Cardioid

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CARDIOID

The figure depicted on the first page of the cover represents one of the replacement curves - the cardioid. The word “cardioid” means “heart-shaped” in Greek.

The properties of this curve were first studied in 1674 by the French academician Ole Roemer.

Cardionda has many “relatives”. These are, first of all, various cyclones, epicyclones and hypocyclondes.

If a circle rolls along a straight line, then a point fixed on this circle leaves a trace on the plane - a curve called a cyclonde (Fig. 1).

If a movable circle rolls along the inside of a fixed circle, then a point fixed on the first of them describes a curve on the plane, which is called an epicyclonde (Fig. 2).

If a movable circle rolls on the outside of a fixed circle, then the curve described by a point lying on the movable circle is called an epicyclonde (Fig. 3).

A cardioid is an epicyclone whose movable and stationary circles are of the same radius (see Fig. 4).

You can read about the features of all these cyclones, including the cardionde, in G. N. Berman’s book “Cycloid” (Gostekhidat, 1954).

figure 5 shows one of the possible constructions of the cardionda. The fixed circle is divided into sections by points. From each such point, as from the center, circles are drawn that also pass through one fixed point on the same circle. Try to prove that with this construction you really get a cardiond (circumscribed around all these circles).

In conclusion, a fun interpretation of the cardionda. Let’s consider a non-slippery, stationary waist, round in cross section. A hula hoop with twice the radius is placed next to it. Then each point of the hoop is described near the waist of the cardiond.

M. L. Smolyansky