Definition and Analysis of the Modulus Length: Based on the Envelope Interface Framework

Within the framework of the Equal Expansion Rate Evolving Envelope Surface, the modulus length is defined as the radial modulus of the generating curve: M(theta) = r(theta) = r0 * e^(b * theta) It is not a global constant, but a quantity that evolves at a fixed rate along the envelope surface. This paper proves two structural relations under the condition that the growth rate b and the angular speed omega are both constant. First, the tangential, vertical, and radial velocity components maintain constant ratios: s(theta) / v(theta) = bd(theta) / v(theta) = b Second, the sum of the squared vertical and tangential velocities is proportional to the square of the modulus length: s(theta)^2 + v(theta)^2 = (1 + b^2) * omega^2 * M(theta)^2 with the proportionality factor K = (1 + b^2) * omega^2 constant throughout the evolution. This paper further shows that this constancy is conditional: if b changes, then K changes, the equal expansion rate condition is lost, and the structure exits the definition domain of the equal expansion rate evolving envelope surface. Keywords: modulus length, equal expansion rate evolving envelope surface, survey line spiral, radial modulus, velocity ratio, velocity-modulus relation