Maths in Programing Video Games

Question

I've just finished second year at Uni doing a games course, this is always been bugging me how math and game programming are related. Up until now I've been using Vectors, Matrices, and Quaternions in games, I can under stand how these fit into games.

This is a General Question about the relationship between Maths and Programming for Real Time Graphics, I'm curious on how dynamic the maths is. Is it a case where all the formulas and derivatives are predefined(semi defined)?

Is it even feasible to calculate derivatives/integrals in realtime?

These are some of things I don't see how they fit inside programming/maths As an example.

1. MacLaurin/Talor Series I can see this is useful, but is it the case that you must pass your function and its derivatives, or can you pass it a single function and have it work out the derivatives for you?

   MacLaurin(sin(X)); or MacLaurin(sin(x), cos(x), -sin(x));
   

2. Derivatives /Integrals This is related to the first point. Calculating the y' of a function done dynamically at run time or is this something that is statically done perhaps with variables inside a set function.

   f = derive(x); or f = derivedX;
   

3. Bilnear Patches We learned this as a way to possible generate landscapes in small chunks that could be 'sewen' together, is this something that happens in games? I've never heard of this (granted my knowlages is very limited) being used with procedural methods or otherwise. What I've done so far involves arrays for vertex information being processesed.

Sorry if this is off topic, but the community here seems spot on, on this kinda thing.

Thanks.

Answer 1

Skizz's answer is true when taken literally, but only a small change is required to make it possible to compute the derivative of a C++ function. We modify skizz's function f to

template<class Float> f (Float x)
{
  return x * x + Float(4.0f) * x + Float(6.0f); // f(x) = x^2 + 4x + 6
}

It is now possible to write a C++ function to compute the derivative of f with respect to x. Here is a complete self-contained program to compute the derivative of f. It is exact (to machine precision) as it's not using an inaccurate method like finite differences. I explain how it works in a paper I wrote. It generalises to higher derivatives. Note that much of the work is done statically by the compiler. If you turn up optimization, and your compiler inlines decently, it should be as fast as anything you could write by hand for simple functions. (Sometimes faster! In particular, it's quite good at amortising the cost of computing f and f' simultaneously because it makes common subexpression elimination easier for the compiler to spot than if you write separate functions for f and f'.)

using namespace std;

template<class Float>
Float f(Float x)
{
  return x * x + Float(4.0f) * x + Float(6.0f);
}

struct D
{
  D(float x0, float dx0 = 0) : x(x0), dx(dx0) { }
  float x, dx;
};

D operator+(const D &a, const D &b)
{
  // The rule for the sum of two functions.
  return D(a.x+b.x, a.dx+b.dx);
}

D operator*(const D &a, const D &b)
{
  // The usual Leibniz product rule.
  return D(a.x*b.x, a.x*b.dx+a.dx*b.x);
}

// Here's the function skizz said you couldn't write.
float d(D (*f)(D), float x) {
  return f(D(x, 1.0f)).dx;
}

int main()
{
  cout << f(0) << endl;
  // We can't just take the address of f. We need to say which instance of the
  // template we need. In this case, f<D>.
  cout << d(&f<D>, 0.0f) << endl;
}

It prints the results 6 and 4 as you should expect. Try other functions f. A nice exercise is to try working out the rules to allow subtraction, division, trig functions etc.

Answer 2

2) Derivatives and integrals are usually not computed on large data sets in real time, its too expensive. Instead they are precomputed. For example (at the top of my head) to render a single scatter media Bo Sun et al. use their "airlight model" which consists of a lot of algebraic shortcuts to get a precomputed lookup table.

3) Streaming large data sets is a big topic, especially in terrain.

A lot of the maths you will encounter in games is to solve very specific problems, and is usually kept simple. Linear algebra is used far more than any calculus. In Graphics (I like this the most) a lot of the algorithms come from research done in academia, and then they are modified for speed by game programmers: although even academic research makes speed their goal these days.

I recommend the two books Real time collision detection and Real time rendering, which contain the guts of most of the maths and concepts used in game engine programming.

