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

Question #145246
Prove that for cardiod r=a(1+cosθ)
ρ^2/r is constant
1
Expert's answer
2020-11-19T16:19:04-0500
  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




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