Binary Degrees

[Jeremy Leach]

This is just a little alternative way of working with angles and Sin and Cos....

I'm working with vectors and typically you have to work out the X and Y components of a Vector, where X = VSinA and Y = VCosA

On the picaxes that support Sin and Cos, the angle argument is in degrees. However, code-wise I feel it's far easier to work in 0 to 256 ('binary degrees')instead of 0 to 360. If you are increasing or decreasing an angle it will 'wrap around' simply using binary degrees.

Also, to get best accuracy when performing vector maths, I think it's handy if Sin and Cos are expressed as N/255, where N is a number from a lookup table.

So, for instance, X = VSinA = V * N /255

If V is a byte value and N is a byte value then this calculation will never overflow in picaxe maths, but it also gets maximum accuracy without getting into 32 bit maths etc.

I call these N values Sin255 and Cos255.

Here's a 64 byte EEPROM lookup for the Sin255 values. (0 to 64 binary degrees = 0 to 90 normal degrees).

EEPROM 0,(0,6,13,19,25,31,37,44,50,56,62,68,74,80,86,92,98,103,109,115,120,126,131,136,142,147,152,157,162,
167,171,176,180,185,189,193,197,201,205,208,212,215,219,222,225,228,231,233,236,238,240,242,244,246,247,
249,250,251,252,253,254,254,255,255,255)

So to obtain the Sin255 of a binary angle, just do:

Read BinAngle,Sin255

In degrees, SinA = Cos(90-A)
So in binary degrees, SinA = Cos(64 - A)

So to obtain the Cos of a binary angle, just do:

Temp = 64 - BinAngle
Read Temp,Cos255

Example:

BinAngle = 20
Vector = 88
 
Read BinAngle,Sin255
Temp = 64 - BinAngle
Read Temp,Cos255
 
Xcomponent = 88 * Sin255 /255
Ycomponent = 88 * Cos255/255

It's just my take on things though !

 

[BeanieBots]

I think you make a very valid point Jez.
Many compass modules already offer a "256 degree circle" for much the same reasoning.

 

[Jeremy Leach]

Yep, it's obvious really, but thought I'd make the point Sometimes we just get so used to numbering systems we forget that they are all invented and we can use whatever is best to get the job done !

Actually, I'm going one further in my own Vector maths ...

Instead of Sin255, I'm using Sin65536 - with a 128 byte lookup table (64 words).

So XComponent = V * N/65536
Where N is the lookup value Word, and V is a Word.

V*N can obviously overflow, but if we think of the result as high and low word:
V*N = (MSW * 65536) + LSW
Therefore V*N/65536 = MSW + (LSW/65536) = MSW (whole part only)

So in code...

CalcVectorComponents:
    Index = BinAngle * 2
    Read Index,Word Sin65536
    Index= 128 - Index
    Read Index, Word Cos65536
    XComponent = V ** Sin65536 
    YComponent = V ** Cos65536
Return

One point about this method though .... N should be 65536 for 90 degrees (64 binary degrees). But we can't store 65536 in a word, so the lookup table sets this to 65535 instead. The error in the calculated component is insignificant though (actually = V/65536 which is always <1).

If anyone's interested, here's the 128 byte EEPROM lookup table of Sin65536 values, in LSW MSW pairs:

EEPROM 0,(0,0,72,6,144,12,213,18,24,25,86,31,144,37,196,43,241,49,23,56,52,62,71,68,80,74,77,80,62,86,34,92,248,97,190,103,
116,109,26,115,173,120,47,126,156,131,246,136,58,142,104,147,128,152,128,157,104,162,54,167,235,171,134,176,5,
181,104,185,175,189,216,193,228,197,209,201,159,205,77,209,219,212,72,216,148,219,190,222,198,225,170,228,108,
231,10,234,131,236,217,238,9,241,20,243,250,244,186,246,84,248,200,249,21,251,59,252,59,253,19,254,196,254,78,255,
177,255,236,255,255,255)