sourceOffset + i + 1, secondCopySize); } float subdeterminant = determinant&lt;size - 1&gt;(temporaryMatrix); total += matrix[i] * subdeterminant * sign; } sign *= -1; } return total; } template&lt;&gt; float determinant&lt;1&gt;(float* matrix) { return matrix[0]; } </code></pre> <p>It seems to work pretty well on relatively small matrices (2x2, 3x3 and 4x4), so I'd expect it to work on 6x6 matrices too. I didn't conduct extensive tests however, and there's a possibility that it's broken.</p> <hr> <h1>The Problem</h1> <p>If I understood correctly the answer from the other question, the determinant should be 0 since the curves intersect. However, feeding my program the Sylvester matrix I made above, it's -2916.</p> <p>Is it a mistake on my end or on their end? What's the correct way to find out if two cubic Bézier curves intersect?</p>
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      1. USEdison Gustavo Muenz
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    1. COPlease don't compute determinants if you don't have to. If you want to check for a singularity compute the lowest and highest singular value. And if you need the determinant for some reason, don't use the Laplace-Expansion! It has exponential time complexity. You can do it on O(n^3)!
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    2. COPlugging your Sylvester Matrix into the matrix calculator at http://www.bluebit.gr/matrix-calculator/ gave -2916 for the determinant. You may need to fix your determinant function.
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      1. USKyle Lutz
    3. CO@Kyle Lutz Yeah I found that about 5 minutes after my post, and I fixed my determinant function.
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