Minimum Bounding Ellipses (MBEs)

Updated 17 November 2025

Minimum bounding ellipses (MBEs) are the unique smallest-area ellipses that enclose a given set in ℝ² by optimizing the center and positive-definite shape matrix.
MBEs showcase rich historical foundations and algorithmic developments, including Welzl’s randomized algorithm and polygonal tightening methods, enabling efficient computation.
Their convex formulation and duality with statistical covariance union underpin practical applications in robust estimation, collision avoidance, and data fusion.

A minimum bounding ellipse (MBE), also termed minimum-volume enclosing ellipsoid (MVEE) or minimum enclosing ellipse (MEE), is the unique ellipse of smallest area that contains a given compact set in $\mathbb{R}^2$ , typically finite point sets, polygons, or convex planar regions. Modern formulations express the MBE as a convex optimization problem over the center and shape matrix of the ellipse. The problem has deep historical roots in the extremal geometry studied by Euler, admits duality with statistical covariance union, and is fundamental in robust estimation, collision avoidance, and data fusion.

1. Mathematical Formulation and Properties

The MBE for a point set $S=\{x_i\}_{i=1}^N\subset\mathbb{R}^2$ is the ellipse

$E=\{\,x\in\mathbb{R}^2: (x-c)^\top A\,(x-c) \leq 1\,\}$

where $c\in\mathbb{R}^2$ is the center and $A\in\mathbb{S}^{2}_{++}$ is a symmetric positive-definite shape matrix. The area of $E$ is proportional to $\sqrt{\det A^{-1}}$ .

The standard convex program is: $\begin{aligned} \text{minimize}_{c,\,A\succ 0}\quad & -\ln\det A \ \text{subject to}\quad & (x_i-c)^\top A\,(x_i-c) \leq 1,\quad i=1,\dots,N \end{aligned}$ This log-determinant barrier is strictly convex and ensures a unique solution for any set $S$ not contained in a proper affine subspace. The primal feasibility condition $A\succ 0$ guarantees the non-degeneracy of the ellipse. The convexity of $-\ln\det A$ and linearity of the matrix-inequality constraints allows solution by interior-point methods, with per-iteration cost dominated by the dimension of $A$ ; in $\mathbb{R}^2$ , this is negligible for moderate $N$ .

Notably, the MEE problem possesses a dual interpretation: the dual program maximizes the log-determinant of a weighted covariance matrix subject to probability simplex constraints, related to the Khachiyan algorithm (Euler et al., 16 Sep 2025). The KKT conditions reduce the optimal MBE to supporting at most $k\leq 3$ or $4$ of the original points in the plane as per John’s theorem; these are the extreme or active constraints (Euler et al., 16 Sep 2025).

2. Classical and Historical Foundations

Euler's studies (1770s) on "extremal" ellipses formalized the problem for families of ellipses passing through prescribed points, notably:

Minimal-area ellipse through three noncollinear points (triangle)—the Steiner (or Löwner–John) ellipse;
Minimal-area ellipse through four general points (ellipse pencil);
Minimal-perimeter ellipse enclosing a rectangle.

Euler produced explicit cubic equations for area-minimizing ellipses in the pencil through four points and closed-form area ratios for triangle cases, for instance, proving that the minimal-area ellipse through any triangle has area exactly $(4\pi/3\sqrt{3})$ times the triangle's area and passes through the centroid with tangents parallel to triangle sides (Euler et al., 16 Sep 2025). These findings prefigure the modern theory of maximal volume ellipsoids and late 20th-century convex-optimization techniques.

3. Algorithmic Approaches

Welzl's Randomized Incremental Algorithm

For point-sets in $\mathbb{R}^2$ , Welzl’s randomized expected-linear-time algorithm constructs the MBE via recursive boundary-set inclusion. Pseudocode (Zhang et al., 26 Feb 2024):

function MVEE_Welzl(P, R):
  if P is empty or |R| = 3:
    return trivialEllipse(R)
  pick p uniformly at random from P
  E = MVEE_Welzl(P \ {p}, R)
  if p lies inside E:
    return E
  else
    return MVEE_Welzl(P \ {p}, R ∪ {p})

function ComputeMVEE(P):
  return MVEE_Welzl(P, ∅)

Here, trivialEllipse(R) recovers the unique ellipse with all points of $R$ on its boundary (closed-form for $|R|{\le}3$ ). The expected runtime is $O(N)$ for $N$ points in $\mathbb{R}^2$ (Zhang et al., 26 Feb 2024).

Polygonal Tightening and Convex Programming

For bounding intersections of multiple ellipses or general convex sets, Yousefi et al. (Yousefi et al., 2017) developed a polygonal outer-approximation followed by convex minimization. The procedure consists of:

Computing all distinct intersections of ellipse boundaries and pruning those not inside all the ellipses;
Sampling on the arc between each pair of adjacent intersection points and assigning arcs to their generating ellipse via inversion to unit-disk;
For each sample, constructing tangent support half-planes and then assembling their pairwise intersections to form the polygon;
Solving the smallest-area ellipsoid enclosing the polygon’s vertices as an SDP:

$\text{minimize}\quad\log\det A^{-1}\quad\text{subject to}\quad(v_i-c)^{\top}A^{-1}(v_i-c)\le1,\quad\forall i$

Having $O(m)$ SDPs of dimension $3\times3$ , feasible for $m\sim 10^2$ .

