Motion, at its best, is not abrupt.
It flows.
From one state to the next, through space, through time, with no harsh turns or sudden breaks.
This is the purpose—and the beauty—of the B-spline formulation.
Where polylines are jagged and polynomials can be wild, B-splines offer something rare: local control with global smoothness.
They are the mathematical shape of grace.
A B-spline—short for basis spline—is a piecewise-defined curve, constructed from a set of control points and a knot vector.
These control points do not lie exactly on the curve, but shape its flow.
Each segment is influenced by a small number of points, giving designers local control without disrupting the entire trajectory.
Formally, a B-spline is defined as:
C(t) = Σ Nᵢ,ₖ(t) · Pᵢ
Where:
– Pᵢ are the control points,
– Nᵢ,ₖ(t) are the basis functions of degree k, and
– t is the parameter, often representing time or arc length.
The result?
– Continuity across segments—up to second or even third derivative, depending on degree.
– Smooth curvature, ideal for vehicles with turning constraints.
– Compact support, meaning changes to one control point affect only nearby sections.
In motion planning and trajectory design, B-splines are used to:
– Design flyable paths for UAVs and fixed-wing aircraft.
– Ensure smooth acceleration profiles that don’t stress actuators.
– Plan collision-free curves in cluttered environments.
– Allow for interactive refinement, where humans or algorithms can shape a path intuitively.
One of the B-spline’s quiet strengths is its parameterization.
The curve exists in a space between points.
It does not demand exact waypoints—it invites guidance.
It is suggestive, not strict.
And that’s what makes it so powerful in intelligent systems:
– It absorbs uncertainty.
– It adapts to constraints.
– It lets motion bend rather than break.
B-splines can also be optimized.
Their control points can be tuned by solvers to minimize energy, time, curvature, or exposure—making them not just elegant, but efficient.
And when the path must be recalculated—due to wind, obstacle, or error—the B-spline formulation allows for real-time adjustment without redoing the whole plan.
Because sometimes, motion must be strong.
But more often, motion must be wise.
And in those moments, the system needs more than a path.
It needs a curve that knows how to listen to its own shape.
That’s what B-splines offer.
Not just geometry—but grace in design.