this curve implicitly. I don't get much insight from this. r=2+sin(θ)
Now consider r=2+sin(θ). Again, the values of sine are all between -1 and 1, so r will be between 1 and 3. Any points on this curve will have distance to the origin between 1 and 3. We can begin (?) the curve at θ=0 when r=2, and spin around counterclockwise. The distance to the origin increases to r=3 at θ=Pi/2 (the positive y-axis). The distance to the origin decreases back to r=2 when θ=Pi (the negative x-axis). The curve gets closest to the origin when θ=3Pi/2 (the negative y-axis) when r=1. Finally, r increases (as θ increases in the counterclockwise fashion) to r=3 again when θ=2Pi.
Here the "deviation" from circularity in the curve is certainly visible. The bottom seems especially dented. r=1+sin(θ)
We decrease the constant a bit more, and look at r=1+sin(θ). The values of sine are all between -1 and 1, so r will be between 0 and 2. The (red) inner circle has shrunk to a point. This curve will be inside a circle of radius 2 centered at the origin. We begin our sweep of the curve at 0, when r is 1. Then r increases to 2, and the curve goes through the point (0,2). In the θ interval from Pi/2 to Pi, sin(θ) decreases from 1 to 0, and the curves moves closer to the origin as r decreases from 2 to 1. Something rather interesting now happens as θ travels from Pi to 3Pi/2 and then from 3Pi/2 to 2Pi. The rectangular graph of 1+sine, shown here, decreases down to 0 and then increases to +1. The polar graph dips to 0 and then goes back up to 1. The dip to 0 in polar form is geometrically a sharp point! I used "!" here because I don't believe this behavior is easily anticipated. The technical name for the behavior when r=3Pi/2 is cusp.
This curve is called a cardioid from the Latin for "heart" because if it is turned upside down, and if you squint a bit, maybe it sort of looks like the symbolic representation of a heart. Maybe. r=1/2+sin(θ)
Let's consider r=1/2+sin(θ). The values of sine are all between -1 and 1, so r will be between -1/2 and 3/2. The (red) inner circle actually had "radius" -1/2, and it consists, of course, of points whose distance to the pole, (0,0), is 1/2. When θ is 0, r is 1/2. In the first two quadrants, 1/2+sin(θ) increases from 1/2 to 3/2 and then backs down to 1/2. In the second two quadrants, when θ is between Pi and 2Pi, more interesting things happen.
The rectangular graph on the interval [0,2Pi] of sine moved up by 1/2 shows that this function is 0 at two values, and is negative between two values. The values are where 1/2+sin(θ)=0 or sin(θ)=-1/2. The values of θ satisfying that equation in the interval of interest are Pi+Pi/6 and 2Pi-Pi/6. The curves goes down to 0 distance from the origin at Pi+Pi/6, and then r is negative until 2Pi-Pi/6. The natural continuation of the curve does allow negative r's, and the curve moves "behind" the pole, making a little loop inside the big loop. Finally, at 2Pi-Pi/6, the values of r become positive, and the curve links up to the start of the big loop.
This curve is called a limacon. The blue lines are lines with θ=Pi+Pi/2 and θ=2Pi-Pi/6. These lines, for the θ values which cross the pole, are actually tangent to the curve at the crossing points. r=0+sin(θ)
Let's try a last curve in this family, with the constant equal to 0. What does r=sin(θ) look like? A graph is shown to the right.

There are several interesting features of this graph. First, this is a polar curve which does have a nice rectangular (xy) description. If we multiply r=sin(θ) by r, we get r2=r·sin(θ), so that x2+y2=y. This is x2+y2-y=0 or, completing the square, x2+y2-2(1/2)y+(1/2)2-(1/2)2=0 so that (x-0)2+(y-1/2)2=(1/2)2. This is a circle of radius 1/2 and center (0,1/2), exactly as it looks.

The moving "picture" of this curve is quite different. Between 0 and π it spins once around the circle but then from π to 2π it goes around the circle another time! So this is really somehow two circles, even though it looks like only one geometrically.