Inflection Points on Polar Curve

Question

For what values of Α > 0 has the polar curve r(θ) = Α +cos(θ) inflection points?

Answer 1

the curve r=A+cosθ does have inflection points if 1<A<2,

the curve r=1 which is the unit circle does not.

correction about inflection. If there is a function y=f(x), then the function has inflection at points where y''=0 (example y=cos(x) at x=pi/2) or y'' is undefined (example y=x^1/3 at x=0) at the point (x₀) and that y'' changes sign after point (x₀+c) compared to before point (x₀-c), where c is an infinitesimal positive number.

However r=A+cosθ is not a function in (x,y) coordinates,,,neither is r=1,,,,x²+y²=1.

for curves that are not functions, the criteria for inflection point(s) is the curvature

changes sign. See Wolfram MathWorld for definition of inflection point.

In polar coordinates curvature=K=(r²+2(r')²-r*r'')/(r²+(r')²)^3/2

See that denominator is always positive

for r=A+cosθ,,,,,the numerator is

A²+3Acosθ+2 after simplification

when θ=π (most negative), numerator is negative

when 1<A<2, and positive otherwise. For any 1<A<2

find when A²+3Acosθ+2 goes from positive to negative

and negative to positive as θ increase for the two inflection points.

For A=1.75 the inflection points are when cosθ=-81/84

The curve has no inflection point(s) close to θ=0.

Obviously unit circle r=1 has K constant,,,K=1. and has no inflection point(s)

Answer 2

You should go to Desmos or similar graphing utility and look at the curve as a function of A.

There will probably be some debate about inflection points, but in my opinion the curves will have inflection points for almost all values of A.

There was recently a brief debate here on Wyzant about whether the unit circle has an inflection point at x=+1 and x=-1. In my opinion there is, but others disagreed.

In any event the cardioid which is the curve of the equation in this problem always has an inflection point at its rightmost point (similar to the unit circle at x=+1)...and depending on A, it has other inflection points as well.