Tangent nomograms with circles

Tangent Circles in the Ratio 2 : 1

In this article we consider the following old Japanese geometry problem (see Figure 1), whose statement in [1, p. 39] is missing the condition that two of the vertices are the opposite ends of a diameter. (The authors implicitly correct the omission in the proof they provide on page 118.) We denote by O(r) the circle with centre O, radius r. Problem [1, Example 3.2]. The squares ACBD and ABCD h...

Labeling Points with Circles

Points with Circles ∗

We present a new algorithm for labeling points with circles of equal size. Our algorithm tries to maximize the label size. It improves the approximation factor of the only known algorithm for this problem by more than 50 % to about 1/20. At the same time, our algorithm keeps the O(n log n) time bound of its predecessor. In addition, we show that the decision problem is NP-hard and that it is NP...

Map Labeling with Circles

We study two geometric optimization problems motivated by cartographic applications: Map Labeling with Uniform Circles (MLUC) and Map Labeling with Uniform Circle Pairs (MLUCP). We show that the decision problems of both MLUC and MLUCP are NP-hard, and that the related optimization problems for maximizing the label sizes are NP-hard to approximate within factor 1.0349. We design approximation a...

The art of nomograms

Background To retrospectively analyse strategies for adjusting refractive surgery plans with reference to the preoperative manifest refraction. Methods We constructed seven nomograms based on the refractive outcomes (sphere, cylinder, axis [SCA]) of 150 consecutive eyes treated with laser in situ keratomileusis for myopic astigmatism. We limited the initial data to the SCA of the manifest ref...