Position Based Dynamics
2006 · paper · Computer science
Presents a simulation approach that works directly on positions rather than forces or velocities. The authors say this gives control over integration and handles collisions easily, and demonstrate it in a real-time cloth simulator for games.
Checked: PDF from matthias-research.github.io, fetched 4 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance Eq. (14) C = (Σ_i (p_t1 × p_t2)·p_t3) − k_pressure V0, and the text says "The sum computes the actual volume of the closed mesh. It is compared against the original volume V0". The sum of triple products is six times the enclosed volume.
- p. 6 Eq. (14)
− kpressureV0
- p. 7 Section 4.4 text
The sum computes the actual volume of the closed mesh.
- p. 6 Eq. (14)
(pti1 ×pti2)·pti3
Only the authors can say what is right here.
Judge's note: On page 6, Eq. (14) shows no 1/6 factor. Page 7 states that the sum is the actual volume and V0 the original volume. My working: the signed volume of the tetrahedron (0, a, b, c) is (a×b)·c/6. For a closed, consistently oriented mesh, the sum over triangles of (p1×p2)·p3 is therefore 6V. Check: for a = e1, b = e2, c = e3, the triple product is 1 while the tetrahedron volume is 1/6. As printed, the constraint is satisfied when 6V = k_pressure·V0, so with V0 the original volume and k_pressure = 1 it is not at rest at the original volume. It would be consistent only if V0 were computed with the same unscaled sum, which the text does not say. Only the authors can say which convention was used.
Second check: another model agreed
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Wording In Sec. 3.5, "In lines (1)-(5) we compute the global linear velocity x_cm and angular velocity ω" names x_cm as the velocity. In the algorithm, x_cm is the centre-of-mass position (line 1) and v_cm is the linear velocity (line 2).
- p. 5 Section 3.5, paragraph after damping algorithm
In lines (1)-(5) we compute the global linear velocity xcm and angular velocity ω of the system.
- p. 5 Section 3.5, paragraph after damping algorithm
In lines (1)-(5) we compute the global linear velocity xcm and angular velocity ω of the system
- p. 5 3.5 Damping, text after algorithm
we compute the global linear velocity xcm and angular velocity ω
- p. 5 Sec. 3.5 text after damping algorithm
we compute the global linear velocity xcm and angular velocity
Judge's note: On page 5, the damping algorithm defines x_cm = (Σ x_i m_i)/(Σ m_i) in line (1) and v_cm in line (2). The following sentence uses 'global motion v_cm + ω × r_i'. The text names the wrong symbol. The fix (v_cm) is evident.
Second check: another model agreed
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23 minor (typos and the like; most checked by the judge only)
Reference or label Sec. 4.4: "These gradients have to be scaled by the scaling factor given in Eq. (7) and weighted by the masses according to Eq. (9)". Eq. (7) is the unweighted factor; the mass-weighted factor that goes with Eq. (9) is Eq. (8).
- p. 7 Section 4.4, after Eq. (15)
These gradients have to be scaled by the scaling factor given in Eq. (7) and weighted by the masses according to Eq. (9)
- p. 7 Section 4.4, after Eq. (15)
scaled by the scaling factor given in Eq. (7) and weighted by the masses according to Eq. (9)
Judge's note: I checked page 4. Eq. (7) is s = C/Σ_j|∇C|² for equal masses. Eq. (8) is s = C/Σ_j w_j|∇C|². Eq. (9) is Δp_i = −s w_i ∇C, using the s from Eq. (8). If masses are weighted per Eq. (9), the factor should come from Eq. (8), so the pointer to Eq. (7) appears to be wrong. With equal masses the two factors differ only by a constant, so I rate this as a minor pointer error.
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Typo CCS descriptors are printed fused without a separator: "and Object ModelingPhysically Based Modeling" and "RealismAnimation and Virtual Reality".
- p. 1 Categories and Subject Descriptors
Object ModelingPhysically Based Modeling
Judge's note: On the page 1 crop, both run-ons appear as printed. The separator (e.g. '—') between the subcategory and the descriptor is missing. Cosmetic.
