Research ArticleMATHEMATICS

The Euler spiral of rat whiskers

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Science Advances  15 Jan 2020:
Vol. 6, no. 3, eaax5145
DOI: 10.1126/sciadv.aax5145
  • Fig. 1 A rat and its whiskers.

    (A) A photograph of a rat (photo credit: Maria Panagiotidi, University of Salford). (B) The mystacial pad with labeled locations of the base points of 30 whiskers at the right side of a rat. The mystacial pad matrix has five rows (A to E) and seven columns (1 to 7); for five entries, the whiskers are absent. (C) A two-dimensional scan of a whisker.

  • Fig. 2 The normalized Euler spiral.

    The crosses mark the limit points x=y=±2π/4.

  • Fig. 3 Mapping whiskers onto the Euler spiral.

    (A) Three exemplar whiskers approximated by Euler spirals (in millimeters) are shown in dashed color over the original data (thick gray). (B) The same three whiskers conformally mapped onto the universal Euler spiral (the base ends are marked with σ0 and the tips with σ1). The green whisker (A) has a noticeably increasing curvature and is mapped onto the right. The blue whisker (B) has the most uniform (but slightly decreasing) curvature, and it appears on the left part of the spiral. The red whisker (C) has an inflection close to its tip and hence passes through the origin. In placing them on the Euler spiral, the three whiskers are individually scaled.

  • Fig. 4 Whisker shapes and shape parameters, 516 whiskers for 15 rats, each animal painted in its reference color.

    (A) Individually layered 516 whisker shapes mapped onto intervals of the universal Euler spiral shown in gray underneath. See fig. S7 for the whisker density map. (B) The parametric plane (σ0, σ1). All points lie above the diagonal σ1 = σ0. Shapes with σ0 < σ1 < 0 (0 < σ0 < σ1) have decreasing (increasing) curvature and have no inflections. Shapes with σ0 ≤ 0 ≤ σ1 are inflectional; among them, there are more with σ1 < − σ0 (below the dashed diagonal), i.e., with the inflection point closer to the tip than to the base.

  • Fig. 5 Euler spiral approximations of whisker planar shapes presented in the form of the mystacial pad matrix (30 entries, 5 rows, and 7 columns) (see Fig. 1B, repeated in the upper right corner).

    Colors mark 15 different animals (same as in Fig. 4). Black curves show mean Euler spiral approximations. The coordinate axes are marked in millimeters. The dark and light gray backgrounds correspond to the matrix entries where the mean Euler spirals have an inflection point (dark) and where their curvature increases from base to tip (light).

  • Fig. 6 Configuration of the right side of the whisker sensory shroud.

    The origin 0,0,0 is placed at the mean position of all whisker basepoint locations (for both mystacial pad vibrissae), the y axis points rostrally, and the negative y axis points caudally, the xy plane is the average whiskerrow plane, and the yz plane is the sagittal plane. (A) Each of the 30 whiskers is represented by a Euler spiral; the blue balls mark the base points at the rat’s mystacial pad, and the pink balls show the tips. (B) The surface spanned by the whisker tips (yellow) is approximated by an ellipsoid (transparent). Arrows show tangent vectors (light blue) at the tips and normals (red) to the ellipsoidal surface at points closest to the tips; the normals are shifted to the corresponding tips. See three-dimensional interactive in figs. S8 and S9.

  • Table 1 The angles (in degrees) between the whisker tip tangents and the normals to the ellipsoidal approximation of the rat’s sensory shroud (see Fig. 6B) for each position on the mystacial pad, averaged across the whole dataset.

    1234567
    A4447504233
    B4246505049
    C36424850495049
    D36434950494848
    E455254555355

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/3/eaax5145/DC1

    Details of Results

    Fig. S1. The RSD graphs for the Euler spiral fits.

    Fig. S2. Comparison of residual mean square graphs for circular arc (red), Euler spiral (green), and quadratic curvature (blue) fits.

    Fig. S3. Distribution of lengths and the curvature coefficients.

    Fig. S4. Distribution of lengths L and coefficients B and A across the mystacial pad matrix.

    Fig. S5. Violin plots showing distributions of the coefficients B (left) and A (right) relative to individual animals.

    Fig. S6. Distribution of 30 average characteristics of shapes for each mystacial follicle.

    Fig. S7. Density of whiskers on the universal Euler spiral.

    Fig. S8. Configuration of the right half of the whisker sensory shroud (interactive three-dimensional image).

    Fig. S9. Configuration of the right half of the whisker sensory shroud (interactive three-dimensional image).

    Fig. S10. A schematic of a whisker in a planar approximation.

  • Supplementary Materials

    This PDF file includes:

    • Details of Results
    • Fig. S1. The RSD graphs for the Euler spiral fits.
    • Fig. S2. Comparison of residual mean square graphs for circular arc (red), Euler spiral (green), and quadratic curvature (blue) fits.
    • Fig. S3. Distribution of lengths and the curvature coefficients.
    • Fig. S4. Distribution of lengths L and coefficients B and A across the mystacial pad matrix.
    • Fig. S5. Violin plots showing distributions of the coefficients B (left) and A (right) relative to individual animals.
    • Fig. S6. Distribution of 30 average characteristics of shapes for each mystacial follicle.
    • Fig. S7. Density of whiskers on the universal Euler spiral.
    • Fig. S8. Configuration of the right half of the whisker sensory shroud (interactive three-dimensional image).
    • Fig. S9. Configuration of the right half of the whisker sensory shroud (interactive three-dimensional image).
    • Fig. S10. A schematic of a whisker in a planar approximation.

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