Cluster Geometry: k-means, DBSCAN, and Shapes That Break Them

Compare centroid and density clustering on blobs, moons, and pure noise — and see why the elbow plot cannot tell you k.

Ch 8 8.4

Comparator

Unsupervised Learning & Text

22 mindifficulty 3/5

Controls

Data geometry

Algorithm

Clustersk2

k-means will always return exactly this many clusters.

Noise0.140

Pointsn400

Random seed

Every result on this page is a deterministic function of the seed and the controls.

Cluster assignment

Crosses mark centroids; boundaries are implicitly straight lines.

Cluster 1Cluster 2

Elbow curve

Within-cluster sum of squares always falls with k.

Monotone decrease is guaranteed, so a low inertia never validates a choice of k.

Clusters found

Labelled noise

DBSCAN only

Silhouette

0.500

assumes compact, convex clusters

Adjusted Rand index

0.346

agreement with the generative labels

Predict before you look

You cluster uniformly random points — no groups whatsoever — with k-means for k = 1…8 and plot inertia. What does the curve look like?

Scenarios

Teacher-ready starting points.

Documentation

What this shows

Five generative geometries and two clustering algorithms, scored against the labels that generated the data. Because the truth is known here, the adjusted Rand index reports what the internal indices cannot.

Two very different objectives

k-means minimizes within-cluster squared Euclidean distance. Its optimal assignment for fixed centroids is the nearest centroid, so the decision boundaries form a Voronoi diagram of straight lines. Any cluster that is not roughly convex and comparably sized is therefore unrecoverable, no matter how much data you supply.

DBSCAN instead grows clusters through density reachability: a point is core if at least minPts neighbours lie within ε, and clusters are maximal chains of reachable core points. This recovers arbitrary shapes and explicitly labels low-density points as noise — but ε must match the data's scale.

Inertia decreases monotonically in k and reaches zero at k = n, so it can never select k on its own. The silhouette rewards compact convex clusters, which quietly penalizes the correct answer on moons and rings; the ARI corrects for chance agreement but needs ground truth.

k-means:  min Σₖ Σ_{i∈Cₖ} ‖xᵢ − μₖ‖²

core(x):  |{y : ‖x−y‖ ≤ ε}| ≥ minPts

ARI = (RI − E[RI]) / (max RI − E[RI])

Try this

Run k-means k = 2 on “moons”, then DBSCAN with ε = 0.25 and compare the ARI readouts.
On “rings”, try every k from 2 to 8 — none of them recovers the two circles.
Choose “Unequal spread” and run k-means k = 3: the wide cluster is carved up to balance the objective.
Select “No structure” and cluster anyway; note that the algorithm still returns clusters.
Raise DBSCAN's ε slowly until all clusters merge, then lower it until everything is noise.

Common misconception

“The elbow in the inertia curve tells me the right number of clusters.”

Inertia falls monotonically with k by construction, and on data with no clusters at all it still produces a smooth curve with a visible bend. An elbow is a heuristic about diminishing returns, not evidence about the number of groups. Use it alongside stability across resamples, a validation index appropriate to the cluster shape, and — decisively — subject-matter meaning.

DSPA3 Chapter 8 — Unsupervised clustering