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Jan
26
awarded  Popular Question
Nov
29
comment Approximate arc length of cubic bezier curve?
Instead of looking at the sub-segment length, you could look at the curvature of the sub-Bezier and stop recursion when the curvature is below some threshold. A couple simple dot-product operations and square roots are needed to check the cosines of the angles involved.
Feb
11
answered What is the pair $(n,k)$ called where $n$ is an integer and $k$ is the ordered factorization index?
Feb
9
asked What is the pair $(n,k)$ called where $n$ is an integer and $k$ is the ordered factorization index?
Sep
30
comment Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]
@RahulNarain — A circle works very nicely when $P_2$ is positioned between $P_1$ and $P_3$, particularly when its close to the $P_1P_3$ line segment. But when $P_2$ is, say, way off to the right or left of $P_3$ and close to collinear with $P_1P_3$, then a circle creates an undesirable tangent that’s almost in a perpendicular direction from what I need. I appreciate the suggestion of trying a circle instead, though! I’m definitely open to possibilities besides ellipses. I was thinking last night that maybe a parabola would work.
Sep
30
revised Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]
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Sep
30
revised Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]
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Sep
29
comment Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]
@Hagen — Well, using a line orthogonal to $P_2-M$, where $M = \frac{1}{2}(P_1+P_3)$, is a decent approximation if $P_2$ is far away from $P_3$. But if $P_2$ is close to $P_3$ (that is, much closer to $P_3$ than it is to $P_1$), then it gives disastrously different line than the tangent to the fitted ellipse. So this is why I’m seeking a solution that actually fits an ellipse, rather than a simpler approximation. Although I won’t be fitting my final curve using the ellipse (I use cubic Bézier curves for that), the tangent of the fitted ellipse supplies a beautiful set of Bézier control points.
Sep
29
awarded  Student
Sep
29
awarded  Editor
Sep
29
revised Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]
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Sep
29
asked Passing an ellipse through 3 points (where 2 two points lie on the ellipse axes)? [Updated with alternative statement of problem and new picture]
Jan
4
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