AI 中文总结
该研究基于欧几里得尺规作图，给出开普勒轨道与牛顿平方反比定律等价性的双向几何证明，采用《原理》式论证避免微分方程，是天体力学核心课题的构造性突破。
AI 中文摘要
开普勒前两大定律指出，行星沿以太阳为一个焦点的椭圆运动，且在相等时间内扫过相等面积（面积速度恒定）。在《原理》中，牛顿阐明了这些定律与万有引力的关联，此后轨道定律与力定律的等价性一直是天体力学的核心课题。我们基于明确的欧几里得尺规作图，给出了双向等价性的完全几何证明。该证明体系结合了有限步作图、切线与三角形几何、仿射变换、局部位移比、圆锥曲线不变量及若干速端曲线实现。在这一更宽泛的框架内，一项贡献是将辅助圆用作位形空间中的主要速端曲线替代物，而非采用半长轴为2a的准线圆归一化。我们的重点是遵循《原理》式的论证，避免使用微分方程，同时保持与欧几里得方法的紧密关联。
英文摘要
Kepler's first two laws state that a planet moves on an ellipse with the Sun at a focus and sweeps out equal areas in equal times (constant areal speed). In the Principia, Newton showed how these laws connect to universal gravitation. Since then, the equivalence between orbital laws and force laws has remained a central topic in celestial mechanics. We present fully geometric proofs, built from explicit Euclidean straightedge-and-compass constructions, of this equivalence in both directions. The proof system combines finite-step constructions, tangent and triangle geometry, affine transport, local displacement ratios, conic invariants, and several hodograph realizations. Within this broader framework, one contribution is to use the auxiliary circle as the primary hodograph proxy in configuration space rather than the directrix-circle normalization of radius 2a. Our emphasis is a Principia-style argument that avoids differential equations while remaining close to Euclidean methods.
Comments37 pages, 10 figures. LaTeX source and GeoGebra construction files are available at https://github.com/CryptoDogAres/AlternativeKeplerToNewton/
