Tile graphics hardware with sprites, like that on the NES, was designed to make scrolling color video-game graphics achievable in 1983 at a reasonable cost. But it also makes certain kinds of visual effects fairly easy to achieve, like wallpapering an area with a repeating pattern or scrolling or globally changing the appearance of a certain kind of tile. Other effects, like changing the on-screen size of anything, are harder.
Sprites themselves are an affordance for making moving objects on a static background; without hardware sprites, given mutable tiles, you could always just use four tile types per sprite, compositing the sprite in the appropriate place in each tile type each frame. I saw MS-DOS text-mode programs that did this for the mouse pointer.
Given enough tile definitions to have a separate definition for each tile on the screen, and enough colors that you can
Nowadays our affordances run more to gradients and alpha-blending, although to some extent that’s only because our program output doesn’t go through a video codec, which has different affordances.
There’s a variety of ways you can express the space of polynomial functions.
You can express a degree-N polynomial as N+1 coefficients, which is convenient for many calculations but doesn’t make it particularly easy to achieve any given effect.
Given some k, you can express it as N+1 coefficients of a polynomial in (x-k). If k happens to be one of the zeroes of the polynomial, you only need N coefficients, since the constant term is zero.
Extending this idea to its limit, you can express it as a scale factor and a set of zeroes: 3(x-1)(x+2)(x+5) has a scale factor of 3 and zeroes at 1, -2, and -5; its representation as zeroes is thus [3, 1, -2, -5]. It works out to 3x³ + 18x² + 9x - 30, so its representation as coefficients is [3, 18, 9, -30], although sometimes people prefer to write that in the other order. (Transforming from zeroes to coefficients is easy; going the other way can be hairy.)
You can express it with its values at some given abscissas and then use Lagrange interpolation. For a given set of abscissas, such as the nonnegative integers, you have a constant set of Lagrange basis polynomials; the polynomial is a weighted sum of its values at these points, which is to say that the transformations between this Lagrange form and the form as coefficients are linear. But the Lagrange form allows you to express the polynomial function in terms of its value at some given points, which is often more convenient.
That of course leaves open the question of the set of abscissas to choose. Sometimes it’s more expressive to be able to choose which abscissas to use, but this is somewhat dangerous with Lagrange interpolation, just as with splines of orders greater than 3; a bad choice of nodes can easily lead to unwanted oscillations (the Runge phenomenon). If you’re approximating some existing smooth curve, using the Chebyshev nodes minimizes this.
The Chebyshev polynomials form another orthogonal basis into which you can linearly transform a polynomial.
For a given step size and starting point, you can express the polynomial in terms of the initial state of the method of divided differences for tabulating values of the polynomial. These values are a linear transform of the coefficients and thus also of the Lagrange form; you can easily derive them by calculating some values of the polynomial and writing a difference table. They are called the “Newton form” of the polynomial. The Newton form is in some sense not very expressive, in that the numbers in it don’t correspond very directly to any interesting feature of the polynomial itself. However, it’s very convenient for calculation.
Order-n polynomials on the real numbers (or even the rationals, or any dense set) are entirely determined by their value and first n derivatives at any arbitrary point; this is the basis of the Taylor-series approximation of a function, which is the analytic equivalent of the Newton form. A polynomial function is equal to its Taylor series as soon as the Taylor series’ order equals or exceeds that of the original polynomial. So the derivatives at some given point are yet another representation of the polynomial, another one which is linearly related to the coefficients.
It isn’t necessary for all the derivatives to be at the same point; it’s adequate to specify N+1 values or derivatives as long as no two of them are the same.
You can specify state machines as regular expressions...