A Geometric Gaussian Mixture Representation of Curves and Trajectories in Euclidean Space

Curves and trajectories in $\mathbb{R}^d$ are basic objects in tracking and sensor data fusion. \acp{GMM} are widely used as flexible parametric \acp{PDF} over the object state space. \acp{GMM} are usually obtained by applying \ac{EM}-based schemes to point samples. When the underlying geometric structure is known deterministically - for example a road centerline in map-assisted tracking, a nominal flight path, or a planned robot trajectory - such iterative fitting is unnecessary, can be slow, and only indirectly reflects the curve or trajectory.

We propose a purely arithmetic, geometry-driven construction that maps a curve or trajectory in $\mathbb{R}^d$ with user-defined transverse uncertainty directly to a \ac{GMM} in a single pass, without \ac{EM} or any iterative optimization.

Starting from a parametrized curve $\alpha \colon I \to \mathbb{R}^d$, we form a \ac{UDPP} and assign each segment a normal-subspace standard deviation, obtaining thin curve-aligned primitives encoding position, tangent, normal, and length scale together with transverse uncertainty. Choosing component weights proportional to segment length distributes probability uniformly over arc length, so all \ac{GMM} parameters follow directly from the discretized curve and uncertainty profile.
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The construction applies to smooth, open, closed, non-regular, and self-intersecting curves and trajectories in $\mathbb{R}^d$. It supports adaptive discretization and spatially varying uncertainty and yields differentiable \acp{GMM} that directly encode geometric structure. We demonstrate the approach on canonical planar curves and nominal trajectories and include a simulation study in higher-dimensional Euclidean spaces comparing approximation quality and runtime against \ac{EM}-based fitting. Applications include map- and road-constrained tracking and sensor data fusion, uncertainty-aware trajectory planning, probabilistic obstacle and boundary modeling in robotics, and uncertainty-aware CAD and digital twins.