The final equation for the x and y of each corner has six terms, combining t, sin(n+mt), cos(n+mt), t*sin(n+mt), t*cos(n+mt), and a constant.

Why are the arguments of since and cosine linear? I'm not seeing why you'd have that just for applying a rotation.

So, most people know what a polynomial is, usually in terms of polynomial functions. E.g., something like f(x) = 5x

^{3}+ x^{2}- 8 or something like that. Here, x is taken to be a real number that can freely vary, so you end up with a function.But something else you can also do is treat x as an

indeterminate, that is as essentially a label. And in that case, x^{0}, x1, x^{2}, etc. are all distinct indeterminates, i.e. distinct labels. Thus, you can just consider arbitrary elements like A = a_{0}x^{0}+ a_{1}x^{1}+ a_{2}x^{2}+ a_{3}x^{3}+ ... + a_{n}x^{n}+ ... for real numbers a_{0}, a_{1}, a_{2}, a_{3}, ..., a_{n}, ....This leads to a natural generalization of addition. Just like when you have f(x) = x

^{3}+ 2 and g(x) = 3x^{3}+ 4x - 1 it makes sense to have (f+g)(x) = 4x^{3}+ 4x + 1 (that is, adding terms with matching powers of x), you can do likewise with these "infinite polynomials": (a_{0}+ a_{1}x^{1}+ a_{2}x^{2}+ a_{3}x^{3}+ ... + a_{n}x^{n}+ ... ) + (b_{0}+ b_{1}x^{1}+ b_{2}x^{2}+ b_{3}x^{3}+ ... + b_{n}x^{n}+ ... ) = (a_{0}+ b_{0}) + (a_{1}+ b_{1})x^{1}+ (a_{2}+ b_{2})x^{2}+ (a_{3}+ b_{3})x^{3}+ ... + (a_{n}+ b_{n})x^{n}+ ... and it works very similar to addition of, say, real numbers or integers or etc (though, because of it being infinite dimensional, you have to be careful about certain things).Likewise, just like for things like f(x) = 2x

^{2}and g(x) = x + 1, you can define (fg)(x) = 2x^{3}+ 2x^{2}by multiplying each term and adding the powers of the x terms, you can likewise do for polynomials: (a_{0}x^{0}+ a_{1}x^{1}+ ... + a_{n}x^{n}+ ...)(b_{0}x^{0}+ b_{1}x^{1}+ ... + b_{n}x^{n}+ ...) = a_{0}b_{0}+ (a_{0}b_{1}+ a_{1}b_{0})x^{1}+ (a_{0}b_{2}+ a_{1}b_{1}+ a_{2}b_{0})x^{2}+ (a_{0}b_{n}+ a_{1}b_{n-1}+ ... + a_{n-1}b_{1}+ a_{n}b_{0})x^{n}+ ... . Once again, it works in a relatively familiar manner (though you still need to be careful about some things when doing it).There are lots of other cool things about polynomials, but this is just a presentation of the algebra of polynomials (more specifically, formal power series) over the real numbers.