High precision trigonometry?

I just tried the Taylor-series function. The deviation between the standard Cos function and mc's version is less than 1 to 100000. At the worst point pi/8 the deviation between standard Cos and DrawCircle is 3 in 1000.

EDIT: I will try to make a function that compensates for DrawCircle's inaccuracy...

Be careful, mc73s' Taylor series gives good results up to 90 degrees.
But at heigher values the difference is enormus.
mc73s' code checks only for the max value but not for the min value !
Setting in the cosd2 routine a value different from 20 gives different results.

Best regards.

klaus said:

Be careful, mc73s' Taylor series gives good results up to 90 degrees.
But at heigher values the difference is enormus.
mc73s' code checks only for the max value but not for the min value !
Setting in the cosd2 routine a value different from 20 gives different results.

Best regards.

Click to expand...

Ah, Klaus, you caught me :sign0079: I forgot to put an abs(), you're absolutely right!
Yes, we have to be careful with such series. In fact, after the 20th power, we can even receive a NAN result.

I tried to analyse the deviation, and it seems to repeat every 45 degrees. And it describes a progress that resembles a Cosine function.

So I tried to set up a cos-function that adds a 3:1000 modifier (maxed at 22.5 deg and minimum at 0 and 45 deg).

It improves, but to get the planets in line I have to repeat the concept several times. Add a 3:10,000 modifer at 0-22.5 deg, add a 3:100,000 modifier at 0-11.25 deg and so on. After 6 of these modifications, the planet's modified radius is close enough to the DrawCircle that you wouldn't notice the difference.

Well at least from 0-22.5 deg. I would need another set of 6 cosine functions for the 22.5-45 deg interval. :BangHead:

--

I have some to the conclusion that it is better to cheat.

When I zoom so close to the orbit that the circle just looks like a straight line, I am going to draw it as a straight line instead of DrawCircle. I know which point it has to go through and the angle it should have, so it is easy to make a line from screen edge to screen edge. Guaranteed precise at all zoom levels, and probably it will be faster as well.

Finally

Update - and solution.

Ok. I adapted my drawing routines so that it uses DrawLine when I have zoomed in really close and DrawCircle for stuff further out. But at the really close zoom rate where you won't notice the change from circle to line; the deviation could still be half a screen when worst.

So back to the drawing board. I investigated further and found that the deviation is mirrored on both X- and Y- axis. So I only needed to measure in the interval 0-90 degrees. I also found that the amount of deviation is the same in the interval 0-45 as 45-90.

In the interval 0-45 the deviation is symmetric mirrored around 22.5 degrees.



I thought the shape would be the same in interval 0-11.25 as in interval 11.25-22.5, just negative. But the first interval is a bit steeper than the second.

So I have modelled an exponential function for this two intervals. (To enhance performance I have put the fixed number into global variables that are only calculated during application start.)

The input in the function is the rotation of the object you want to place on the DrawCircle shape (in degrees).

The output is multiplier in the range 1.000 to 1.003 approx. that extends the radius that the placement is calculated on.

Example of use:

The actual function: