### A Note on Non-Rectangular Boundaries ### Most of the videos on this webpage are solutions to the 2+1 wave equation with u(x,y,t) fixed at zero along the domain of the domain of discourse. In many cases, the domain of discource is not a rectangle. In such a case, it is sufficient to write code as if one was working on a rectangular domain, (this rect domain must be chosen so that it contains the smaller non-rect domain) and then include a subroutine in your code that determines if a point within your rectangular domain is also within your non-rectangular domain. if-else statements should then be added to the code so that only points determined by the subroutine to be within the non-rectangular domain participate in the main body of the code. An unfortunate effect of solving things this way is that the whole enclosing rectangular domain gets shipped to the video display program, not just the non rectangular subset. Consequently, in all videos featuring a non rectangular domain, the true boundary is actually invisible, and the boundary displayed is that of a rectangle. One may try changing coordinates from (x,y)->(u,v) in such a way that the curves u(x,y)=constant and v(x,y)=constant lie on the boundary of the domain of discourse. The modified wave equation in the new coordinate system may then be solved numerically on a rectangular domain in u,v and the results converted back into x,y coordinates. This helps cut down error and also allows the displayed boundary to coincide with the true boundary but in my opinion is alot more trouble than it's worth which is why I haven't bothered doing it.