To the consternation of my companion, I immediately searching for Hogarth curves on my phone. There was no mathematics--the curves had been used to describe the aesthetics in drawing, painting, and flower arrangement, and their name derived from the “line of beauty” extolled by William Hogarth in his 1753 book,

*The Analysis of Beauty*. Hogarth was an 18th-century English painter and writer. He (and many subsequent flower arrangers) saw s-shaped curves as signifying liveliness and activity that cannot be found with straight lines or other curves. I found no mathematical or scientific discussion of Hogarth curves, so I will speculate on an interpretation now.

There is one special point on an s-shaped curve called the point of inflection. It is in the middle of the curve, and is the point where the curvature changes from negative to positive. In other words, it is a point of zero curvature. Imagine looking at an s-shaped curve drawn on a flat sheet of paper, and noting (with a very sharp pencil) the point of inflection. Now change your point of view (e.g., slant the paper away from you) and re-image the figure with a pinhole camera. Surprisingly, the point of inflection of the new image is exactly at the image of the point you noted with the pencil. The point of inflection is an invariant of the perspective projection.

This invariance is a mathematical property of projective geometry (which includes perspective). A projective transformation sends any straight line into another straight line. It is more general than a linear transformation because parallel lines when mapped may converge: the meeting point is called a vanishing point. When you look at an s-shaped curve, there is only one point of that curve whose neighborhood is a straight line, and that is the point of inflection. Three points on the curve can be made as collinear as desired by moving them closer and closer to the point of inflection. Since the neighborhood of the inflection point is a straight line segment, the segment will stay at the same part of the s-curve when that curve is projected to another planar image.

What does this have to do with Hogarth’s praise for s-shaped curves in art? It might be that the inflection point is anchoring a moving viewer to a distinct point in the picture, relative to which all else is swirling around. By being a view-independent feature, it may be an effect opposite to the elusive smile of the Mona Lisa, which (as explained by Margaret Livingstone at Munsell2018) is a low-spatial-frequency rendering that is noticeable only in peripheral vision, which sees with much lower resolution than our fovea (center of vision).

In any event, this tiny vestige of math in the middle of flower arrangements may allow me to revisit the 1960s with a sense of entitlement to its memories.

Michael H. Brill

*Datacolor*