Constant Expansion Rate Evolving Envelope Interface: Definition, Geometric Properties, and Physical Significance

This paper defines a special class of envelope surfaces — the Constant Expansion Rate Evolving Envelope Interface. It is a closed surface generated by the rotational sweep of a Survey Line Spiral at constant angular speed omega and fixed growth rate b. The core characteristics of this envelope surface are: it appears at a fixed frequency, and its front and rear closure points connect end-to-end, forming a continuous evolution chain. Its geometry is completely determined by three parameters: the initial radius r0, the growth rate b, and the angular speed omega. In terms of dynamic properties, this paper proves two structural constraints. First, the tangential, vertical, and radial velocity components of the generating curve maintain constant ratios, satisfying s(theta)/v(theta) = b and d(theta)/v(theta) = b. Second, the sum of squared velocities is proportional to the square of the radial modulus, satisfying s^2 + v^2 = (1 + b^2) * omega^2 * M^2, where M(theta) = r(theta) is the radial modulus. The proportionality coefficient is jointly determined by the growth rate b and the angular speed omega. This paper provides its precise definition, parametric equations, closure point conditions, and geometric and velocity properties. It should be particularly noted that in this paper, M(theta) = r(theta) is the radial modulus, evolving at the fixed rate e^(b*theta). It is not a global constant, nor is it equivalent to the "modulus length" defined elsewhere as the product of minimum length and maximum frequency. The two terms are similar in name but refer to different objects and should not be confused. Keywords: Constant Expansion Rate Evolving Envelope Interface, Survey Line Spiral, fixed frequency, closure point connection, rotational sweep, velocity ratio constancy, velocity-modulus relation