We give a detailed description of the geometry of isotropic space, in
parallel to those of Euclidean space within the realm of Laguerre geometry.
After developing basic surface theory in isotropic space, we define spin
transformations, directly leading to the spinor representation of conformal
surfaces in isotropic space. As an application, we obtain the Weierstrass-type
representation for zero mean curvature surfaces, and the Kenmotsu-type
representation for constant mean curvature surfaces, allowing us to construct
many explicit examples.Comment: 30 pages, 9 figure