# Citing examples and making graphs, how do you explain that #f(r, theta;a,b,n,alpha )=r-(a + b cos n(theta- alpha) )=0# represents a family of limacons, rose curves, circles and more?

##### 3 Answers

See explanation. To be continued, in my 2nd answer.

#### Explanation:

families of curves, with similar characteristics.

( i ) a = 0 and n = 1.

produce circles through the pole r = 0.

Example: b = 2 and alpha = pi/3:

diameter inclined at

graph{ (x^2+y^2- 2(1/2x+sqrt3/2 y))(y-sqrt3 x)=0[-0.8 3.8 -0.3 2 1.8] }

Concentric circles come from r =a, when b = 0.

graph{(x^2+y^2-1/4)(x^2+y^2-1)=0}

(ii)

Here, as r is non-negative, I do not count r-negative petals.

Pixels do not glow for r-negative points.

Example:

graph{(x^2+y^2)^3.5-(x^6-15x^2y^2(x^2-y^2)-y^6)=0[-4 4 -2 2]}

(iii)

#r = a ( 1 +- cos (theta - alpha ) produce Cardioids.

Example:

Cardioid Couple:

graph{(x^2+y^2-2sqrt(x^2+y^2)+sqrt2(x+ y))(x^2+y^2-2sqrt(x^2+y^2)-sqrt2(x+ y))=0}

Continuation, for the 2nd part.

#### Explanation:

(iv) n =1

Let

The double here includes both ( pole-on and pole-not-on ) cases

Graph for the limacons

graph{ (x^2+y^2-3sqrt(x^2+y^2)-x) (x^2+y^2-sqrt(x^2+y^2)-3 x)=0[- 16 16 -8 8]}

With

graph{ (x^2+y^2-3sqrt(x^2+y^2)-1/sqrt2 (x+y)) (x^2+y^2-sqrt(x^2+y^2)-3/sqrt2 ( x+y))=0[- 20 20 -10 10]}

The innie of the dimple weakens, as

Rosy Limacon, from limacon-like equation, giving rosy graph.

If

Example:

graph{((x^2+y^2)^2-(x^2+y^2)^1.5-1/sqrt2(x^3+3xy(x-y)-y^3))((x^2+y^2)^2-(x^2+y^2)^1.5-x^3+3xy^2)=0}

Continuation,for the 3rd part of my answer.

Ref: My answers to related Socratic questions.

#### Explanation:

(vi)

The count for the open loops is p = 2.

Example 1:

For 3 complete rotations through

giving two

graph{(x^2-y^2) - (x^2+y^2)^1.5 sqrt ( 1-(x^2+y^2))(4(x^2+y^2)-3) =0[-6 6 -3 3]}

Example 2:

graph{(x^4- 6 x^2y^2 +x^4) - (x^2+y^2)^2.5 sqrt ( 1-(x^2+y^2))(4(x^2+y^2)-3) =0[-6 6 -3 3]}

The count ( 2 open and 2 closed )of idiosyncratic loops is p = 4.