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Let a be a regular curve parametrized by arc length with > 0 and 70. Denote n and b the principal normal and the binormal of a. (i) If a lies on a sphere of center c E R³ and radius r > 0, show that a c = -pn - p'ob, where p= 1/K and σ = = -1/7. Deduce that r² = p² + (po)². (ii) Conversely, if p² + (p'o)2 has constant value r² and p' #0 show that a lies on a sphere of radius r.

Let a be a regular curve parametrized by arc length with > 0 and 70. Denote n and b the principal normal and the binormal of a. (i) If a lies on a sphere of center c E R³ and radius r > 0, show that a c = -pn - p'ob, where p= 1/K and σ = = -1/7. Deduce that r² = p² + (po)². (ii) Conversely, if p² + (p'o)2 has constant value r² and p' #0 show that a lies on a sphere of radius r.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Regular curve

Let a be a regular curve parametrized by arc length with к > 0 and 70. Denote n and b the principal normal and the binormal of a. (i) If a lies on a sphere of center c € R³ and radius r > 0, show that a c = -pn - p'ob, where p= 1/k and o= -1/7. Deduce that r² = p² + (p'o)². (ii) Conversely, if p² + (p'o)² has constant value r² and p′ ‡ 0 show that a lies on a sphere of radius r.
Transcribed Image Text:Let a be a regular curve parametrized by arc length with к > 0 and 70. Denote n and b the principal normal and the binormal of a. (i) If a lies on a sphere of center c € R³ and radius r > 0, show that a c = -pn - p'ob, where p= 1/k and o= -1/7. Deduce that r² = p² + (p'o)². (ii) Conversely, if p² + (p'o)² has constant value r² and p′ ‡ 0 show that a lies on a sphere of radius r.
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