This interactive visualization compares two complementary metrics for polygon similarity:
PoLiS (Polygon-to-Line-Segment) distance [1] measures the average distance between vertices of one polygon and the boundary of another polygon.
The metric is computed as:
$$\text{PoLiS}(A,B) = \frac{1}{2} \left( \frac{1}{|A|}\sum_{a_j \in A} \text{dist}(a_j, \partial B) + \frac{1}{|B|}\sum_{b_k \in B} \text{dist}(b_k, \partial A) \right)$$
where \(\text{dist}(a_j, \partial B)\) is the shortest distance from vertex \(a_j\) to any edge of polygon \(B\).
Properties of PoLiS (copied from [1]):
- Compares polygons, not only point sets, with different numbers of vertices
- Insensitive to additional points on polygon edges
- Monotonic with linear response to small changes in translation, rotation, and scale
- A true metric in the mathematical sense (satisfies triangle inequality)
[1] Avbelj, J., Müller, R., & Bamler, R. (2014). A metric for polygon comparison and building extraction evaluation.
IEEE Geoscience and Remote Sensing Letters, 12(1), 170-174.
[PDF]
IoU (Intersection over Union) measures the overlap between two polygons as the ratio of their intersection area to their union area:
$$\text{IoU}(A,B) = \frac{\text{Area}(A \cap B)}{\text{Area}(A \cup B)} = \frac{\text{Area}(A \cap B)}{\text{Area}(A) + \text{Area}(B) - \text{Area}(A \cap B)}$$
IoU ranges from 0 (no overlap) to 1 (perfect overlap) and is widely used in object detection and segmentation tasks.
When they agree vs. differ: Both metrics detect large-scale misalignment and shape differences. However,
PoLiS is more sensitive to positional shifts and boundary irregularities even when overlap remains high, while IoU primarily
reflects area-based similarity and can remain stable despite shape deformations. Try rotating or translating the polygons
slightly—you'll notice PoLiS increases while IoU may stay relatively constant.
Here the
blue rectangle is polygon A and the
purple polygon is polygon B. Drag and reshape B, or rotate A, to explore
how both metrics respond.