All great questions! Lets start at the top...
even for todays computers?
Most of today's general purpose
computers (laptops and desktops) have intrinsic trigonometry in their processors. Having said that, there are some issues with x86's trig and Java which I'll hold til the end 'cause it complciates the picture slightly.
However today's computers are in more then just desktops and laptops, they are everywhere. In game consoles, in cell phones, in your pocket, in your set top box, your NAT gateway, and so on. Which leads nciely to the next question...
What's an embedded processor?
An embedded processor is one that is intended for use in an embedded application. That doesn't help much as a definition, I know, but ist a starting point. Embedded applications are all those devices that aren't desktops and laptops. They have very differnt needs. A desktop, and to a lesser degree a laptop, is almost all about processing speed. These other areas though have other over-riding needs such as low power consumption or low processor cost. As a result, embedded processors are generally not as fast or as capable as the general purpose processors.
Most processors intended for embedded work do *not* have intrinsic trig and do it all in software.
How do you profile your application?
With a profiler! A profiler is a program that watches your program run and gives you a detailed report on what parts are using what percentage of the processing time, what parts are generating what memory usage, etc. A good profiler is an essential Java performance coding tool. Both Eclipse and Netbeans now have free Profilers you can download that plug right into the IDE. The Netbeans one is quite good, almost as good as some very expensive commercial tools I've used in the past. I haven't tried the Eclipse one yet so I can't comment on it..
You can find a whole chapter on profilign in Steve and my performacne book. The book is a bit old and some of the "tactical" stuff is a bit outdated but the "strategy" section is still quite appropriate to today's VMs. (As are msot of the tctics. The biggestchnage is that we no longer have to be as afraid of creating garbage collection runs as we used to be.)
It's all on line here:http://java.sun.com/docs/books/performance
The profiler chapter is here:http://java.sun.com/docs/books/performance/1st_edition/html/JPMeasurement.fm.html#30000
I strongly recommend yo uread the entire measurmeent section though, not just the Profiler section. People new to Java often get themselves in trouble trying to write "microbenchmarks" and the section talks about that, too.
What is meant by linear "interpolation" in the context used here?
When you say a trig-lookup table do you mean some sort of an array of the values of sin, cos precalculated?
Okay, Il lanswer these together and yes, yo uare on the right tarck here, you create an array of precaculated sin values (or cos, either will work.)
You only need one because Sin and Cos are the same with a phase shift (+/- 90 degrees).
On top of that, they repeat every 180 degrees, so you can do a mod(180) and only have 180 dgrees in your table.
You can actually get fancier then that by rcognizing that the table refelcts around 90 degrees so IF180>=N>=90, sin(n) == sin(180-n), tha reduces your table size to 90 degrees. Finally, it reflects again around 45, so IF 90>=N>=45, sin(n) == sin (90-n)
These tables can be a useful aid in gorkking all this: http://www.math2.org/math/trig/tables.htm
The end result of all this, though, is all you need is a 45 degree table, some IFs and some addition or subtraction (and one mod) to get the Sin of any number.
SO lets say you build a table with 45 entries, one for each degree. How do you get the values inj betweem, say sin (1.75). The answer is interpolation. In general a simple linear interpolation will suffice for the accuracy required for most games. A linear interpolation betwene two points is just a weighted average,
So lets say our SIn array is called "sinarray", sin(1.75) == ((2-1.75)*sinarray)+((1.75-1)*sinarray))
or in the general case, assuming a 45 degree table with 45 entries and 45>=n>=0
long highval = Math.ceiling(n);
long lowval = Math.floor(n);
long result = (((highval-n)*sinarray[highval)+((n-lowval)*sinarray[lowval]))Note that I am an old guy and think in "degrees". Java's trig is actually in radians so if you want to mimic it you will have to do conversions
or arrange your table in radians rather then degrees.
NOW I said Java compilcates things. The long explaination is here:http://blogs.sun.com/roller/page/jag?entry=transcendental_meditation
But the short hand is that, for angles under -45 or over 45 degrees, at the accuracy that Java by definition must calculate flaoting point math, the intrinsic trig in the x86 processor is broken
. For that reason Java defaults to a software implementastion outside of that range BUT using the reflection tricks I just showed you, you should never have to leave that range