Empirical comparison in (Yousefi et al., 2017) demonstrates that this approach outperforms S-procedure LMI and two-stage expansions in tightness, with only modest computational overhead.

4. Theoretical Equivalence and Connections

The equivalence between the geometric MEE and the General Covariance Union (GCU) is established via linear matrix inequalities (LMIs) (Bochardt et al., 2010). Given mean–covariance pairs $(a_i,A_i)$ , the GCU seeks $U\succeq A_i + (u-a_i)(u-a_i)^T$ , so that the “output” ellipse $\mathcal{E}(u,U)$ contains all input ellipsoids $\mathcal{E}(a_i,A_i)$ . When $A_i=0$ , the containment conditions reduce to the MEE over the points $a_i$ . This establishes a duality between robust geometric estimation and statistical covariance combination.

The duality extends to broader intersections: the smallest ellipsoid containing several ellipsoids, with each input either a point or a higher-rank covariance, via S-lemma–derived LMI conditions (Bochardt et al., 2010). The result unifies analytic geometry, robust filtering, and convex programming.

5. Practical Applications

Robotics and Perception

In online obstacle avoidance, MBEs are used to encapsulate clustered LiDAR returns, providing compact, convex obstacle models for trajectory planning and safety certification (Zhang et al., 26 Feb 2024). The pipeline includes:

Clustering raw 2D LiDAR with DBSCAN;
Applying MVEE to each cluster for tight elliptical approximation;
Expressing clusters in axis-aligned or canonical form, extracting center, semi-axes, and orientation by diagonalization;
Assigning and tracking ellipses between time-steps (Hungarian assignment);
Distinguishing static and dynamic objects by Kalman filter–based state prediction on MBE parameters;
Using the MBE for dynamic control barrier function (D-CBF) constraints, where for each obstacle ellipse $E_{t,k}$ , the signed distance

$l_{t,k} = \sqrt{\frac{a_{t,k}^2b_{t,k}^2(1+\tan^2\Theta_{t,k})}{b_{t,k}^2 + a_{t,k}^2\tan^2\Theta_{t,k}}}$

quantifies closest approach, yielding safety CBFs and buffer-zone gating. Only the proximate MBEs activate barrier constraints, ensuring real-time feasibility even in dense multi-obstacle scenes (Zhang et al., 26 Feb 2024).

Data Fusion and Estimation

The MBE/MEE appears as a robust-uncertainty region in data fusion, equivalent to merging multiple covariance estimates into a single bounding ellipse containing all possible input ellipsoids (Bochardt et al., 2010). The LMI structure enables efficient implementation with standard convex programming solvers.

Poly-ellipse Intersections

For the outer-approximation of the intersection of $M$ ellipses, the polygonal tightening method offers the tightest (lowest-area) bounds among polynomial-time algorithms, outperforming S-procedure and naive expansion in typical instances (Yousefi et al., 2017). This is significant for conservative overbounding in uncertainty propagation, feasible regions, and estimation.

6. Computational Aspects and Performance

The following table summarizes key complexity characteristics for selected MBE computation algorithms as outlined in the literature:

Algorithm	Setting / Input	Complexity (per call)
Welzl’s algorithm (Zhang et al., 26 Feb 2024)	$N$ points, $\mathbb{R}^2$	$O(N)$ expected
Polygonal/tangent-SDP (Yousefi et al., 2017)	$M$ ellipses	$O(M^3)$ (intersections) + SDP in $O(m_p^{1.5})$
GCU/MEE via Maxdet (Bochardt et al., 2010)	$\le30$ ellipsoids/points	Polynomial, milliseconds per instance

In practical robotic applications, the end-to-end pipeline (DBSCAN, MBE, Hungarian match, per-obstacle QM/QP) supports 100 Hz update rates and handles up to 20 obstacles in real time on TurtleBot3-class hardware (Zhang et al., 26 Feb 2024).

7. Special Cases, Uniqueness, and Geometric Invariants

For $N=3$ (triangle), the Steiner ellipse provides a closed-form solution: center at the centroid, tangents at vertices parallel to opposite sides, area ratio $4\pi/(3\sqrt{3})$ times that of the triangle (Euler et al., 16 Sep 2025). For $N=4$ , Euler’s cubic suffices for the minimal-area ellipse in the pencil. For $N < 3$ , degenerate or trivial cases arise. For arbitrary $N$ , the MBE is unique, always convex, affine-invariant, and contacts the convex hull at at most $4$ points in the plane, as per John’s and Löwner’s theorems (Euler et al., 16 Sep 2025).

A plausible implication is that for large $N$ or high aspect-ratio point distributions, the supporting set may degenerate, but convexity guarantees exist regardless of configuration.

In summary, minimum bounding ellipses encapsulate a convergence of geometric extremal theory, convex optimization, and modern control/sensing practice. Their efficient computation and tightness critically impact numerous applications from real-time perception and safety-critical robotics to robust estimation and geometric data fusion (Zhang et al., 26 Feb 2024, Bochardt et al., 2010, Yousefi et al., 2017, Euler et al., 16 Sep 2025).