The s-plane and the z-plane are related by a *conformal mapping* specified
by the analytic complex function

where

The mapping is continuous, i.e., neighboring points in s-plane are mapped to neighboring points in z-plane and vice versa. Consider the mapping of these specific features:

- The origin of s-plane is mapped to on the real axis in z-plane.
- Each vertical line
in s-plane is mapped to a circle
centered about the origin in z-plane. In particular,
- Leftmost vertical line is mapped as the origin
- Imaginary axis is mapped as the unit circle
- Rightmost vertical line is mapped as a circle of infinite radius .

- Each horizontal line in s-plane is mapped to , a ray from the origin in z-plane of angle with respect to the positive horizontal direction.
- A right angle formed by a pair vertical and horizontal lines in s-plane is conserved by the mapping, as the corresponding circle and ray in z-plane also form a right angle. (In fact any angle is conserved, an important property of the conformal mapping.)