A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Antonio Sauci
Date: 2018 Jan 21, 13:49 -0800
I downloaded this evening the new 2017 edition of publication no. 9, published by the National Geospatial.Intelligence Agency. While scrolling down volume 2,for a possible print-out of its tables, I was surprised to find in the entry for "Cardiod", in the section entitled "Glossary of Marine Navigation",on page 312, no actual figure which depicts what a cardioid looks like. The figure in the book only depicts three concentric circles in different colours and radii and another fourth circle tangent to the innermost one and of the same radius and with center also laying on the same x-axis.
Regarding the definition,although accurate, is only part of the problem. Let me expand on this topic. If a generating circle of radius r rolls along the exterior (or the interior) circunference of a fixed circle of radius r1,then each point of the first circle describes a simple (general,normal) epicycloid (or hypocycloid),respectively.The parametric equations are:
x = (r1+ or -)· cos(t) - or + r ·cos ((r1+r)/r) t and y= (r1+ or -)· sin(t) - r· sin((r1+r)/r)t
The (+) signs for epicycloid; the (-)sign for hypocycloid.. Only when r1 = r, we get the cardioid of equations: x = r(2cos(t) -cos(2t)) and y=r(2sin(t) - sn(2t)) , while if we let r1=2r,then we get another epicycloid,called a nephroid, and whose equations are: x =r(3cos(t)-cos(3t)) and y=r(3sin(t) -sin(3t)).
For someone interested in visualizing a real cardioid,it is much easier to use its polar coordinates equation: ro=2r(1+cos(phi)).
I infer from my very quick inspection of the edition that the two books need a good proof-reading.