Stem surface as a Quasisphere

Abstract

Consider a tree trunk. One may imagine a broken line consisting of two straight segments. The one is a perpendicular ascending from the centre of its base disk to the centre of the disk at the height z, and the other a horizontal stretching from there to the stem surface of radius r. We will call the line a "S-S broken line". If the distance between two points A and B on the same S-S broken line is defined in terms of time necessary for the tree tissue to spread over the interval (A, B), all the points of the stem surface are equidistant from the centre of the base disk. This means a stem surface a kind of a sphere. We will call it a stem quasisphere Q. An equation of the quasisphere Q at age t is

in which H, R(z) are limiting values of the stem height and of the radius r of the height z, and , the inverses of the growth curves of the stem height z and of the radius r at the height z, respectively. Let us define a kind of index function u (t, z, r) of three variables age t , height z and the radius r as

We call it a stem Heaviside function. Then u is a 'weak solution' of a linear homogeneous partial differential equation

in which is a gradient of its stem curve, and the stem surface is a characteristic surface of the equation. In other words, the solution u = 1 jumps into another solution u = 0 discontinuously on the stem surface. Therefore the stem surface can be regarded as a kind of 'wave front' of its growth.