This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure we study is the set of trajectories obtained by integrating this vector field, which we call P -trajectories. In order to study P -trajectories we also study a related vector field on the linear programming polytope, which we call the affine scaling vector field, and its associated trajectories, called A -trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. These affine and projective scaling vector fields are each defined for liner programs of a special form, called strict standard form and canonical form, respectively. This paper defines and presents basic properties of P -trajectories and A -trajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for P -trajectories and A -trajectories. It presents Karmarkar's interpretation of A - trajectories as steepest descent paths of the objective function ⟨c, x⟩ with respect to the Riemannian geometry ds 2 = c = 1 Σ n