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    <p>The elliptical curve arc link you posted includes a <a href="http://www.w3.org/TR/SVG/implnote.html#ArcImplementationNotes" rel="nofollow">link to elliptical arc implementation notes</a>.</p> <p>In there, you will find the equations for <a href="http://www.w3.org/TR/SVG/implnote.html#ArcConversionEndpointToCenter" rel="nofollow">conversion from endpoint to centre parameterisation</a>.</p> <p>Here is my JavaScript implementation of those equations, taken from <a href="http://users.ecs.soton.ac.uk/rfp07r/interactive-svg-examples/arc.html" rel="nofollow">an interactive demo of elliptical arc paths</a>, using <a href="http://sylvester.jcoglan.com/" rel="nofollow">Sylvester.js</a> to perform the matrix and vector calculations.</p> <pre><code>// Calculate the centre of the ellipse // Based on http://www.w3.org/TR/SVG/implnote.html#ArcConversionEndpointToCenter var x1 = 150; // Starting x-point of the arc var y1 = 150; // Starting y-point of the arc var x2 = 400; // End x-point of the arc var y2 = 300; // End y-point of the arc var fA = 1; // Large arc flag var fS = 1; // Sweep flag var rx = 100; // Horizontal radius of ellipse var ry = 50; // Vertical radius of ellipse var phi = 0; // Angle between co-ord system and ellipse x-axes var Cx, Cy; // Step 1: Compute (x1′, y1′) var M = $M([ [ Math.cos(phi), Math.sin(phi)], [-Math.sin(phi), Math.cos(phi)] ]); var V = $V( [ (x1-x2)/2, (y1-y2)/2 ] ); var P = M.multiply(V); var x1p = P.e(1); // x1 prime var y1p = P.e(2); // y1 prime // Ensure radii are large enough // Based on http://www.w3.org/TR/SVG/implnote.html#ArcOutOfRangeParameters // Step (a): Ensure radii are non-zero // Step (b): Ensure radii are positive rx = Math.abs(rx); ry = Math.abs(ry); // Step (c): Ensure radii are large enough var lambda = ( (x1p * x1p) / (rx * rx) ) + ( (y1p * y1p) / (ry * ry) ); if(lambda &gt; 1) { rx = Math.sqrt(lambda) * rx; ry = Math.sqrt(lambda) * ry; } // Step 2: Compute (cx′, cy′) var sign = (fA == fS)? -1 : 1; // Bit of a hack, as presumably rounding errors were making his negative inside the square root! if((( (rx*rx*ry*ry) - (rx*rx*y1p*y1p) - (ry*ry*x1p*x1p) ) / ( (rx*rx*y1p*y1p) + (ry*ry*x1p*x1p) )) &lt; 1e-7) var co = 0; else var co = sign * Math.sqrt( ( (rx*rx*ry*ry) - (rx*rx*y1p*y1p) - (ry*ry*x1p*x1p) ) / ( (rx*rx*y1p*y1p) + (ry*ry*x1p*x1p) ) ); var V = $V( [rx*y1p/ry, -ry*x1p/rx] ); var Cp = V.multiply(co); // Step 3: Compute (cx, cy) from (cx′, cy′) var M = $M([ [ Math.cos(phi), -Math.sin(phi)], [ Math.sin(phi), Math.cos(phi)] ]); var V = $V( [ (x1+x2)/2, (y1+y2)/2 ] ); var C = M.multiply(Cp).add(V); Cx = C.e(1); Cy = C.e(2); </code></pre>
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