PMC

An example of CONICS analysis on scRNA-seq and exome-seq of a glioblstoma biopsy. (A) CONICS quantifies CNVs in single cells with a controlled error rate: ScRNA-seq read-count correlations with CNV status (top left); scRNAseq read-count distributions for an example CNV (top right); FDR estimates in assigning CNV status to individual cells, computed via cross validation (bottom left) and via comparison to a control dataset (bottom right). (B–D) CONICS estimates CNV allele frequency, which, when compared to the expression of canonical markers and clustering of CNV status, enables the rigorous separation of stromal /immune cells from neoplastic cells. (E) Co-expression network of PTEN, produced by CONICS, compared between cells with a chr. 10 loss and wild-type

Horopteric conics for anthropomorphic parameters. The conics shown in solid lines: the line horopter for F_a at the abathic distance of 99.61 cm, the hyperbola for fixation F₁, the ellipse for fixation F₂, and the parabola for fixation F₃ on the abathic iso-subtense curve (AIS) (solid line) through F_a are constructed in section 4. Each fixation point is on the corresponding Vieth-Müller circle (VMC) (dot-dash line). The conics obtained for the same fixation points by the method from Turski () are shown as dashed lines. As we can see, the hyperbola and ellipse for both methods overlay each other nearly perfectly. The difference between the parabola for F₃ and the tangent line to the VMC at F₃ is explained in the next section. The conics' orientations are given by the version angles in Binocular Conics Construction of section 4.

(a) The Juno trajectory in a coordinate system with the z‐axis aligned with Jupiter's magnetic dipole. (b) The Juno footprint mapped to Jupiter using magnetic field models along with ultraviolet images of the auroral emissions, with labels for the distinct regions and solar direction. The grey boxes indicate where Juno observed dispersive proton conics.

Two arbitrarily placed non-coplanar conics do not intersect one another. The common 3D line formed by the intersection of their planes intersects each of the two conics in two places, creating four (usually distinct) points. The two intersection points for a particular conic are complex valued if does not physically intersect that conic. The four colinear intersection points (possibly complex valued) may be used to form a cross-ratio, which is the single unique 3D invariant for a pair of 3D conics