<P> When b> a (\ displaystyle b> a) the limaçon is a simple closed curve . However, the origin satisfies the Cartesian equation given above so the graph of this equation has an acnode or isolated point . </P> <P> When b> 2 a (\ displaystyle b> 2a) the area bounded by the curve is convex and when a <b <2 a (\ displaystyle a <b <2a) the curve has an indentation bounded by two inflection points . At b = 2 a (\ displaystyle b = 2a) the point (− a, 0) (\ displaystyle (- a, 0)) is a point of 0 curvature . </P> <P> As b (\ displaystyle b) is decreased relative to a (\ displaystyle a), the indentation becomes more pronounced until, at b = a (\ displaystyle b = a), the cardioid, it becomes a cusp . For 0 <b <a (\ displaystyle 0 <b <a), the cusp expands to an inner loop and the curve crosses itself at the origin . As b (\ displaystyle b) approaches 0 the loop fills up the outer curve and, in the limit, the limaçon becomes a circle traversed twice . </P> <P> The area enclosed by the limaçon r = b + a cos ⁡ θ (\ displaystyle r = b + a \ cos \ theta) is (b 2 + a 2 2) π (\ displaystyle (b ^ (2) + ((a ^ (2)) \ over 2)) \ pi). When b <a (\ displaystyle b <a) this counts the area enclosed by the inner loop twice . In this case the curve crosses the origin at angles π ± arccos ⁡ b a (\ displaystyle \ pi \ pm \ arccos (b \ over a)), the area enclosed by the inner loop is (b 2 + a 2 2) arccos ⁡ b a − 3 2 b a 2 − b 2 (\ displaystyle (b ^ (2) + ((a ^ (2)) \ over 2)) \ arccos (b \ over a) - (3 \ over 2) b (\ sqrt ((a ^ (2)) - (b ^ (2))))), the area enclosed by the outer loop is (b 2 + a 2 2) (π − arccos ⁡ b a) + 3 2 b a 2 − b 2 (\ displaystyle (b ^ (2) + ((a ^ (2)) \ over 2)) (\ pi - \ arccos (b \ over a)) + (3 \ over 2) b (\ sqrt ((a ^ (2)) - (b ^ (2))))), and the area between the loops is (b 2 + a 2 2) (π − 2 arccos ⁡ b a) + 3 b a 2 − b 2 . (\ displaystyle (b ^ (2) + ((a ^ (2)) \ over 2)) (\ pi - 2 \ arccos (b \ over a)) + 3b (\ sqrt ((a ^ (2)) - (b ^ (2)))).) </P>

When does a limacon have an inner loop
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