The Power Crust, Unions of Balls, and the Medial Axis Transform
Computational Geometry, 2001 · paper · Computer science
Approximates an object's medial axis transform with a finite union of balls chosen from the Voronoi balls of surface samples, and defines the power crust surface from it. It bounds the geometric error and shows topological correctness, giving a surface reconstruction algorithm.
Checked: PDF from cs.ucdavis.edu, fetched 4 Oct 2026. Page numbers are those of that PDF.
14 findings
Substance Proof of Lemma 29: "If u is inside B_c,μ, then it is inside B_p,ρ" and "this ball must also intersect B_p,ρ (Observation 1)" both require B_c,μ to be of the same type (inner/outer) as p. However, B_c,μ is defined as "the smaller of the two polar balls at s", which may be of the other type.
- p. 28 Statement and proof of Lemma 29
If u is inside Bc,μ, then it is inside Bp,ρ, and the lemma is trivial.
- p. 28 Proof of Lemma 29
Since this ball must also intersect Bp,ρ (Observation 1)
Only the authors can say what is right here.
Judge's note: Lemma 29 (page 28) defines p as the inner (outer) pole of minimum power distance to u, and B_c,μ as the smaller polar ball at s, with no type restriction. Observation 1 (layout text, line 389) compares only balls within one set B. So w ≤ λ follows only if B_c,μ belongs to p's family, and so does 'inside B_c,μ implies inside B_p,ρ'. Counterexample to the first step: let p be inner and the smaller ball at s be the outer polar ball. A point u inside that outer ball and outside every inner ball satisfies the premise but is not inside B_p,ρ. The proof does not cover this case. The ball at s that has p's type is then the larger one, and Lemma 28 gives no bound for it. The lemma may still be true by another argument, but as written its proof is incomplete for this case. Observation 30, Lemma 31 and Theorem 32 depend on it. The authors would need to restrict the statement or supply the missing case.
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Substance Remark after Lemma 34 (p.33): "a similar statement could be made for any γ = o(r^{1/2})". The proof needs γ large relative to r^{1/2} (γ = ω(r^{1/2})), and the lemma's own case γ = Ω(r^{1/3}) is not o(r^{1/2}).
- p. 33 Remark after Lemma 34
similar statement could be made for any γ = o(r^{1/2})
- p. 32 Lemma 34
with γ = Ω(r^{1/3})
- p. 33 Remark after Lemma 34
a similar statement could be made for any γ = o(r1/2)
- p. 33 Remark after Lemma 34
any γ = o(r1/2)
Only the authors can say what is right here.
Judge's note: My own reworking from page 33: ε ≤ arcsin(r/(2 sin(α/2 − r))) ~ r/α, with α ≥ γ. The bound on d(t,p) divides by sin(α/2 − β − ε), so it needs ε ≪ α/2, i.e. r/γ ≪ γ, i.e. γ ≫ r^{1/2}. Take γ = o(r^{1/2}), for example γ = r^{0.6} with α = γ: then ε ~ r^{0.4} ≫ γ and the denominator turns negative. Since r^{1/3}/r^{1/2} → ∞ as r → 0, the stated case γ = Ω(r^{1/3}) is not included in o(r^{1/2}) either. The asymptotic direction in the remark is reversed.
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Substance Proof of Lemma 34 (p.33): "O(ρ (r + r^{1/3})/(r^{1/3} − r − r^{2/3})) = O(ρ(1 + r^{2/3}))", then "d(q,p) = O(ρr^{2/3})". The ratio is 1 + Θ(r^{1/3}), so the stated O(ρr^{2/3}) bound of Lemma 34 is not delivered by this step.
- p. 33 Proof of Lemma 34, bound on d(t,p)
= O(ρ(1 + r^{2/3}))
- p. 33 Proof of Lemma 34, conclusion
we have d(q, p) = O(ρr^{2/3}) and finally d(p, c) = O(ρr^{2/3}).
- p. 33 Proof of Lemma 34, display bounding d(t,p) and the conclusion
we have d(q, p) = O(ρr2/3) and
- p. 32 Statement of Lemma 34
Then the distance from c to the inner (outer) pole p of t is O(ρr2/3).
Only the authors can say what is right here.
