0 <= 4 sqrt (x^2 + y^2 ) = 4 cos 3theta in [ 0, 4 ]0≤4√x2+y2=4cos3θ∈[0,4]
The period for this cosine function = ( 2 pi )/3=2π3.
r >= 0r≥0 for theta in [ 0, pi/6 ], [ pi/2, 2/3pi ], [ 7/6pi, 3/2pi ] andθ∈[0,π6],[π2,23π],[76π,32π]and
[ 11/6pi, 2pi ][116π,2π], in one revolution theta in [ 0, 2pi ]θ∈[0,2π]. The first
and the last are halves, from the same loop. For subsequent
revolutions, these three three loops are redrawn.
For Socratic graphic utility that keeps off r-negative pixels, the
Cartesian form.
sqrt ( x^2 + y^2 )√x2+y2
= 4 = cos 3theta =4 ( cos^3theta - 3 cos theta sin^2theta )=4=cos3θ=4(cos3θ−3cosθsin2θ)
= 4 ( x^3/r^3- (3xy^2)/r^3)=4(x3r3−3xy2r3), giving
( x^2 + y^2 )^2 = 4 ( x^3 - 3 xy^2 )(x2+y2)2=4(x3−3xy2)
is used. See graph.
graph{( x^2 + y^2 )^2 - 4 ( x^3 - 3 xy^2 )=0}
