IMA Journal of Mathematics Applied in Medicine & Biology (1992) 9, 289-294
On the Penrose hypothesis on fingerprint patterns K. V. MARDIA, Q. LI, AND T. J. HAINSWORTH
Department of Statistics, University of Leeds, Leeds, UK [Received 11 July 1992]
Keywords: curvature; dermatoglyphic; differential equation; fingerprints; pattern recognition; Penrose hypothesis.
1. Introduction
Since Galton (1892), different types of fingerprint patterns have been well established. Similar patterns appear in the palm and generally all the patterns are called dermatoglyphic patterns (Penrose, 1969). Penrose hypothesized (Penrose & O'Hara, 1973; Smith, 1979) that the dermatoglyphic ridges might correspond to the lines of curvature of the skin of the embryo at the time when the ridges were being formed, although they no longer do so everywhere after birth. Under the above hypothesis, Smith (1979) derived some forms (triradius, tented arch, loop) from a differential equation. We give the exact solution of the equation. Also, we generalize his work to include the remaining important forms: symmetrical whorl, spiral whorl, double loop, accidental whorl, and pocket whorl. Hence, this work supports the hypothesis of Penrose on the formation of ridge patterns. However, the dermatoglyphic ridges might follow what were originally lines of stress rather than lines of curvature.
2. The exact solution
Assuming that ridges follow the lines of curvature, it can be shown (Smith, 1979) that the tangent to the curve passing through a point with Cartesian coordinates (x, y) can be modelled as the minor axis of an (arbitrarily small) ellipse centred at (x, y). Let (X, Y) be the Cartesian coordinates of a point lying on the ellipse with centre at (x, y) having equation aii(X
- x)2 + 2a12(X - x)(Y-y)
+ a22(Y - y)2 = k2,
(1)
where k is 'small' and the {atJ} are the coefficients of the ellipse (a12 = a2l). Suppose that the ellipse coefficients are a linear function of the position of the 289
On the Penrose hypothesis on fingerprint patterns K. V. MARDIA, Q. LI, AND T. J. HAINSWORTH
Department of Statistics, University of Leeds, Leeds, UK [Received 11 July 1992]
Keywords: curvature; dermatoglyphic; differential equation; fingerprints; pattern recognition; Penrose hypothesis.
1. Introduction
Since Galton (1892), different types of fingerprint patterns have been well established. Similar patterns appear in the palm and generally all the patterns are called dermatoglyphic patterns (Penrose, 1969). Penrose hypothesized (Penrose & O'Hara, 1973; Smith, 1979) that the dermatoglyphic ridges might correspond to the lines of curvature of the skin of the embryo at the time when the ridges were being formed, although they no longer do so everywhere after birth. Under the above hypothesis, Smith (1979) derived some forms (triradius, tented arch, loop) from a differential equation. We give the exact solution of the equation. Also, we generalize his work to include the remaining important forms: symmetrical whorl, spiral whorl, double loop, accidental whorl, and pocket whorl. Hence, this work supports the hypothesis of Penrose on the formation of ridge patterns. However, the dermatoglyphic ridges might follow what were originally lines of stress rather than lines of curvature.
2. The exact solution
Assuming that ridges follow the lines of curvature, it can be shown (Smith, 1979) that the tangent to the curve passing through a point with Cartesian coordinates (x, y) can be modelled as the minor axis of an (arbitrarily small) ellipse centred at (x, y). Let (X, Y) be the Cartesian coordinates of a point lying on the ellipse with centre at (x, y) having equation aii(X
- x)2 + 2a12(X - x)(Y-y)
+ a22(Y - y)2 = k2,
(1)
where k is 'small' and the {atJ} are the coefficients of the ellipse (a12 = a2l). Suppose that the ellipse coefficients are a linear function of the position of the 289