Answer 3

I think there's a fundamental problem with your understanding of the C++ language itself. Functions in C++ are not the same as mathmatical functions. So, in C++, you could define a function (which I will now call methods to avoid confusion) to implement a mathmatical function:

float f (float x)
{
  return x * x + 4.0f * x + 6.0f; // f(x) = x^2 + 4x + 6
}

In C++, there is no way to do anything with the method f other than to get the value of f(x) for a given x. The mathmatical function f(x) can be transformed quite easily, f'(x) for example, which in the example above is f'(x) = 2x + 4. To do this in C++ you'd need to define a method df (x):

float df (float x)
{
  return 2.0f * x + 4.0f; //  f'(x) = 2x + 4
}

you can't do this:

get_derivative (f(x));

and have the method get_derivative transform the method f(x) for you.

Also, you would have to ensure that when you wanted the derivative of f that you call the method df. If you called the method for the derivative of g by accident, your results would be wrong.

We can, however, approximate the derivative of f(x) for a given x:

float d (float (*f) (float x), x) // pass a pointer to the method f and the value x
{
  const float epsilon = a small value;
  float dy = f(x+epsilon/2.0f) - f(x-epsilon/2.0f);
  return epsilon / dy;
}

but this is very unstable and quite inaccurate.

Now, in C++ you can create a class to help here:

class Function
{
public:
  virtual float f (float x) = 0; // f(x)
  virtual float df (float x) = 0; // f'(x)
  virtual float ddf (float x) = 0; // f''(x)
  // if you wanted further transformations you'd need to add methods for them
};

and create our specific mathmatical function:

class ExampleFunction : Function
{
  float f (float x) { return x * x + 4.0f * x + 6.0f; } // f(x) = x^2 + 4x + 6 
  float df (float x) { return 2.0f * x + 4.0f; } //  f'(x) = 2x + 4
  float ddf (float x) { return 2.0f; } //  f''(x) = 2
};

and pass an instance of this class to a series expansion routine:

float Series (Function &f, float x)
{
   return f.f (x) + f.df (x) + f.ddf (x); // series = f(x) + f'(x) + f''(x)
}

but, we're still having to create a method for the function's derivative ourselves, but at least we're not going to accidentally call the wrong one.

Now, as others have stated, games tend to favour speed, so a lot of the maths is simplified: interpolation, pre-computed tables, etc.

Answer 4

Most of the maths in games is designed to to as cheap to calculate as possible, trading speed over accuracy. For example, much of the number crunching uses integers or single-precision floats rather than doubles.

Not sure about your specific examples, but if you can define a cheap (to calculate) formula for a derivative beforehand, then that is preferable to calculating things on the fly.

Answer 5

In games, performance is paramount. You won't find anything that's done dynamically when it could be done statically, unless it leads to a notable increase in visual fidelity.

Answer 6

You might be interested in compile time symbolic differentiation. This can (in principle) be done with c++ templates. No idea as to whether games do this in practice (symbolic differentiation might be too expensive to program right and such extensive template use might be too expensive in compile time, I have no idea).

However, I thought that you might find the discussion of this topic interesting. Googling "c++ template symbolic derivative" gives a few articles.

Answer 7

There's many great answers if you are interested in symbolic calculation and computation of derivatives.

However, just as a sanity check, this kind of symbolic (analytical) calculus isn't practical to do at real time in the context of games.

In my experience (which is more 3D geometry in computer vision than games), most of the calculus and math in 3D geometry comes in by way of computing things offline ahead of time and then coding to implement this math. It's very seldom that you'll need to symbolically compute things on the fly and then get on-the-fly analytical formulae this way.

Can any game programmers verify?

Answer 8

1), 2)

MacLaurin/Taylor series (1) are constructed from derivatives (2) in any case.

Yes, you are unlikely to need to symbolically compute any of these at run-time - but for sure user207442's answer is great if you need it.

What you do find is that you need to perform a mathematical calculation and that you need to do it in reasonable time, or sometimes very fast. To do this, even if you re-use other's solutions, you will need to understand basic analysis.

If you do have to solve the problem yourself, the upside is that you often only need an approximate answer. This means that, for example, a series type expansion may well allow you to reduce a complex function to a simple linear or quadratic, which will be very fast.

For integrals, the you can often compute the result numerically, but it will always be much slower than an analytic solution. The difference may well be the difference between being practical or not.

In short: Yes, you need to learn the maths, but in order to write the program rather than have the program do it for you.