Judge's note: Dividing through by r^{1/3}, the ratio is (1 + r^{2/3})/(1 − r^{1/3} − r^{2/3}) = 1 + r^{1/3} + O(r^{2/3}). The r^{1/3} term comes from ε = O(r^{2/3}) divided by α/2 ~ r^{1/3}. Since d(t,c) ≥ ρ, d(t,p) − d(t,c) is then bounded only by O(ρ r^{1/3}), not O(ρ r^{2/3}). For r = 1e-6 the ratio is 1.0103, against 1 + r^{2/3} = 1.0001. Convergence in Theorem 35 survives, but the rate stated in Lemma 34 (p.32) is left without proof.
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Reference or label Section 10 says "From Observation 2 above, we see that T is a subset of Pow(B_V)" (p.37) and "Again, this follows from Observation 2" (p.38). No Observation 2 exists; the result meant is Lemma 2 (Pow(B) of the Voronoi balls is DT(S)).
- p. 37 Section 10
From Observation 2 above, we see that T is a subset of Pow(BV).
- p. 38 Section 10, after Observation 43
Again, this follows from Observation 2.
- p. 11 Lemma 2
Lemma 2 Let S be a set of points in general position
- p. 37 Section 10, before Observation 41
From Observation 2 above, we see that T is a subset of Pow(B_V).
Judge's note: Searching the text layer for observations finds Observations 1, 3, 4, 5, 9, 14, 22, 27, 30 and 39-43, all on one shared counter. Item 2 is Lemma 2 on p.11: "Let S be a set of points in general position ... B the set of Voronoi balls ... Pow(B) is the Delaunay triangulation DT(S)". That lemma is what supports both statements. The page images of p.37 and p.38 show the "Observation 2" wording.
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Reference or label "First, an observation, illustrated by Figure 14 below.": the figure directly below is captioned "Figure 3:".
- p. 17 Section 6, Shallow intersections
First, an observation, illustrated by Figure 14 below.
- p. 17 Section 6, Shallow intersections
illustrated by Figure 14 below
Judge's note: On the p.17 image, the figure right after the sentence is labelled "Figure 3:", and the observation it illustrates is Observation 14. The figure number should be 3.
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Reference or label Proof of Lemma 20 says "let s be the closest sample to u; see Figure 6", but the points s_x, x, u and s are drawn in Figure 5. Figure 6 shows the tubular neighborhood.
- p. 21 Proof of Lemma 20
let s be the closest sample to u; see Figure 6
- p. 22 Figure 5 caption
Figure 5: The point u is closer to x than sx
- p. 21 Proof of Lemma 20
see Figure 6
- p. 22 Figure 5 caption
The point u is closer to x than sx
Judge's note: Page 21 has the pointer to Figure 6. Figure 5 on page 22 labels s_x, x, u and s, and its caption begins 'The point u is closer to x than s_x'. Figure 6 on page 23 shows the unions and the tubular neighborhood and labels none of these points. The pointer should be to Figure 5.
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Reference or label Proof of Theorem 32 cites "the tubular neighborhood around W (Lemma 26)", but item 26 is Corollary 26.
- p. 30 Proof of Theorem 32
(Lemma 26)
- p. 27 Corollary 26
Corollary 26 Any point u on a face of the power crust lies within
- p. 30 Proof of Theorem 32
around W (Lemma 26)
- p. 30 Proof of Theorem 32
tubular neighborhood around W (Lemma 26)
Judge's note: Page 27 labels the item as 'Corollary 26: Any point u on a face of the power crust lies within O(r)LFS(x) of some point x ∈ W', which is the result cited. Item numbering is shared across result types, and there is no Lemma 26. The cited content is right; only the label is wrong.
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Reference or label Reference [8] names the journal "IEEE Transactions on Vision and Computer Graphics". Reference [26] names the same journal correctly as "IEEE Transactions on Visualization and Computer Graphics".
- p. 39 Reference [8]
IEEE Transactions on Vision and Computer Graphics.
- p. 41 Reference [26]
IEEE Transactions on Visualization and Computer Graphics
- p. 39 Reference [8]
IEEE Transactions on Vision and Computer Graphics
Judge's note: The p.39 image shows [8] with 'Vision'; the p.41 image shows [26] with 'Visualization'. The paper's own list is inconsistent, so the fix can be made from internal evidence without outside knowledge.
