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Registered Member #261
Joined: Mon Feb 27 2006, 12:34AM
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Posts: 22
Hello all, recently I have been investigating some interesting curves expressed as simple trigonometric parametric ordered pairs. Consider, (Sin(t), Sin(2t)) {t,0,2pi} makes a pretty pi-orbital or dumbell shaped curve depending how you look at it. Considering that the Cartesian equivalent is: y = (+ or -)2x(1-x^2)^(1/2) {x,-1,1}, it seems convenient to just express it parametrically. I am looking at curves that are expressed using simple trig functions with multiple angles. When one of the values has a t coeffecient of 1, they are trivial to convert to Cartesian equations.
However, when they both have t coeffecients >1, it is extremely difficult. Consider (Cos(2t), Cos(3t)) which makes a pretty little loop thingy. I have used a page or two of notebook paper trying to just crunch it using identities and substitution. I have faith that it can be done, but there has to be a better way....
Registered Member #261
Joined: Mon Feb 27 2006, 12:34AM
Location:
Posts: 22
Entirely possible-- I found a solution. Reserving this place to edit and post it later, yet I would still like ideas for alternate routes.
The solution is: y=(+or -)[(1/2+x/2)^(1/2)](2x-1) {x,-1,1} and is obtainable in less than 10 steps using trig identities. I was taking the wrong approach earlier, of trying to work with Cos(2t) in addition to doing the expected work with Cos(3t).
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Math: Parametric to Cartesian Conversions
