Geometry is full of familiar friends: Euclidean circles, taxicab diamonds, and max-norm squares. What happens when you stretch, tilt or otherwise remix the notion of distance itself? A 2019 paper on arXiv takes a clean, elegant run at exactly that question by proposing a generalized Minkowski distance in n-dimensional space and exploring the geometric consequences in the plane — including a fresh way to think about ellipses.
“In this paper, we generalize the Minkowski distance by defining a new distance function in n-dimensional space, and we show that this function determines also a metric family as the Minkowski distance.” — Harun Barışoğlu (abstract)
I’ll walk you through the key ideas in plain language, show how the work extends the classical Lp (Minkowski) family, explain the geometric shapes (generalized circles) you get in the plane, and highlight the surprising connection to ellipses. If you want the formal proofs, there’s a link to the full paper at the end.
What is the generalized Minkowski distance and how does it differ from the classical Minkowski norm? (generalized Minkowski distance vs Minkowski norm)
The classical Minkowski distances — often called Lp norms — measure distance between points x and y in R^n by taking the p-th power of absolute coordinate differences, summing them, and taking the p-th root:
Classical Minkowski (Lp): d_p(x,y) = ( Σ_i |x_i – y_i|^p )^(1/p)
This formula gives the familiar special cases: p = 1 is the taxicab (L1), p = 2 is Euclidean, and p → ∞ gives the maximum (L∞) metric.
The authors define a new distance function that generalizes this family. At a conceptual level, they let the way you combine coordinate differences be more flexible — introducing parameters or structure that can weight, distort, or couple coordinates differently. Rather than being limited to the isotropic, coordinate-wise sum of powered differences, the generalized Minkowski distance lets you transform and aggregate those differences in a broader family while still satisfying the core properties of a distance.
Key difference: where the classical Minkowski norm treats coordinates uniformly and independently, the generalized Minkowski distance introduces structure that can make the metric anisotropic (direction-sensitive) or weighted — but it is constructed so the result remains a valid metric (nonnegative, symmetric, satisfies triangle inequality, and zero only on identical points).
How does the new distance define a metric family and what are its properties? (metric family generalization of Minkowski distance properties)
A metric family means a continuous collection of distance functions parameterized in some way (for instance by p in classical Lp). The paper shows the generalized function also forms such a family — you can vary parameters and get a smooth family of metrics, each satisfying:
- Non-negativity: distances are ≥ 0.
- Identity of indiscernibles: distance is zero iff points are identical.
- Symmetry: d(x,y) = d(y,x).
- Triangle inequality: d(x,z) ≤ d(x,y) + d(y,z).
The technical meat of the paper checks these properties under the proposed generalization and isolates parameter ranges that preserve metric axioms. Intuitively, the construction preserves the convexity and homogeneity features you need for a valid norm-like distance while allowing more degrees of freedom in how coordinate contributions are combined.
Implication: You now have a toolbox of metrics that behave like Minkowski distances in many ways but can be skewed, stretched, or rotated to reflect directional costs or asymmetric measurement contexts — useful when modeling anisotropic phenomena (e.g., urban travel with one-way streets or directional friction in physical systems).
How are the taxicab, Euclidean and maximum metrics generalized in this work? (generalization of taxicab Euclidean and maximum norms)
The paper highlights three special cases inside the generalized family that correspond to natural extensions of:
- Taxicab (L1) generalization: Instead of an unweighted sum of absolute coordinate differences, you get weighted or transformed sums that produce diamond-like unit balls that may be skewed or elongated along new axes.
- Euclidean (L2) generalization: The quadratic (sum of squares) combination is generalized, leading to ellipsoidal or rotated quadratic forms as unit balls — these look like stretched/rotated circles in the plane.
- Maximum (L∞) generalization: The infinity norm’s square-shaped unit ball becomes a possibly rotated/rectangular polygon in the generalized metric, reflecting directional maxima that depend on the transformation.
Put plainly: for each classical norm, the generalized construction gives a family that recovers the classical shape when parameters are specialized, but otherwise produces modified shapes that capture weighted or direction-dependent distance.