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Wording Proof of Lemma 23 prints "d(u, y) = sqrt(d²(m, y) − d²(m, u)) = O(sqrt((1 + O(r))² − (1 − O(r²))²))". The first radicand has its terms reversed; it should be d²(m,u) − d²(m,y).
- p. 24 Proof of Lemma 23, last display
d(u, y) = sqrt(d²(m, y) − d²(m, u)) = O(sqrt((1 + O(r))² − (1 − O(r²))²))
- p. 24 Proof of Lemma 23, last display
d(u, y) = √(d2(m, y) − d2(m, u))
Only the authors can say what is right here.
Judge's note: In the left part of Figure 7 (page 24), y lies on segment cm on B′, whose radius is R(1−O(r²)). The boundary of B_c,ρ (with ρ = ∞) is the line through y perpendicular to cm, and u lies on that line. So m–y–u is a right triangle with the right angle at y, and d(m,u)² = d(m,y)² + d(u,y)². Since d(m,u) = 1+O(r) > d(m,y) = 1−O(r²), the printed radicand d²(m,y) − d²(m,u) is negative. The paper's next expression already puts (1+O(r))², i.e. d(m,u), first. The slip is in the first expression only, and the O(√r) bound stands.
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Wording Proof of Lemma 23 bounds β = ∠cmu (the angle at m) but states the result as "We get ∠ucm = arcsin(d(u,y)/d(m,u)) = O(√r)". The next sentence then defines "γ = ∠ucm" (the angle at c).
- p. 24 Proof of Lemma 23
Angle β = ∠cmu will depend on the ratio of the two radii
- p. 25 Proof of Lemma 23, first line
We get ∠ucm = arcsin(d(u,y)/d(m,u)) = O(√r).
- p. 25 Proof of Lemma 23
We use a similar argument to bound γ = ∠ucm.
- p. 25 Proof of Lemma 23, first line
We get ∠ucm = arcsin
Only the authors can say what is right here.
Judge's note: Page 24 writes α = ∠ucm + ∠umc, says 'We begin by bounding ∠umc', and introduces 'Angle β = ∠cmu'. In the right triangle m–y–u (right angle at y, with y on segment cm), arcsin(d(u,y)/d(m,u)) is the angle at m, which is ∠umc. Page 25 nevertheless prints '∠ucm' for this result and then bounds 'γ = ∠ucm' separately. The same symbol therefore stands for both summands of α; the first should read ∠umc. The bound itself is unaffected.
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Wording Proof of Lemma 23 prints "Distance d(x, u) = sqrt(d(c,u)² − d(c,z)²) = O(sqrt(1 − (1 − r)²)) = O(√r)". By the figure this leg is d(z,u), and the same paragraph has just used d(u,x) = O(r).
- p. 25 Proof of Lemma 23, bound on γ
Distance d(x, u) = sqrt(d(c, u)² − d(c, z)²) = O(sqrt(1 − (1 − r)²)) = O(√r)
- p. 25 Proof of Lemma 23, bound on γ
Distance d(x, u) =
Only the authors can say what is right here.
Judge's note: In the right part of Figure 7 (page 24), z lies on the axis c–y, and the segment z–u is perpendicular to that axis. x is a distinct point on B_mR below u. In the right triangle c–z–u, sqrt(d(c,u)² − d(c,z)²) = d(z,u). Page 25 states just before this that d(z,y) is at most the lune thickness plus d(u,x), i.e. O(r²)+O(r). That treats d(u,x) as O(r), which conflicts with the printed d(x,u) = O(√r). Reading d(z,u) instead, the O(√r) bound on γ goes through.
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Wording Proof of Lemma 29: "By Lemma 28, d(u, B_p,ρ) ≤ k(r²/μ)". Lemma 28 bounds the distance to the smaller polar ball at s, which in Lemma 29 is B_c,μ, not B_p,ρ.
- p. 29 Proof of Lemma 29
By Lemma 28, d(u, Bp,ρ) ≤ k(r²/μ)
- p. 28 Statement of Lemma 29
Let Bc,μ be the smaller of the two polar balls at s.
Only the authors can say what is right here.
