My common sense tells me something similar, that only straight lines could be used. I played around a bit stretching lines into curves to fit onto the tops and bottoms. But, it soon became clear it was possible to draw any curve to fit any set of tops or bottoms, and if they don't follow some set of rules, they are most likely not going to hold up or provide any meaningful interpretation over different cases or instances, across time and instruments. But that's what my common sense tells me, so I'm obligated to seek out contradicting theories or models, if there is an alternative that exists to get different perspectives. Perhaps collinearity and ratios are the key, but rather the price must be viewed outside of time? I read one article that said sometimes time is not the invariant, so should not be the independent variable in some experiments. I don't know, I'll keep googling for something.
Do the following test: Draw a trendline on a chart and a circle some place. Then zoom in. The straight line does not change shape but the circle becomes an ellipse.
the vast majority of tls are straight lines, channels including pitchfork which have been systemized by Dr. Alan Andrews, then Bollinger Bands, then a teatsy few using spirals, plus Gann who used angled lines - 45° 72° etc I've seen 'curve indicators' used on MetaTrader and MetaStock charts and generally they've been applied on longer timeframe charts, but imo nothing beats the shortest distance between two points ;
You could use a bezier curve to fit around low points (support) to give you a curve but I don't see the point. Presumably you are interested in where this support line is going to intersect to some future time so you can use that in an order so you might as well use linear regression i.e. a straight line as it will be better than an extrapolated curve. Both bezier and spline curves are doubtful for extrapolating beyond their last known point. Also, I think S/R is rarely a fine line but more of a zone so it will probably make little difference whether it's a curve or a line.
Research Clif Droke, A Testimony to Parabolic Analysis. Michael Jenkins also works with arcs for S/R and timing exhaustion. Question, can the equation of a future parabola be determined from reversal points.........
Sure, as few as 3 points are enough: "Three points uniquely determine one parabola" from: http://mathworld.wolfram.com/Parabola.html (after the 6. equation) or: http://en.wikipedia.org/wiki/Parabola under the headline of "Derivation of the focus" you see how 3 points are used for finding the equation.
Parabolic SAR is curved. <img src=http://www.elitetrader.com/vb/attachment.php?s=&postid=3015172 \img>
here's Advanced GET's Fibo Circle tool applied to the SP500 index quite interesting how the 2009 low hit the 1.618 line and how the price in July came down to hit the same line and then climbed along it how this tool is used is to drag from a point, which works fine on this chart since it's progressed to the point it has, guess it would have to be continually re-applied - no, that doesn't work . . .hmmmmm if you google: 'metatrader 4 spiral' then click Images you'll find some charts of the applied MT4 spiral indicator, and others
So many people spend so much time trying to confuse themselves. Oh well...actually I'm glad they exist.