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wrong result when using `beside` with circle and vector (1 & (-1)) #46

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byorgey opened this Issue Sep 12, 2012 · 3 comments

2 participants

@byorgey
diagrams member
byorgey commented Sep 12, 2012

Observe:

beside (1 & (-1.01)) (square 1) (circle 1)

Looks OK. And now:

beside (1 & (-1)) (square 1) (circle 1)

wat.

@byorgey
diagrams member
byorgey commented Dec 12, 2012

Tracking this down a bit further, I've reduced it to a smaller test case:

p :: Trail R2
p = Trail [last $ trailSegments (circle 1)] False

main = defaultMain $ beside (1 & (-1)) (strokeT p) (square 1 # rotateBy (1/8))

The problem is that the envelope for the given cubic segment returns the wrong value for v = 1 & (-1). Ultimately, the envelope ends up solving the quadratic equation

-3.3306690738754696e-16 x^2 - 3.3137084989847607 x + 1.6568542494923806 = 0

The correct solution is x = 1/2, but the quadForm solver does a really poor job here and returns [-9.949077574161476e15,-0.0]. The problem seems to be the extreme tininess of the x^2 coefficient which introduces all sorts of numerical instability.

Sigh. Maybe we ought to just switch to iterative rootfinding? What do "real" graphics programs (or other programs that have to solve "real world" quadratic or cubic equations) do?

@fryguybob
diagrams member

I thought I looked at quadForm before to check this, but I guess I hadn't. There is a way to avoid the problems with small values given here http://www.mpi-hd.mpg.de/astrophysik/HEA/internal/Numerical_Recipes/f5-6.pdf.

My understanding is that for cubic and quartic you are better off with some iterative method (some are given in that book which I have around here somewhere).

@byorgey
diagrams member
byorgey commented Dec 12, 2012

Ah, thanks for the link! Works like a charm. That PDF also has some stuff about solving cubics, I wonder if we should be using that too. Though I've never (yet) noticed any problems with cubForm.

@byorgey byorgey closed this in be58994 Dec 12, 2012
@byorgey byorgey pushed a commit that referenced this issue Dec 12, 2012
Brent Yorgey note #46 fix in CHANGES [ci skip] 9f22d7a
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