English: Explanatory image, nucleophile approach trajectories analogized to the celestial (horizontal) coordinate system. The image is a three-dimensional representation of the approach of a nucleophile, Nu, with lone pair electrons (red ":"), to an en electrophile, here a carbonyl functional group of an organic compound (an aldehyde, ketone, or carboxylic acid derivative, though the following concepts also extend to carbon-carbon double bond reactions). The Bürgi–Dunitz angle (${\displaystyle \alpha _{BD}}$; Bürgi, Dunitz, Lehn & Wipff, 1974) that describes the elevation of the attacking Nu above the plane of the organic compound is analogized to the altitude parameter in a celestial (horizontal) coordinate system, and the Flippin-Lodge angle (${\displaystyle \alpha _{FL}}$; Heathcock, 1990) that describes the displacement of the attacking Nu toward one or the other substituent of the carbonyl group is analogized to the azimuth parameter in that coordinate system. The organic compound is an aldehyde when one substituent is a hydrogen (e.g., R' = H), a ketone when neither is a hydrogen or non-carbon atom, or may be a carboxylic acid derivative (e.g., esters or amides, with one substituent being an O- or N-substituent, respectively). Bond wedges are used to indicate that the attached groups R and R' project toward and away from the viewer, and the carbon atom of the carbonyl group is implied at the apex of the bond wedges. In this analogy, the organic compound is presented in the equatorial plane of a sphere, plane I, represented as an ellipse—really a circle, like all plane-sphere intersections, but appearing elliptical for purposes of illustration. This plane bisects the sphere into upper and lower hemispheres, and is viewed from slightly above the near "edge" of the ellipse. Reference plane I contains a horizontal diameter of the sphere (maroon line segment, solid and dashed), and a further diameter of the sphere perpendicular to it that projects before and behind the plane of the page (green line segment). The C=O bond and the atoms attaching the compound's constituents (R and R') lie fully on this first plane: the carbonyl lies on the maroon diameter, and the substituent atoms lie in-plane on either side of that diameter. A second bisecting plane, plane II, contains the plane of the page, and divides the sphere into hemispheres before and behind the plane of the page. Reference plane II contains the same horizontal diameter (maroon segment), and a further vertical diameter of the sphere perpendicular to the first (coincident maroon-green line segment, solid and dashed). Plane II contains the carbonyl C=O bond, and bisects the R-C-R' bond angle. Finally, a further vertical plane, plane III, that is orthogonal to both preceding planes divides the sphere into left and right hemispheres. Reference plane III is defined by the same vertical diameter (maroon-green segment), and the diameter projecting before and behind the page (green segment); like the green segment, this plane projects before and behind the plane of the page. Plane III contains only the carbon atom of the carbonyl group. The black line connecting the carbonyl carbon atom to the lone pair of the electronegative atom of the Nu (red ":") represents the approach trajectory of this relevant electronegative Nu atom to the carbonyl. The Bürgi–Dunitz angle, ${\displaystyle \alpha _{BD}}$, describes the elevation of this "attacking" Nu atom above plane I, analogous to altitude in the celestial coordinate system. Because the Nu atom does not necessarily lie in plane II—it can be off-plane, displaced at its elevation toward one or the other substituent of the carbonyl group (R, R')—a further angle is needed for a complete description of the trajectory. The Flippin-Lodge angle, ${\displaystyle \alpha _{FL}}$, describes the angle between plane II and the vertical plane on which the the Nu atom actually is found/inferred. (As illustrated, the Nu plane intersects Planes II and III at the vertical, maroon-green diameter described earlier.) A stated convention for ${\displaystyle \alpha _{FL}}$ is that it is positive when deviating in the direction away from the larger substituent (or away from the more electron-rich substituent, which is similarly repulsive). Hence, if R'=H, R=alkyl (i.e., the electrophile is an aldehyde, positioned with H substituent behind the plane of the page), the Nu would be expected to approach in a plane on the opposite side of Plane II, relative to what is shown. The important concepts and the utility associated with these angles (and further formal mathematical ways of determining or measuring them) are expected to appear in the relevant Wikipedia articles. Note: Drawing created for "Bürgi-Dunitz angle" and "Flippin-Lodge angle" articles at English Wikipedia by delta0349 in 2014. REFERENCES: (1) H. B. Bürgi, J. D. Dunitz, J. M. Lehn & G. Wipff (1974) Stereochemistry of reaction paths at carbonyl centres. Tetrahedron 30 (12):1563–1572. [DOI:10.1016/S0040-4020(01)90678-7]. (2) C.H. Heathcock (1990) Understanding and controlling diastereofacial selectivity in carbon-carbon bond-forming reactions, Aldrichimica Acta 23(4):94-111, esp. p. 101, and references to Flippin et al. and Lodge et al. articles therein. [Le Prof]