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

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, fZ 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.