, , If we introduce a more general and consistent notation of then = = and = whereas = 1. Note that is invariant with respect to reflections α α111, β α110 γ α113, α100 Vn and permutations of the axes. We shall further refer to as the growth parameter of the n plane. αn The equation of the plane at time is n t where is a dimensionless parameter which controls the initial position of the plane (at = 0) and is dn t l0 taken to be the initial size of the crystal along the -axis. In fact, the distance from the origin to the plane at x n time = 0 is equal to t . It is now natural to introduce dimensionless such that normalized Cartesian coordinates rˆ, Obviously, division by is not practical in computer calculations, but using dimensionless coordinates makes t sense in a theoretical treatment. Then the equation of the plane becomes n πn which is equivalent to where We see that the set of parameters and completely control the shape of the crystal at all times , with L, dn t αn being the only parameter with dimensions (L controls the scaling of the time axis and the initial size of the L crystal). Computationally, the easiest way to specify a plane is by providing its normal vector and the point on that n plane closest to the origin, which is the intersection of the plane with the straight line going along n. Suppose that this point is Then, from the equation of the plane,
, , If we introduce a more general and consistent notation of then α = α111, β = α110 and γ = α113, whereas α100 = 1. Note that Vn is invariant with respect to reflections and permutations of the axes. We shall further refer to αn as the growth parameter of the n plane. The equation of the n plane at time t is where dn is a dimensionless parameter which controls the initial position of the plane (at t = 0) and l0 is taken to be the initial size of the crystal along the x -axis. In fact, the distance from the origin to the n plane at time t = 0 is equal to . It is now natural to introduce dimensionless normalized Cartesian coordinates rˆ, such that Obviously, division by t is not practical in computer calculations, but using dimensionless coordinates makes sense in a theoretical treatment. Then the equation of the n plane πn becomes which is equivalent to where We see that the set of parameters L, dn and αn completely control the shape of the crystal at all times t , with L being the only parameter with dimensions (L controls the scaling of the time axis and the initial size of the crystal). Computationally, the easiest way to specify a plane is by providing its normal vector n and the point on that plane closest to the origin, which is the intersection of the plane with the straight line going along n. Suppose that this point is Then, from the equation of the plane,
GeometricModel-revised2.docx