High precision trigonometry?

gerth6000 · Aug 14, 2012

Not sure if my problem is because of the Sin/Cos functions or with the DrawCircle function....

Is there a library with Sin/Cos functions with high precision (10 digits or more)??

Or a DrawCircle function with high precision?

Explanation:
I'm working on an interactive 2-dimensional star map.

A particular star is placed in space with a set of x,y coordinates.

Its planet is then placed by using the radius of its orbit and its place in the orbit in radians, ie. I use Cos and Sin functions to calculate the planet's x,y coordinates.

I use the same star's coordinates and the orbit's radius to DrawCircle the orbit.

When zoomed out, that I can see planet and star on screen it looks perfect. Planet is on the circle. When zoomed in, that I can see the planet and its moons, the planet is off the circle by up to half a screen width.

I can calculate the misplacement as roughly 0.2% (2 in 1000) but it is not consistant. Bigger orbits have bigger misplacements.

For average applications I would think this error margin would be acceptable, but I use a zoom factor of up to 1 to 10 million (space is pretty big). Where 2 in 1000 is quite noticable.

gerth6000 · Aug 14, 2012

I'm thinking that DrawCircle doesn't draw a perfect circle, but a many-pointed polygon... That way the precision can never be guaranteed...?

klaus · Aug 14, 2012

If you use Double variables the precision is at least 15 digits.
How do you draw the big circles ?
Could you post your project as a zip file (IDE menu Files / Export As Zip) so we could have a look at it ?
Are you sure that the orbits are circles and not ellipses ?

Best regards.

mc73 · Aug 15, 2012

Never checked the precision of sin/cos etc, but if they're just approximations, why not recalculating them using Taylor's series? Just a thought.

klaus · Aug 15, 2012

We can't do anything with the code snippets.
Couldn't you post your project ? So we could test it in the same conditions as you do.

Best regards.

Hubert Brandel · Aug 16, 2012

mc73 said:
Never checked the precision of sin/cos etc, but if they're just approximations, why not recalculating them using Taylor's series? Just a thought.

could you explain how this works (building own sin, cos functions) because within a not b4a programm I do need this.

klaus · Aug 16, 2012

Mit Taylor Serien.
Siehe hier Kapitel Motivation durch Taylorreihen.

Beste Grüsse.

With Taylor series.
Look here chapter Series definition.

Best regards.

gerth6000 · Aug 16, 2012

Ok. Here goes.

gerth6000 · Aug 16, 2012

It starts fully zoomed out. Then try to pinch-zoom (repeatedly) to get to one of the green planets.

gerth6000 · Aug 16, 2012

You should remove the 0.99763 factor in lines 270-280.

It was an attempt to get closer to the orbit, but it still wasn't good enough.

mc73 · Aug 16, 2012

True, no reason for this factor. I see now the problem, I'll have a closer look, I hope other friends here can help.

mc73 · Aug 16, 2012

Just a thought: if it is a matter of precision, perhaps you could try setting factors as long you zoom in, manually. I mean, the way you set the 0.99763 factor, but altering it for different zoom scales.

gerth6000 · Aug 16, 2012

I used the factor because I thought it was a consistant deviation. What I found is the deviation was different on small and large orbits, not really related to the zoom. If i match one of the outer planets, the inner planets get thrown off orbit.

gerth6000 · Aug 16, 2012

Of course this could mean that the deviation is a quadratic function of the circle radius....

klaus · Aug 17, 2012

I would suggest following:
- can you reduce the number of planets to draw let's say two green planets.
- in the orbit drawing routine draw also every one or two degrees a small dot to see if there is a difference in position between the circle function and sin/cos drawing.

Best regards.

gerth6000 · Aug 18, 2012

I haven't tried your suggestion yet, Klaus, but I will.

I took screenshots of various planets so I could measure the deviation in pixels and calculate how much.

The data shows no correlation between the circle's radius and the amount of deviation.

gerth6000 · Aug 18, 2012

Here is your "version", Klaus.

At every pi/4 (45 degrees) the planet meets the orbit. In between the planets form a circle-arc with a radius smaller than the orbit.

So one of the two sets is not a perfect circle.

EDIT: I *guees* I could measure the deviation at pi/8 and assume the deviation is linearly progressive between 0 and pi/8 and pi/4.... but to do a correction such as this is definately not pretty programming

mc73 · Aug 18, 2012

Playing a bit with the Taylor's cos series, I've noticed some very tiny deviation. Could this help?

B4X:

'Activity module
Sub Process_Globals
    'These global variables will be declared once when the application starts.
    'These variables can be accessed from all modules.

End Sub

Sub Globals
    'These global variables will be redeclared each time the activity is created.
    'These variables can only be accessed from this module.
Dim lv As ListView 
End Sub

Sub Activity_Create(FirstTime As Boolean)
lv.Initialize ("lv")
Dim maxD As Double
maxD=0
For k=0 To 359
Dim a,b,curD As Double 
a=CosD(k)
b=cosd2(k)
curD=a/b
lv.AddSingleLine (k & "/" & a & " / " & b & "/" & curD)
maxD=Max(maxD,curD)
Next
Activity.AddView (lv,0,0,Activity.Width,Activity.Height )
Msgbox(maxD,"maxD")
End Sub

Sub Activity_Resume

End Sub

Sub Activity_Pause (UserClosed As Boolean)

End Sub

Sub cosd2(degrees As Double) As Double 
Dim n As Int 
Dim sumTaylor,x As Double 
x=cPI*degrees/180
sumTaylor=0
For n=0 To 20
    Dim nf As Long
    nf=1
    If n>0 Then
        For i=1 To 2*n
            nf=nf*i
        Next
    End If
    sumTaylor=sumTaylor+Power(-1,n)*Power(x,2*n)/nf
Next
Return sumTaylor
End Sub
Sub lv_itemclick(position As Int,value As Object)
Msgbox(value,"ok")
End Sub

klaus · Aug 18, 2012

I tried the test version you posted but it doesn't draw any orbit nor dots?
But from what you are telling I'm afraid that it's the circle drawing routine that is not accurate.

Best regards.

gerth6000 · Aug 18, 2012

Zoom in on the white star (dot) and they appear, it still has a minimum size before it draws the orbit.

High precision trigonometry?

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