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Modeling Height Distributions in Tilted Ellipsoids With Applications to Pennate Muscle Geometry.

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Title: Modeling Height Distributions in Tilted Ellipsoids With Applications to Pennate Muscle Geometry.
Authors: Rockenfeller, Robert1 (AUTHOR) rrockenfeller@uni-koblenz.de, Youssri, Youssri Hassan1 (AUTHOR) youssri@cu.edu.eg
Source: Journal of Applied Mathematics. 6/18/2026, Vol. 2026, p1-8. 8p.
Subjects: Ellipsoids, Probability density function, Biomechanics, Distribution (Probability theory), Mathematical symmetry
Abstract: This study presents closed‐form expressions for the height distribution of uniformly sampled ellipsoids and proves their invariance under tilt about a principal axis. Starting from the circle and the sphere, we extend the geometric–probabilistic framework to the triaxial ellipsoid with semiaxes a, b, c. For the upright configuration, the probability density function (PDF) of the height in the z‐direction is found to depend solely on the semiaxis length c, namely, f_Z (z) = 2 · z · c−2, for 0 ≤ z ≤ c. When the ellipsoid is rotated by an angle α about the x‐axis, the orthogonal projection of the body remains an ellipse, and its effective half‐axis length along the new height direction is λ=b·c·b2cos2α+c2sin2α−1. Accordingly, the tilted height distribution retains the same analytical form, fZ∧z∧=2·z∧·λ−2, for 0≤z∧≤λ. These expressions constitute, to our knowledge, the first explicit closed forms for the height PDFs of upright and tilted ellipsoids under area‐uniform sampling. The results unify geometric and probabilistic treatments of ellipsoidal domains and provide direct applications in biomechanics, where tilted ellipsoids serve as idealized models of pennate muscle architecture, enabling analytical estimates of mean fascicle length and its variability as a function of pennation angle. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Applied Mathematics is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Modeling Height Distributions in Tilted Ellipsoids With Applications to Pennate Muscle Geometry.
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Applied+Mathematics%22">Journal of Applied Mathematics</searchLink>. 6/18/2026, Vol. 2026, p1-8. 8p.
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  Data: <searchLink fieldCode="DE" term="%22Ellipsoids%22">Ellipsoids</searchLink><br /><searchLink fieldCode="DE" term="%22Probability+density+function%22">Probability density function</searchLink><br /><searchLink fieldCode="DE" term="%22Biomechanics%22">Biomechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Distribution+%28Probability+theory%29%22">Distribution (Probability theory)</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+symmetry%22">Mathematical symmetry</searchLink>
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  Label: Abstract
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  Data: This study presents closed‐form expressions for the height distribution of uniformly sampled ellipsoids and proves their invariance under tilt about a principal axis. Starting from the circle and the sphere, we extend the geometric–probabilistic framework to the triaxial ellipsoid with semiaxes a, b, c. For the upright configuration, the probability density function (PDF) of the height in the z‐direction is found to depend solely on the semiaxis length c, namely, f_Z (z) = 2 · z · c−2, for 0 ≤ z ≤ c. When the ellipsoid is rotated by an angle α about the x‐axis, the orthogonal projection of the body remains an ellipse, and its effective half‐axis length along the new height direction is λ=b·c·b2cos2α+c2sin2α−1. Accordingly, the tilted height distribution retains the same analytical form, fZ∧z∧=2·z∧·λ−2, for 0≤z∧≤λ. These expressions constitute, to our knowledge, the first explicit closed forms for the height PDFs of upright and tilted ellipsoids under area‐uniform sampling. The results unify geometric and probabilistic treatments of ellipsoidal domains and provide direct applications in biomechanics, where tilted ellipsoids serve as idealized models of pennate muscle architecture, enabling analytical estimates of mean fascicle length and its variability as a function of pennation angle. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
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  Data: <i>Copyright of Journal of Applied Mathematics is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1155/jama/3099113
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      – Code: eng
        Text: English
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        PageCount: 8
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      – SubjectFull: Ellipsoids
        Type: general
      – SubjectFull: Probability density function
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      – SubjectFull: Biomechanics
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      – SubjectFull: Distribution (Probability theory)
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      – SubjectFull: Mathematical symmetry
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      – TitleFull: Modeling Height Distributions in Tilted Ellipsoids With Applications to Pennate Muscle Geometry.
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            – D: 18
              M: 06
              Text: 6/18/2026
              Type: published
              Y: 2026
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