Answer to Question #145246 in Differential Geometry | Topology for Dolly

Question

Prove that for cardiod r=a(1+cosθ)
ρ^2/r is constant

Answer 1

1. For parametric function ρ - radius of curvature is [r² + (r') ²]^3/2 / |r²+2*(r') ² - r * r"|
2. r' - first deritative r' = -a * sinθ , r'' - second deritative = -a * cosθ
3. ρ =[r² + (r') ²]^3/2 / |r²+2*(r') ² - r * r"| =

[a² * (1+ cosθ)² + a² * (sinθ)²]^3/2 /a²(1 + cosθ)² + 2 * a² * (sinθ)² + a² * (cosθ)(1 + cosθ) =

[a² * (1 + 2cosθ+ (cosθ)²) + a² * (sinθ)²]^3/2 / a²(1 + 2cosθ+ (cosθ)² )+ 2 * a² * (sinθ)² + a² * (cosθ) + a² *(cosθ)²) =

[a² * (1 + 2cosθ+ (cosθ)²) + a² * (sinθ)²]^3/2 / a²(1 + 3cosθ+ 2*(cosθ)² + 2* (sinθ)²)

= [(a²)^3/2 * (2+2* cosθ)^3/2 ]/a²(1 + 3cosθ+ 2) =

[a³ * (2)^3/2 *(1 + cosθ)^3/2] /3*a²(1 + cosθ)

Then 1 + cosθ = r/a and

ρ = a³ * (2)^3/2 *(1 + cosθ)^3/2 / 3*a²(1 + cosθ) = a * (2)^3/2 * (r/a)^3/2 /(3* (r/a)) =

a *( 2 * r/a)^3/2 / (3 * r/a);

Then search ρ² = a² *( 2 * r/a)³ / (3 * r/a)²

ρ² = (a² *8*r³ /a³ )/(9*r²/a² ) = a * 8r³ / 9 * r² = 8 * a * r /9

Then ρ² /r = 8a/9. a = const, then ρ² /r = const