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Deciding which is the “best” approximation to a quarter circle depends on your choice of error metric. Just choosing the one with the correct endpoints and endp
by jacobolus 8y ago
Deciding which is the “best” approximation to a quarter circle depends on your choice of error metric. Just choosing the one with the correct endpoints and endpoint tangents which intersects the circle halfway along is not the best you can do by most choices of error metric.
A more traditional choice might be the cubic curve where the maximum distance between a curve point and the circle is minimized. Or it might instead be desirable to preserve the arclength of the circle and minimize curvature deviation. Or it might be desirable to preserve the circle’s area. Or ...
Depending on the application different choices might be better. For some purposes it is not strictly essential that the endpoints of the curve directly fall on the circle.
- raphlinus 8y agoOr in the case of my recent Euler explorer, minimizes the L2 norm of tangent error. That said, most of the error norms will give similar results, as long as you're actually minimizing something. You're absolutely right about this though, the main reason people pick that one is that the math comes out easy, it's just a bit of trig.
- ttoinou 8y agoThe best could also be... a true quarter circle :)
- ginko 8y agoOr you could describe the quarter circle exactly using a rational spline.
- jacobolus 8y agohttps://beta.observablehq.com/@jrus/svg-elliptical-arcs https://beta.observablehq.com/@jrus/svg-elliptical-arcs