What are the geometric shapes (generalized circles) produced by these generalized metrics in the plane? (circles of generalized Minkowski distance in the plane)
“Circles” in a metric are the set of points at a fixed distance r from a center — the unit ball scaled by r. The authors work out these shapes in the real plane (R^2) for their generalized versions of L1, L2, and L∞. The takeaway:
- Generalized taxicab circles: Instead of the symmetric diamond you expect from L1, you often get a convex polygon that may have different vertex positions, producing a skewed diamond or an asymmetric polygon depending on weights or directional transforms.
- Generalized Euclidean circles: These typically become ellipses or rotated ellipses — which is intuitive because any positive-definite quadratic form defines an ellipse as a level set. The novelty is showing how that ellipse arises naturally as a member of the generalized Minkowski family.
- Generalized maximum circles: These can produce rotated, axis-aligned, or rectangular shapes whose sides reflect the dominant coordinate or direction at that position.
Across cases, the generalized circles remain convex and closed — properties inherited from the construction. What changes is the symmetry: many classical norms give highly symmetric unit balls; the generalized versions typically break some symmetry to reflect the anisotropy introduced by the parameters.
How does this lead to a new definition or characterization of the ellipse via generalized distance functions? (new definition of the ellipse via generalized distance)
One striking connection in the paper is that certain generalized “circles” (level sets of the new metric) coincide with Euclidean ellipses. That observation suggests a new, metric-based way to characterize ellipses:
Ellipse as a generalized metric circle: For appropriate choices in the generalized Minkowski family, the set of points at fixed generalized distance from a center is exactly a Euclidean ellipse.
Why is that idea neat? Classic characterizations of ellipses include:
- Conic definition: set of points with constant eccentricity ratio to a focus-directrix pair.
- Two-focus definition: set of points whose sum of Euclidean distances to two fixed points is constant.
The metric-based viewpoint adds a third, elegant characterization: an ellipse can be seen as a unit ball (scaled) of a properly chosen generalized Minkowski metric. This reframing can be useful conceptually and computationally — it gives a lens that ties together normed geometry and classical conic sections.
In other words: instead of thinking of an ellipse only as a special kind of conic, you can think of it as the natural “circle” for a particular anisotropic notion of distance. That viewpoint makes it easier to translate geometric intuition between normed spaces and classical Euclidean shapes.
Practical takeaway: Why the generalized Minkowski distance and new ellipse definition matter (applications of generalized Minkowski distance and ellipse)
There are several practical and theoretical reasons to care about these generalizations:
- Modeling anisotropy: Many real-world systems are not isotropic (directionally uniform). These generalized metrics let you model directional costs naturally — e.g., travel time with wind/current, or material stress that varies with direction.
- Optimization and data science: Norm choice affects optimization landscapes. A richer family of norms lets you design objective functions or regularizers that align with problem geometry.
- Geometry and teaching: Recasting classical shapes like ellipses as metric circles builds intuition and bridges metric geometry with conic sections, which can be pedagogically valuable.
For a bit of cultural flavor, thinking about how ideas transform across contexts — like myths or shapes morphing under different lenses — reminds me of how narratives can shift when viewed from another perspective, much like in some of my other writing on myth and change.
Where to go next to explore the generalized Minkowski distance further (resources and next steps)
If you’re mathematically inclined, dive into the arXiv paper for formal definitions, lemmas, and proofs. If you prefer intuition and visual exploration, try plotting unit balls for a few parameter choices: start by visualizing classical L1, L2, L∞ and then introduce directional weights or linear transforms to see how the shapes deform.
You can also relate this to other areas: if you work in optimization, try replacing standard norms in a regularized loss with a generalized metric and observe the effect; if you work in computational geometry or robotics, consider how anisotropic distance affects path planning.
Finally, if you enjoy cross-disciplinary analogies and reading about how perspectives mutate across contexts, there’s an interesting piece that interweaves transformation and narrative that I found relevant in tone and approach.
Demeter – Goddess Of Agriculture And Harvest, But Also Prone To Fits Of Anger And Grief
If you want the original technical source, read the full paper on arXiv:
https://arxiv.org/abs/1903.09657
Final note: The paper is a tidy reminder that “distance” is not a monolith — by tweaking how we measure, we unlock a family of geometries with fresh insights and useful applications, including a crisp, metric-flavored way to think about the classical ellipse.
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