Question 2. The osculating circle. A unit speed parametrisation of a circle in R³ may be written as y: RR³ where y(s) = c +rcos where r > 0 is a constant real number, c, V₁, V₂ are constant vectors and v¡· Vj = dij. Let ß: I → R³ be a unit speed curve with (0) > 0. Show that there is precisely one unit speed circle Y that agrees with to second order at (0), i.e. such that B(0) = y(0), š(0) =(0) and (0)= ï(0). ¹ (²) V₂ V2 V1 + r sin Show that this circle has radius 1/K(0). The circle y is called the osculating circle and c the centre of curvature of 3 at 3(0).
Question 2. The osculating circle. A unit speed parametrisation of a circle in R³ may be written as y: RR³ where y(s) = c +rcos where r > 0 is a constant real number, c, V₁, V₂ are constant vectors and v¡· Vj = dij. Let ß: I → R³ be a unit speed curve with (0) > 0. Show that there is precisely one unit speed circle Y that agrees with to second order at (0), i.e. such that B(0) = y(0), š(0) =(0) and (0)= ï(0). ¹ (²) V₂ V2 V1 + r sin Show that this circle has radius 1/K(0). The circle y is called the osculating circle and c the centre of curvature of 3 at 3(0).
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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Transcribed Image Text:Question 2. The osculating circle. A unit speed parametrisation of a circle in R³
may be written as y: RR³ where
y(s) = c + r cos
where r > 0 is a constant real number, c, V₁, V₂ are constant vectors and v¡· Vj = dij.
Let ß: I → R³ be a unit speed curve with (0) > 0. Show that there is precisely
one unit speed circle that agrees with to second order at (0), i.e. such that
(0) and (0) = ï(0).
B(0) = y(0), 8(0) =
Show that this circle has radius 1/K(0). The circle y is called the osculating circle
and c the centre of curvature of 3 at 3(0).
¹ (²) V₂
V2
V1 + r sin
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