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Typo Word choice: "is similar to the one proposed in [GHDS03] and [BMF03] but adopted to the point based approach". 'adapted' is meant.
- p. 2 Section 2, paragraph on constraint functions
but adopted to the point based approach
Judge's note: The page 2 crop shows 'adopted to'. In context the bending term is modified (adapted) for the approach.
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Typo Section 4 opens "We have used the point based dynamics framework", while the method is called 'position based dynamics' in the title and elsewhere.
- p. 6 Section 4, first sentence
We have used the point based dynamics framework
- p. 2 Section 2
adopted to the point based approach
- p. 4 Section 3.3
point based approach is more direct
Judge's note: The page 6 crop shows 'point based dynamics framework'. Pages 2 and 4 use the informal 'point based approach'. The meaning is recoverable, but the framework name is inconsistent with the title and could be confused with point-based (meshless) methods. This is a consistency point only.
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Typo Grammar: "It does this by repeatedly project each constraint in a Gauss-Seidel type fashion". It should read 'projecting'.
- p. 3 Section 3.1, after algorithm
does this by repeatedly project each constraint
- p. 3 Sec 3.1
It does this by repeatedly project each constraint
Judge's note: The page 3 crops show the wording as quoted. It is a plain verb-form slip.
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Typo Grammar: "The original Gauss-Seidel algorithm (GS) can only handle linear system." It should read 'linear systems'.
- p. 3 Section 3.2
can only handle linear system
Judge's note: The page 3 crop shows the text as quoted.
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Typo Missing word: "application of the method in the subsequent and in the results section". A noun such as 'section' is missing after 'subsequent'.
- p. 3 Section 3, first paragraph
in the subsequent and in the results section
Judge's note: The page 3 crop (Section 3, first paragraph) shows the text as quoted.
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Typo Notation slip: "If the correction ∆p is chosen to be along ∇C_p". Everywhere else the subscript p is on the operator (∇_p C).
- p. 4 Section 3.3, left column
If the correction ∆p is chosen to be along ∇Cp
Judge's note: On the page 4 crop, the subscript p is attached to C here. In the preceding sentence and in Eqs. (3) and (4) it is ∇_p C.
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Typo Grammar: "ghost forces which act like external forces dragging and rotation the object". It should read 'rotating'.
- p. 4 Section 3.3
dragging and rotation the object
- p. 4
external forces dragging and rotation the object
Judge's note: The page 4 crop shows the text as quoted.
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Typo Garbled phrase: "In this case a point with infinite mass, i.e. w_i = 0, does not move for example as expected." 'for example' is misplaced.
- p. 4 Section 3.3, right column
does not move for example as expected
Judge's note: The page 4 crop (right column) shows the text as quoted. The intended order appears to be 'for example, a point with infinite mass ... does not move, as expected'.
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Typo Number agreement: "if a point q of one objects moves through a triangle". It should read 'one object'.
- p. 5 Section 3.4, right column
if a point q of one objects moves through a triangle
Judge's note: The page 5 crop shows the text as quoted.
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Typo Figure 4 caption: "The actual dihedral angle φ is measure as the angle between the normals". It should read 'measured'.
- p. 6 Figure 4 caption
is measure as the angle between the nor-
- p. 6 Figure 4 caption
is measure as the angle between the nor-mals
- p. 6 Figure 4 caption
is measure as the angle
Judge's note: The page 6 crop of the Figure 4 caption shows the text as quoted.
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Typo Duplicated word in the Figure 5 caption: "stays above the triangle p1,p2,p3 by the the cloth thickness h."
- p. 6 Figure 5 caption
by the the cloth thickness h
- p. 6
bove the triangle p1,p2,p3 by the the cloth thickness h.
Judge's note: On page 6, the Figure 5 caption shows 'the the' on one line.
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Typo Notation inconsistency: Section 4.1 defines k_stretch and k_bend, but the Figure 3 caption writes "(k_stretching, k_bending) = (1,1)".