Judge's note: Lemma 28 (page 27) uses B_p,ρ for 'the smaller of the two polar balls at s'. In Lemma 29 (page 28) that ball is B_c,μ, while B_p,ρ is the minimum-power ball whose distance is what the lemma sets out to bound. The computation of λ on page 29, sqrt(μ² + 2(kr²/μ)μ + k²r⁴/μ² − μ²), is the radius of the ball orthogonal to B_c,μ at distance kr²/μ. So the cited bound should read d(u, B_c,μ). Lemma 28's notation appears to have been carried over unchanged.
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Wording Proof of Lemma 31 claims f_I is "monotonically decreasing as u goes from the outer end of g to the inner end" because x moves "at an angle of at most π/6 from the the vector from the nearest pole p to x ... to ensure that x is moving away from p". Moving away from p increases the power distance to p.
- p. 29 Proof of Lemma 31
We claim that this function is monotonically decreasing as u goes from the outer
- p. 30 Proof of Lemma 31
any angle less than π/2 would suffice to ensure that x is moving away from p.
- p. 30 Proof of Lemma 31, first lines
would suffice to ensure that x is moving away from p
- p. 29 Proof of Lemma 31
We claim that this function is monotonically decreasing
Only the authors can say what is right here.
Judge's note: Pages 29–30: f_I(u) is the minimum of |u−p|² − ρ² over inner poles. If the velocity v makes an angle below π/2 with x−p, then d/dt|x−p|² = 2(x−p)·v > 0, so the power distance increases. That contradicts the claimed decrease. Observation 30 gives the vector u→c (towards the pole) within π/6 of the normal n_x, and Lemma 23 orients n_x as the inner normal for inner balls. Moving inward is therefore moving towards p, within π/6 of the vector from x to p. The direction words are reversed; the lemma is unaffected. The same sentence also repeats a word: 'the the'.
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Wording Algorithm Step 4 (p.35) propagates a label when the balls meet "at an angle of more than π/4", but the proof of Lemma 37 (p.36) analyses the case "they meet at an angle of more than π/2".
- p. 35 Algorithm, Step 4
at an angle of more than π/4:
- p. 36 Proof of Lemma 37
but they meet at an angle of more than π/2
- p. 36 Proof of Lemma 37
they meet at an angle of more than π/2
- p. 35 Algorithm Step 4
at an angle of more than π/4
Only the authors can say what is right here.
Judge's note: Both thresholds are confirmed in the images of pages 35 and 36. The case analysis in Lemma 37 does not use the algorithm's own threshold, so it leaves angles in (π/4, π/2] uncovered. Lemma 16's O(r) bound would still exclude these for small r, so the result stands, but the proof and the algorithm disagree.
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37 minor (typos and the like; most checked by the judge only)
Wording The Introduction says "the power crust, and the surfaces of the two unions of polar balls ... are all close to the surface of the original object, that their surface normals are close", but Section 7 gives no normal result for the power crust.
- p. 4 Introduction, final paragraph
We show that the power crust, and the surfaces of the two unions of polar balls, inner and outer, are all close to the surface of the original object, that their surface normals are close
Only the authors can say what is right here.
Judge's note: Read all of Section 7 (pp.26-30). It gives proximity (Corollary 26), Observation 27, Lemmas 28-29, Observation 30, Lemma 31 and Theorem 32. Observation 30 bounds the angle between the surface normal n_x and the vector from u to the nearest pole, not the normal of a power crust face. Section 6 (p.17) promises a normal result only for U_I and U_O, which Theorem 24 gives. As written, the intro's claim about power crust normals is not backed by a stated result. The authors should either add the result or narrow the wording to the union boundaries.
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Wording The Introduction says "the power shape, like the medial axis, is homotopy equivalent to the original object", but Theorem 33 states "The power shape is homotopy equivalent to Q − W."
- p. 4 Introduction, final paragraph
the power shape, like the medial axis, is homotopy equivalent to the original object
- p. 31 Theorem 33
The power shape is homotopy equivalent to Q − W.
- p. 14 after Theorem 6
the medial axis is homotopy equivalent to the object [12]
Only the authors can say what is right here.
