Braid pictures for Artin groups

We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams $A_n$, $B_n=C_n$ and $D_n$ and the affine diagrams $\tilde{A}_n$, $\tilde{B}_n$, $\tilde{C}_n$ and $\tilde{D}_n$ as subgroups of the braid groups of various simple orbifolds. The cases $D_n$, $\tilde{B}_n$, $\tilde{C}_n$ and $\tilde{D}_n$ are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except $\tilde{A}_n$ the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type $D_n$.