Conics on a Sphere

Graphic Statics offers a geometrical framework for the design & analysis of structural systems with reciprocity between the ‘form’ and ‘force’ diagrams being a fundamental notion. In this paper we provide a detailed illustration and explanation of Maxwell’s reciprocal construction for 2-dimensional trusses through a 3-dimensional paraboloid of revolution. Even though Maxwell briefly mentions this method there is no record of the detailed construction. The main focus of this research is placed on the synthesis of Poncelet duality in projective geometry and of reciprocity in graphic statics resulting in the definition of a generalised dual statement for the construction of reciprocal diagrams (Maxwell 2D & Rankine 3D). This is based on the construction of their higher dimensional polyhedral Airy stress functions, using conic sections. We generalise the construction for other conics, such as the sphere and ellipsoid, and we extend the definition for 3-dimensional trusses and corresponding higher-dimensional conic sections.

Self-Similar Expansion (SSE) fitting models the leading edge of the Coronal Mass Ejection (CME) as a self-similar expansion sphere with a fixed propagation direction and a fixed angular width. Cone angle of the heliocentric cone which is tangent to the sphere usually refers to the angular width of the CME, ranging from 0 to 90 degrees. Angular widths of Fixed-Φ (FΦ) and Harmonic-mean (HM) fitting are defined, corresponding to 0 and 90 degrees of SSE model, respectively. Modified SSE (MSSE) fitting discussed in this paper models the leading edge of the CME as a self-similar expansion semi-sphere. It is able to extract CME parameters using MSSE fitting, including propagation direction, velocity and cone angle of the heliocentric cone, whose conic section is across the center of the semi-sphere, ranging from 0 to 90 degrees. Cone angles of FΦ and HM fitting are defined, corresponding to 0 and 45 degrees of MSSE model, respectively. Leading edge of the CME can be assumed as a heliocentric semi-sphere when cone angle is assumed to be 90 degrees. MSSE fitting expands the morphology of leading edge of the CME and considers cone model as a reference to determine whether and when the CME will reach a satellite. 23 CMEs were analyzed based on the observations by Heliospheric Imager (HI) from one spacecraft of STEREO, combined with in-situ observations of solar wind on STEREO and ACE. We found that the error of the CME propagation direction, arrival speed and theoretical arrival time analyzed by MSSE fitting decreased, compared to FΦ, HM and SSE fitting.