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LIMACON OF
PASCAL
BY FRANCIS
CHAN
 
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HISTORY
The Limacon of Pascal is sound by Roberval between 1630 and 1640.
He developed a method of drawing tangents by considering a curve as being
described by the resultant of two or more multaneous movements, and one
of them is which he calls the Limacon of Pascal. And this curve is also
refers to Etinne Pascal, the father of Blaise Pascal.
PROPERTIES
Limacon of Pascal is a special case of epitrochoid, when the rolling and
fixed circles has equal radius. i.e., it is the trace of a point Q fixed
to a circle that rolls around another circle of the same size.
POLAR EQUATION OF PASCAL
Figure 2
Let the diameter AB=2a, and DC=k.
From the Figure 2,
The polar equation can be derived by using the Trig identity.
Since the segment AC is the path to be parametric in the
polar coordinate, the polar equation is
PARAMATRIC EQUATION OF PASCAL
The parametric equation can be
derived by converting the polar equation by substitute the equation 
to
and
.
Therefore, the parametric equation of the Limacon of Pascal
is
Here is the animation of the Limacon of Pascal
RELATED WEB SITES
Key Curriculum Press Inc., Web Site, http://www.keypress.com/sketchpad/java_gsp/index.html
Lockwood, E.H. (1961) A Book of Curves. Great Britian:
Cambridge University Press, 1967
MacTutor
Famous Curve Index
http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/limacon.html
Xah:
Special Plane Curves
http//www.best.com/~wah/SpecialplaneOfPascal_dir/limaconOfPascal.html