Tetrahedral Mesh Generation by Delaunay Refinement
Proceedings of the fourteenth annual symposium on Computational geometry - SCG '98, 1998 · paper · Computer science
Presents an algorithm that generates Delaunay tetrahedral meshes with circumradius-to-shortest-edge ratios no greater than two, with sizes grading from small to large. An implementation generally exceeds the theoretical bounds and helps eliminate badly shaped tetrahedra.
Checked: PDF from cs.jhu.edu, fetched 4 Oct 2026. Page numbers are those of that PDF.
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None that passed a second check.
6 minor (typos and the like; most checked by the judge only)
Wording Proof of Lemma 2 concludes "the smallest ball centered at v that contains two nonincident features of X has radius no larger than lfs(u) + |uv|", whereas lfs is defined by the smallest ball that "intersects two nonincident features of X" and the preceding proof sentences use "intersects".
- p. 6 Proof of Lemma 2, last sentence
the smallest ball centered at v that contains two nonincident features of X has radius no larger than
- p. 6 Section 7, definition of local feature size
is the radius of the smallest ball centered at p that intersects two nonincident features of X
Only the authors can say what is right here.
Judge's note: Viewed the page 6 image. The definition and the first two sentences of the proof use "intersects"; the last sentence uses "contains". My reworking: the ball of radius lfs(u)+|uv| about v contains the ball of radius lfs(u) about u, so it intersects the same two features. It need not contain them, for example when the features are long segments. The final sentence is therefore not what the argument establishes if "contains" is read literally. With "intersects" the step is correct, and Lemma 2 stands. Wording slip in a proof.
Second check: another model agreed
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Typo Figure 2 caption drops an article: "Every new edge has length at least B times that of shortest edge of the poor triangle." (should read "that of the shortest edge").
- p. 2 Figure 2 caption, last sentence
Every new edge has length at least B times that of shortest edge of the poor triangle.
- p. 2 Section 3, second paragraph
every new edge has length at least B times that of the shortest edge of t
Judge's note: Viewed the page 2 image. The caption reads "that of shortest edge of the poor triangle"; the parallel body sentence in Section 3 reads "that of the shortest edge of t". Missing word, cosmetic.
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Typo Section 2 says "tetrahedra with poor angles have circumcircles much larger than their shortest edges"; for tetrahedra the paper's own term is circumsphere (circumradius).
- p. 2 Section 2, third paragraph
tetrahedra with poor angles have circumcircles much larger than their shortest edges
- p. 2 Section 2, first paragraph (definition)
The circumsphere of a simplex is the unique circle or sphere that passes through all its vertices.
Judge's note: Viewed the page 2 image. The sentence is about tetrahedra (needle, wedge, cap). The same section defines "circumsphere" and "circumradius" for simplices, and the paper elsewhere uses circumsphere for tetrahedra. Terminology slip; the meaning is clear.
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Typo Figure 5 caption: "all equatorial spheres (included the two illustrated) are empty after the split"; "included" should be "including".
- p. 4 Figure 5 caption, part (b)
all equatorial spheres (included the two illustrated) are empty after the split
Judge's note: Viewed the page 4 image; the caption reads "included the two illustrated". Grammatical slip, cosmetic.
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Typo Proof of Lemma 3 says "so lfs(v) ≤ r_v, and the theorem holds." although the statement being proved is a lemma.
- p. 7 Proof of Lemma 3, first sentence
and the theorem holds
- p. 7 Section 7, Proof of Lemma 3, first sentence
and the theorem holds.
Judge's note: Viewed the page 7 image. The first sentence of the proof of Lemma 3 ends "and the theorem holds". The result is labelled Lemma 3, and no theorem has been stated before this point. Naming slip; the meaning is unaffected.
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Typo Proof of Theorem 5 states its induction hypothesis as "assume for the sake of induction that the lemma is true for p", although the statement being proved by induction is Theorem 5.
- p. 8 Proof of Theorem 5, first paragraph
assume for the sake of induction that the lemma is true for p
- p. 7 Proof of Lemma 3, first paragraph
and the theorem holds.
- p. 8 Proof of Theorem 5, first paragraph
assume for the sake of induction that the lemma is true for
Judge's note: Viewed the page 8 image. The sentence reads "the lemma is true for p. Hence, D_p ≤ max{D_T, D_F, D_S}", which is the conclusion of Theorem 5 applied to p; neither Lemma 3 nor Lemma 4 gives this. Later in the same proof the text says "the theorem holds". Naming slip. The companion slip in the proof of Lemma 3 ("the theorem holds") is handled under C007; this cluster is confirmed for the Theorem 5 instance.
Second check: none (judge only)