Monday Math: Forms
Aug. 27th, 2012 07:39 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowWe're ready to set up the functions that will allow us to define our new classes of polygons. Under the cut.
We've defined the z- and w-functions, and looked at the effects of translations and of shifting the initial vertex on those functions. We still have to worry about the effects of a few more kinds of transformation. First, what happens if we rotate our polygon, through (say) a counterclockwise angle of t? (Note: we measure angles in radians, as always in advanced math.) The effect on one of the z-functions is simple: it is multiplied by eit=cos(t) + i sin(t). The w-functions are the complex conjugates of the z-functions, and the effect of rotation through t on one of them is multiplication by the conjugate, e-it=cos(t) - i sin(t). It follows that, for example, rotation has no effect on the product of a z-function with a w-function; more generally, if we have a product of z- and w-functions, with just as many of the former as of the latter, rotation doesn't affect it.
What about scaling the polygon by a factor of some real number a? All of the functions, z- and w- alike, are multiplied by a. A product of k z-functions with the same number of w-functions is, therefore, multiplied by a2k.
There remain two more operations: reflecting the polygon, and reversing the order of the vertices. The latter is easy; in the case of quadrilaterals, z1 and z3 exchange places, and z2 is left alone. Note that 1+3=2+2=4; in general, if we're working with n-gons, zk swaps places with zn-k, and likewise for the w-functions.
Reflection is a little more complicated; it turns out that it interchanges zk with wn-k.
All of the above leads to the following. Let k be any positive integer. The "space of k-forms" is the vector space of polynomials in the z- and w-functions, in which every monomial is the product of k z-functions and k w-functions; and it is in these spaces that the functions defining our classes of polygons are to be found.
As an example, again with quadrilaterals: the space of 1-forms is generated by the nine monomials zrws, where r and s take any of the values 1,2,3. (The space of 1-forms for triangles is four-dimensional; for pentagons, 16-dimensional; and so forth.)
Next time, we'll actually start identifying classes of quadrilaterals. Stay tuned.
We've defined the z- and w-functions, and looked at the effects of translations and of shifting the initial vertex on those functions. We still have to worry about the effects of a few more kinds of transformation. First, what happens if we rotate our polygon, through (say) a counterclockwise angle of t? (Note: we measure angles in radians, as always in advanced math.) The effect on one of the z-functions is simple: it is multiplied by eit=cos(t) + i sin(t). The w-functions are the complex conjugates of the z-functions, and the effect of rotation through t on one of them is multiplication by the conjugate, e-it=cos(t) - i sin(t). It follows that, for example, rotation has no effect on the product of a z-function with a w-function; more generally, if we have a product of z- and w-functions, with just as many of the former as of the latter, rotation doesn't affect it.
What about scaling the polygon by a factor of some real number a? All of the functions, z- and w- alike, are multiplied by a. A product of k z-functions with the same number of w-functions is, therefore, multiplied by a2k.
There remain two more operations: reflecting the polygon, and reversing the order of the vertices. The latter is easy; in the case of quadrilaterals, z1 and z3 exchange places, and z2 is left alone. Note that 1+3=2+2=4; in general, if we're working with n-gons, zk swaps places with zn-k, and likewise for the w-functions.
Reflection is a little more complicated; it turns out that it interchanges zk with wn-k.
All of the above leads to the following. Let k be any positive integer. The "space of k-forms" is the vector space of polynomials in the z- and w-functions, in which every monomial is the product of k z-functions and k w-functions; and it is in these spaces that the functions defining our classes of polygons are to be found.
As an example, again with quadrilaterals: the space of 1-forms is generated by the nine monomials zrws, where r and s take any of the values 1,2,3. (The space of 1-forms for triangles is four-dimensional; for pentagons, 16-dimensional; and so forth.)
Next time, we'll actually start identifying classes of quadrilaterals. Stay tuned.