- p. 6 Section 4.1
stiffness kstretch and type equality
- p. 7 Figure 3 caption
(kstretching,kbending) = (1,1)
Judge's note: Page 6 (Sec. 4.1) uses k_stretch and k_bend. The page 7 Figure 3 caption uses k_stretching and k_bending. These are the same parameters under different subscripts; the meaning is clear.
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Typo Figure 4 caption: "The actual dihedral angle φ is measure as the angle between the normals". "measure" should be "measured".
- p. 6 Figure 4 caption
The actual dihedral angle ϕ is measure as the angle
Judge's note: I opened the crop, and the image of the Figure 4 caption shows "is measure as". This is a simple grammatical slip.
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Typo Sec. 4.1: "The scalar ł0 is the initial length of the edge". The image shows ł (l with a stroke), but the constraint just above uses l0.
- p. 6 Sec. 4.1
The scalar ł0 is the initial
Judge's note: The crop confirms that the printed glyph is ł with a stroke, so this is not only a text-layer artefact. The equation C_stretch = |p1−p2| − l0 directly above uses a plain l0.
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Typo Figure 11 ("Three inflated characters experience multiple collisions and self collisions.") is never cited in the text. Figures 1–10 and 12 are all cited.
- p. 9 Figure 11 caption
Three inflated characters experience multiple collisions and self collisions.
Judge's note: I searched the full text for figure citations. 'Figure 11' appears only in its own caption on page 9. The Results section cites Figures 1, 6, 9, 10 and 12, but not 11. I opened page 9: the figure is present but is never introduced.
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Typo Appendix A: the equation number "(20)" is printed on a line below Eq. (19) with no equation beside it.
- p. 10 Appendix A, after Eq. (19)
(20)
Judge's note: The crop of page 10 shows (20) alone on an empty line between Eq. (19) and 'where p̃ is the matrix...'. This is a typesetting leftover.
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Typo Reference [MHTG05] lists "TESCHER M.", but the same author is spelled TESCHNER M. in [THM*03] and [THMG04].
- p. 10 Reference [MHTG05]
TESCHER M.
Judge's note: The crop shows 'TESCHER M.' in [MHTG05]. The same page has 'TESCHNER M.' in [THM*03] and [THMG04], so the inconsistency is internal to the paper.
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Typo Reference [THM*03] lists the co-author as "POMERANERTS D.", which is apparently a misspelling of Pomeranets.
- p. 10 Reference [THM*03]
POMERANERTS D.
- p. 10
POMERANERTS D., GROSS M.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the co-author's name is spelled 'Pomeranets'. The crop shows 'POMERANERTS D., GROSS M.:' in [THM*03]. The name appears nowhere else in the paper to compare against.
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Typo Reference [Rat04]: "Rocket - a viewer for real-time phyics simulations". "phyics" should be "physics".
- p. 10 Reference [Rat04]
real-time phyics sim-
- p. 10 References, [Rat04]
a viewer for real-time phyics sim-ulations
- p. 10
a viewer for real-time phyics sim-
Judge's note: The crop shows 'phyics' in the [Rat04] entry, so this is a printed misspelling. The paper itself calls Rocket 'a game-like environment for physics simulation'.
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Typo A stray glyph (Ű/Ů) is printed where a dash belongs in two references: [Jak01] "Advanced character physics Ű the fysix engine" and [Pro95] "(1995), 147Ű–154".
- p. 10 Reference [Jak01]
Advanced character physics
- p. 10 Reference [Pro95]
147
Judge's note: The crops of page 10 show the stray accented-U glyph in both entries. The glyph is visible in the rendered PDF, so it is an encoding error in the bibliography, not just an extraction artefact.
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Typo The same author is spelled "SCHRÖDER P." in [DSB99] but "SCHRODER P.:" without the umlaut in [GHDS03].
- p. 10 Reference [GHDS03]
SCHRODER P.:
- p. 10 Reference [DSB99]
SCHRÖDER P.
- p. 10 References, [GHDS03]
SCHRODER P.
Judge's note: The crops confirm the umlaut in [DSB99] and its absence in [GHDS03]. This is a minor inconsistency; either form may be acceptable, but the two entries should match.
Second check: none (judge only)