Equation of Limaçon of Pascal/Cartesian Form

Theorem

The limaçon of Pascal can be defined by the Cartesian equation:

$\paren {x^2 + y^2 - a x}^2 = b^2 \paren {x^2 + y^2}$

Proof

\(\ds r\)

\(=\)

\(\ds b + a \cos \theta\)

Equation of Limaçon of Pascal: Polar Form

\(\ds \leadsto \ \ \)

\(\ds r^2\)

\(=\)

\(\ds r b + r a \cos \theta\)

\(\ds \leadsto \ \ \)

\(\ds x^2 + y^2\)

\(=\)

\(\ds b \sqrt {x^2 + y^2} + a x\)

Conversion between Cartesian and Polar Coordinates in Plane

\(\ds \leadsto \ \ \)

\(\ds x^2 + y^2 - a x\)

\(=\)

\(\ds b \sqrt {x^2 + y^2}\)

rearranging

\(\ds \leadsto \ \ \)

\(\ds \paren {x^2 + y^2 - a x}^2\)

\(=\)

\(\ds b^2 \paren {x^2 + y^2}\)

rearranging

$\blacksquare$

Also see

• Equation of Limaçon of Pascal/Polar Form