Judge's note: Theorem 33 (p.31) and its proof give homotopy equivalence to Q − W, which contains both the inside and the outside of W. On p.8 the medial-axis fact is stated as homotopy equivalence "to the complement of W, Q − W [12]". Yet p.4 and p.14 ("the medial axis is homotopy equivalent to the object [12]") say "object", which the paper elsewhere uses for the solid 𝒲. As worded, the intro's claim is not the one the theorem proves. Only the authors can confirm that Q − W is meant.
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Wording "its medial axis will consist of piecewise smooth two-dimensional patches, with singular edges and vertices contained in two and three patches respectively": the patch counts look too low.
- p. 8 Section 3, Medial axis transform
singular edges and vertices contained in two and three patches respectively
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that for a surface in general position, singular (Y-junction) medial-axis edges are where three sheets meet, and vertices are where more sheets meet. Checked the p.8 image: the text says "two and three". The paper gives no definition that would make an edge in two patches singular. The authors should confirm what they intended.
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Wording Algorithm Step 3 (p.35): "Label its infinite outer pole with O′", whereas Section 4 (p.12) adds eight box points Z "so that all the Voronoi vertices of each sample in S are finite points".
- p. 35 Algorithm, Step 3
Label its infinite outer pole with O′ and the opposite inner pole I′.
- p. 12 Section 4
so that all the Voronoi vertices of each sample in S are finite points.
Only the authors can say what is right here.
Judge's note: With the box points added, no sample has an infinite pole, and the convex-hull samples would be the box vertices. The algorithm (Input: r-sample S, Step 1: Delaunay triangulation of S) does not say whether Z is included. This ambiguity needs clarification from the authors.
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Wording Proof of Lemma 17 reads "Since d(z, c) ≤ ρ_m, we can simplify to (1 − r)d(x, z) ≤ rρ_m", but no c is defined in this proof. The displays just above use d(z, m).
- p. 20 Proof of Lemma 17
Since d(z, c) ≤ ρm, we can simplify to
- p. 20 Proof of Lemma 17
Since d(z, c) ≤ ρ_m
Only the authors can say what is right here.
Judge's note: Page 19 defines c as the centre of the smaller ball only in the proof of Lemma 16. The proof of Lemma 17 (pages 19–20) uses the centres p and m, and both displays on page 20 contain d(z,m). Reworked with d(z,m) ≤ ρ_m: the left side, sqrt(ρ_m² − d²(z,m) + d²(x,z)), is at least d(x,z), and the right side is at most r d(x,z) + rρ_m. That gives (1−r)d(x,z) ≤ rρ_m, so the printed result stands. The symbol c looks like a slip carried over from Lemma 16.
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Wording Proof of Lemma 25 states that the portion of l(x) lies in B_c,ρ "for distance at least 2ρ cos α = O(LFS(x))". The argument needs a lower bound, Ω(LFS(x)), not an O(·) bound.
- p. 26 Proof of Lemma 25
for distance at least 2ρ cos α = O(LFS(x))
- p. 26 Proof of Lemma 25, one-to-one argument
for distance at least 2ρ cos α =
Only the authors can say what is right here.
Judge's note: Page 26 says ρ = Ω(LFS(x)) by Observation 22 and α = O(√r), so 2ρ cos α = Ω(LFS(x)). The following sentence ('every point ... closer than O(r)LFS(x) from x lies in B_c,ρ') needs exactly that lower bound. The printed O(LFS(x)) is an upper bound and does not support the step; nothing in the proof bounds ρ above by a multiple of LFS(x). This is probably loose use of O for Ω or Θ. Writing Ω makes the step valid, and the lemma stands.
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Wording Observation 30 introduces the pole p with ball B_p but states "The vector u→c forms an angle of at most α = π/6 with n_x". No c is defined there; the vector should be u→p.
- p. 29 Observation 30
The vector ~uc forms an angle of at most α = π/6 with nx
Judge's note: Page 29: Observation 30 defines only p, B_p, x and n_x. The symbol c comes from Lemma 23, where it is the centre of B_c,ρ. Lemma 31 (page 30) applies the observation to 'the nearest pole p'.
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Wording Page 31 says "the set P_I of poles converges, geometrically, to the true medial axis of W", while Theorem 35 (p.33) states only that inner poles converge to the inner medial axis (outer to outer).
