In computer graphics, trajectory planning, and vector fields, constraining degrees of freedom to exactly two custom control points (plus two fixed boundary anchors) produces the canonical 4-point Cubic Hermite/Bézier formulation: $B(t) = (1-t)^3 P_0 + 3(1-t)^2 t P_1 + 3(1-t)t^2 P_2 + t^3 P_3$.
Two custom internal points allow independent control over the initial departure angle and the terminal arrival velocity, guaranteeing continuous first and second derivatives across transitions.
Treating the two custom points as electrical or gravitational dual-charges creates an instantaneous potential gradient $\nabla \Phi(x,y)$, visualizing how dual influence regions shape physical force flows.
Robotic end-effectors and UAV guidance algorithms utilize two customizable intermediate tangent points to compute smooth deceleration, obstacle avoidance windows, and jerk-minimized profiles.