  We develop a new method to construct explicit, regular minimal surfaces in
Euclidean space that are defined on the entire complex plane with controlled
geometry. More precisely we show that for a large class of planar curves
$(x(t), y(t))$ one can find a third coordinate $z(t)$ and normal fields $n(t)$
along the space curve $c(t)=(x(t), y(t), z(t))$ so that the Bj\"orling formula
applied to $c(t)$ and $n(t)$ can be explicitly evaluated. We give many
examples.