- p. 31 Section 8, paragraph after Figure 10
we show that the set P_I of poles converges, geometrically, to the true medial axis of W
- p. 33 Theorem 35
The set of inner (outer) poles of S_i converges to the inner (outer) medial axis of W.
Only the authors can say what is right here.
Judge's note: Page 12 defines P_I as the inside poles. The overview sentence on page 31 pairs P_I with the whole medial axis, which does not match the statement of Theorem 35 on page 33. It may be a slip for P (all poles) or for 'inner medial axis'.
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Wording Proof of Lemma 34 (p.32): "From the sampling criterion we have that d(u,s) ≤ rLFS(s) ≤ rρ". The r-sample definition (p.15) gives d(u,s) ≤ r LFS(u), not r LFS(s).
- p. 32 Proof of Lemma 34
d(u, s) ≤ rLFS(s) ≤ rρ
Only the authors can say what is right here.
Judge's note: The definition on page 15 bounds the distance from w to its closest sample by r·LFS(w), so here it gives r·LFS(u). Since u lies on the medial ball centred at c on the medial axis, LFS(u) ≤ ρ, so d(u,s) ≤ rρ holds. Neither d(u,s) ≤ rLFS(s) nor LFS(s) ≤ ρ follows from the cited criterion. Lemma 8 gives only LFS(s) ≤ LFS(u)+d(u,s) ≤ ρ(1+r). The intermediate term is probably a slip for LFS(u). The end inequality, and the result, stand.
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Wording Proof of Lemma 34 (p.33): "both d(t,s) and d(t,u) are at least 2 sin(α/2 − r)", a dimensionless quantity; the factor ρ is missing.
- p. 33 Proof of Lemma 34, bound on ε
are at least 2 sin(
Only the authors can say what is right here.
Judge's note: Reworked: ∠cxu is acute, so d(t,u) ≥ d(x,u) = 2ρ sin(α/2), and d(t,s) ≥ 2ρ sin(α/2) − rρ. Both bounds are ≥ 2ρ sin(α/2 − r) for α below about 2π/3, so the bound holds once multiplied by ρ. The arcsin expression that follows is consistent with ρ cancelling against d(s,u) ≤ ρr. The factor ρ is dropped in the printed bound; the stated result stands.
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Typo The abstract's last sentence ends "...the set of poles, the centers of the polar balls, converges to the medial axis" with no full stop.
- p. 1 Abstract, last line
converges to the medial axis
Judge's note: Checked the p.1 image: there is no period after "medial axis" at the end of the abstract.
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Typo "but also that it's interior is homeomorphic to the solid object itself": the possessive should be "its".
- p. 4 Section 1, first paragraph
but also that it's interior is homeomorphic
- p. 4 Introduction, final paragraph
but also that it's interior is homeomorphic to the solid object itself
- p. 4 Section 1 end
also that it's interior is homeomorphic
Judge's note: Checked the p.4 image, last paragraph of the Introduction: "it's interior" is printed as quoted.
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Typo The text cites "Rajan and Fournier [25]", but reference [25] reads "V. Ranjan and A. Fournier."
- p. 7 Section 2, Medial axis approximation
Rajan and Fournier [25]
- p. 41 Reference [25]
V. Ranjan and A. Fournier.
- p. 41 Ref [25]
V. Ranjan and A. Fournier
Judge's note: Checked p.7 (text: "Rajan") and p.41 (reference [25]: "Ranjan"). The same cited work is spelled two ways. The fix is to make the text match the reference list.
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Typo "we assume that surface 𝒲 is contained in an open, bounded region Q" uses the object symbol 𝒲 for the surface, which was just defined as W.
- p. 7 Section 3, Surfaces and balls
we assume that surface 𝒲
Judge's note: On the p.7 image, the previous sentence defines W as the surface and calligraphic 𝒲 as the object. The next sentence then writes "surface 𝒲" in calligraphic type (continuing on p.8). The meaning is still clear, so this is a notation slip.
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Typo "union of balls, which is a also a useful shape approximation" has a stray "a" before "also".
- p. 7 Section 2, first line
which is a also a useful shape approximation
- p. 7 Related work, first line
union of balls, which is a also a useful shape approximation
Judge's note: Checked the first line of the p.7 image: the text reads as quoted.
