Linear Addivity Theorem

This paper describes a detailed mathematical structure that examines how boundary properties remain constant, add linearly and scale within divided conic systems. The standard rules of Euclidean geometry mostly focus on single, undivided geometric shapes - but the process of dividing those curves into any series of smaller parts that lie on the same line shows that certain laws of conservation do not change. The authors organized Chapter 1 to focus on divided circular systems. In this section we use basic real analysis and trigonometric identities to provide exact proofs for specific behaviors. As a large circle with a measured radius R is divided into a number of smaller circles along its centre line or diameter, the total length of the small arcs, chords, segment boundaries and sector boundaries that share a single central angle θ is equal to the measurements of the large circle. To expand this study, we wrote Chapter 2 to focus on divided elliptic systems - those theorems now apply to conic sections where the curvature changes. By using parametric equations, coordinate geometry and calculus integrals that measure lines, we show that boundary lengths add together exactly. It is clear that the properties stay the same across similar elliptic parts that we place along the primary axis.