This paper investigates the kinematics of a fixed point on the connecting bar of an ellipsograph (Archimedes' trammel) under different driving motion regimes: uniform, uniformly accelerated, and Keplerian (elliptical). Based on differential constraint equations, analytical expressions for the velocity and acceleration vectors of the trajectory point are derived. A numerical approach, modeling the division of the orbital quadrant into equal time intervals, is utilized to calculate the corresponding areal velocities. It is rigorously demonstrated that the constancy of the areal velocity and the fulfillment of Kepler's second law occur exclusively under one specific law of angular velocity—the Keplerian regime—whereas under uniform and uniformly accelerated motions, the law of areas is violated. This work establishes a link between the mechanical modeling of trajectories and the author's fundamental theoretical research in the focal and central kinematics of the ellipse.