By writing the momentum equation in the system of normal coordinates (in the direction of flow and perpendicular to the flow direction) for a volume control of the fluid, the equations (4) and (5) can be obtained:
To establish relations between the equations of flow and the spillway geometry, the normal coordinates system of the flow is expressed as Cartesian coordinate system.
The position of vertex v in the fisheye view is a function of its position in normal coordinates and the position of the focus f:
The size of a vertex in the fisheye view is a function of its size and position in normal coordinates, the position of the focus, and the vertex's API:
Finally, the visual worth of a vertex depends on the distance between the vertex and the focus in normal coordinates (found by examining the positions of the vertex and the focus in normal coordinates) and on the vertex's API:
norm] from normal coordinates to fisheye coordinates, using focus position [P.
x] is the horizontal distance between the point being transformed and the focus, also in normal coordinates.