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Topic: improvement on sin(x) = −0.4176x^2 +1.3122x -
0.0504 ;; we need semicircle sine, not the fakery of sinusoi
d sine

Replies: 15   Last Post: Feb 25, 2017 3:31 AM

 Messages: [ Previous | Next ]
 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
sine is -1.44x^2 + 2.44x + 0 = Y
Posted: Feb 20, 2017 3:18 AM

Now I do not know why I have this mischeviousness within me to never follow instructions until absolutely necessary-- all frustration out.

So, I employed the tool:
The Polynomial Generator is this tool::

For 2 coordinate points, (x0, y0) (x1, y1), we produce the 1st degree polynomial, a straightline or line segment

P(x) = y0(x-x1) / (x0-x1)
+ y1(x-x0) / (x1-x0)

For 3 coordinate points, (x0, y0) (x1, y1) (x2, y2), we produce the 2nd degree polynomial, a compilation-curve

P(x) = y0(x-x1)(x-x2) / (x0-x1)(x0-x2)
+y1(x-x0)(x-x2) / (x1-x0)(x1-x2)
+y2(x-x0)(x-x1) / (x2-x0)(x2-x1)

For 4 coordinate points, (x0, y0) (x1, y1) (x2, y2) (x3, y3), we produce the 3rd degree polynomial, a compilation curve

P(x) = y0(x-x1)(x-x2)(x-x3) / (x0-x1)(x0-x2)(x0-x3)
+y1(x-x0)(x-x2)(x-x3) / (x1-x0)(x1-x2)(x1-x3)
+y2(x-x0)(x-x1)(x-x3) / (x2-x0)(x2-x1)(x2-x3)
+y3(x-x0)(x-x1)(x-x2) / (x3-x0)(x3-x1)(x3-x2)

Obviously I need the tool for 3 coordinate points:

sine (0,0) (.5,.86) (1,1)

And what I end up with is this polynomial

-1.44x^2 + 2.44x + 0 = Y

Now, was that so much easier than playing around-- I think so. For that equation gives me exactly sine at those sought for 3 points. And, the bonus is that it is still quadratic.

And it looks as though the cosine, which I have not yet worked out, but is also quadratic since the last term of y2 is 0 and thus ending as a power of 2 quadratic.

So, let me work that one out, ....

AP