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Typo "(equivilantly, a set of balls)" should read "equivalently".
- p. 10 Definition of WDT
(equivilantly, a set of balls)
- p. 10 Definition of WDT
equivilantly
Judge's note: Checked the WDT definition on the p.10 image: the misspelling is in the print.
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Typo The proof of Observation 1 reads "We know that d_pow(B, x)) ≤ d_pow(B′, x)", with an extra closing parenthesis.
- p. 11 Proof of Observation 1
dpow(B, x)) ≤ dpow(B′, x)
Judge's note: The text layer did not find the quote, but the p.11 image clearly shows "d_pow(B, x))" with two closing parentheses in the first line of the proof.
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Typo "d-simplicies in DT(S)" should read "d-simplices".
- p. 12 Proof of Lemma 2
d-simplicies in DT(S)
- p. 12 Proof of Lemma 2
d-simplicies
Judge's note: Checked the proof of Lemma 2 at the top of the p.12 image: the word is printed as quoted.
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Typo "an elegant correspondance" should read "correspondence".
- p. 13 Dual shapes paragraph
an elegant correspondance
- p. 13 Dual shapes
correspondance
Judge's note: Checked the last paragraph of the p.13 image: the word is printed as quoted.
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Typo "The dual shape of Y is the union of the dual faces of of every face in Pow(B) − Y" repeats "of".
- p. 14 Definition of dual shape of Y
the union of the dual faces of of every face
- p. 14 Definition of dual shape of Y
dual faces of of every face
- p. 14
s the union of the dual faces of of ev ery face in P ow
Judge's note: On the p.14 image, "of" ends one line and "of" starts the next, so the repeated word is printed, not an extraction artefact.
Second check: code check
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Typo "the dual shape of Y is analgous to the medial axis" should read "analogous".
- p. 14 Before Theorem 7
analgous to the medial axis
- p. 14
is analgous to the medial axis
Judge's note: Checked the line before Theorem 7 on the p.14 image: the word is printed as quoted.
Second check: none (judge only)
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Typo Section 6 opening paragraph repeats a word: "the boundary U_O of the union of the outer outer polar balls".
- p. 17 Section 6, first paragraph
the union of the outer outer polar balls
- p. 17 Section 6 intro
union of the outer outer polar balls
- p. 17
ndary U O of the union of the outer outer p olar balls, are b
Judge's note: The page 17 image shows 'outer' at the end of one line and again at the start of the next. This is a duplicated word.
Second check: code check
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Typo End of the proof of Lemma 17 is missing a word: "the angle between the two balls is most 2 arcsin 2r" should read 'is at most'.
- p. 20 End of Lemma 17 proof
the angle between the two balls is most 2 arcsin 2r
Judge's note: The page 20 image shows 'is most 2 arcsin 2r'. The word 'at' is missing.
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Typo Proof of Theorem 21: "The either u ∈ 𝒰_I and u ∈ 𝒰_O" should read 'Then either'.
- p. 22 Proof of Theorem 21
The either u ∈ UI and u ∈ UO
- p. 22 Proof of Theorem 21
The either u 2 UI and u 2 UO
Judge's note: The page 22 image reads 'The either u ∈ 𝒰_I and u ∈ 𝒰_O'. This is a typo for 'Then'.
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Typo Duplicated word in proof of Lemma 31: "at an angle of at most π/6 from the the vector from the nearest pole".
- p. 30 Proof of Lemma 31
from the the vector from
- p. 30 Proof of Lemma 31
from the the vector
- p. 30
t an angle of at most =6 from the the v ector from the ne
Judge's note: The page 30 image, line 2, shows 'from the the vector'.
Second check: code check
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Typo Misspelling in Section 8 opening: "a good appoximation to the medial axis" should be "approximation".
- p. 30 Section 8, first sentence
a good appoximation to the medial axis
- p. 30 Section 8 intro
good appoximation
Judge's note: Seen in the last line of page 30.
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Typo Duplicated article on page 31: "we can construct a a very small, shallow bump".
- p. 31 Section 8
we can construct a a very small, shallow bump
- p. 31 text after Figure 10
we can construct a a very small
Judge's note: Seen in the paragraph after Figure 10 on page 31.
