  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 illustrate the value of our braid calculus by performing
with pictures a nontrivial calculation in the Artin groups of type D_n.