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Typo Remark on p.33: "the value γ = Ω(r^{1/3}) in the theorem above" refers to Lemma 34, which is stated as a lemma.
- p. 33 Remark after Lemma 34
Note that the value γ = Ω(r^{1/3}) in the theorem above is not crucial
Judge's note: The result above the remark is Lemma 34, and Theorem 35 follows the remark. The reference is informal but unambiguous.
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Typo Misspelling in proof of Theorem 35 (p.34): "we assiciate a value γx" should be "associate".
- p. 34 Proof of Theorem 35
First, we assiciate a value γx
- p. 34 Proof of Theorem 35
we assiciate a value
Judge's note: Seen on page 34.
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Typo Garbled sentence at end of proof of Theorem 35 (p.34): "there is some i such that small enough so that, if s ∈ Sj for any j > i, c is within distance ε if the interior pole of s".
- p. 34 End of proof of Theorem 35
there is some i such that small enough so that, if s ∈ Sj for any j > i, c is within distance ε if the interior pole of s
- p. 34 end of proof of Theorem 35
c is within distance if the interior pole of s
Judge's note: The page 34 image confirms the wording: 'such that small enough so that' is ungrammatical, and 'if the interior pole' should read 'of the interior pole'.
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Typo Proof of Lemma 36 (p.35): "Let p be any point on the f inside the tubular neighborhood" has a stray article (should be "on f" or "on the face f").
- p. 35 Proof of Lemma 36
Let p be any point on the f inside the tubular neighborhood
Judge's note: Seen in the first line of page 35.
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Typo Page 36: "corresponding to the inside an outside of W" should read "inside and outside".
- p. 36 Before Lemma 37
corresponding to the inside an outside of W
- p. 36 before Lemma 37
the inside an outside of W
Judge's note: The page image of p.36 (line before Lemma 37) reads "the inside an outside of W". This is a missing letter, so it is cosmetic.
Second check: none (judge only)
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Typo Page 36, proof of Lemma 38: "a label will be propogated across this sample" misspells "propagated".
- p. 36 Proof of Lemma 38
a label will be propogated across this sample
- p. 36 Proof of Lemma 38
propogated
- p. 36
prop ogated across this sample
Judge's note: The p.36 image confirms the spelling "propogated" in the third paragraph of the proof of Lemma 38.
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Typo Page 38: "unwanted “hairs” on the anticrust A" is unhyphenated, while everywhere else, including the Section 10 title, the term is written "anti-crust".
- p. 38 Section 10, penultimate paragraph
unwanted “hairs” on the anticrust A
- p. 37 Section 10 heading and definition
The anti-crust A is the set of Voronoi faces
Judge's note: The p.38 image shows "anticrust A" in the second paragraph. Page 37 uses "Anti-Crust" in the heading and "anti-crust" in the text. The point is only spelling consistency.
Second check: none (judge only)
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Typo Section 11: "then the generalization of their algorithms to three dimensions produce the power crust". The subject 'generalization' is singular, so the verb should be 'produces'.
- p. 38 Section 11, after Conjecture 44
then the generalization of their algorithms to three dimensions produce the power crust
Judge's note: Confirmed on the p.38 image, after Conjecture 44. It is a grammar slip with no change in meaning.
Second check: none (judge only)
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Typo Reference [8] lists the author "H. Rushmeir", which appears to be a misspelling of "Rushmeier". "J. Mittleman" appears correct.
- p. 39 Ref [8]
J. Mittleman, H. Rushmeir
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the co-author's surname is spelled Rushmeier (Holly Rushmeier). The p.39 image shows "H. Rushmeir". Nothing in the files settles the spelling.
Second check: none (judge only)
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Typo Reference [17] reads "Computational Geometry, pp, 218-231". The comma after 'pp' is a stray and should be a period. The "S893–4" range in [15] looks deliberate.
- p. 40 Ref [17]
pp, 218-231
- p. 40 Ref [15]
pages S893{4
Judge's note: The p.40 image confirms "pp," in [17]. In [15] the range "pages S893–4" uses the S-prefixed page numbers of the SODA short-paper section; that is a plausible original style, not a defect. Only the punctuation in [17] is confirmed.
Second check: none (judge only)