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Question 2. The osculating circle. A unit speed parametrisation of a circle in R³ may be written as 7: R → R³ where y(s) = c+rcos ( (²) V₁ + r sin (²) V2 where r> 0 is a constant real number, c, V1, V2 are constant vectors and vi. Vj = dij. Let : 1 R³ be a unit speed curve with K(0) > 0. Show that there is precisely one unit speed circle y that agrees with 3 to second order at 3(0), i.e. such that B(0) = y(0), 8(0) = (0) and and (0) = 7(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 B at B(0).

Question 2. The osculating circle. A unit speed parametrisation of a circle in R³ may be written as 7: R → R³ where y(s) = c+rcos ( (²) V₁ + r sin (²) V2 where r> 0 is a constant real number, c, V1, V2 are constant vectors and vi. Vj = dij. Let : 1 R³ be a unit speed curve with K(0) > 0. Show that there is precisely one unit speed circle y that agrees with 3 to second order at 3(0), i.e. such that B(0) = y(0), 8(0) = (0) and and (0) = 7(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 B at B(0).

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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Question 2. The osculating circle. A unit speed parametrisation of a circle in R³ may be written as y: R → R³ where y(s) = c + r cos * (²³) · +r sin V₁ + r sin ¹ ( ² ) ₁ V2 where r > 0 is a constant real number, c, V₁, V2 are constant vectors and vi. Vj = dij. Let 3: IR³ be a unit speed curve with K(0) > 0. Show that there is precisely one unit speed circle y that agrees with 3 to second order at (0), i.e. such that B(0) = y(0), 8(0) = (0) and (0) = 7(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 ß at B(0).
Transcribed Image Text:Question 2. The osculating circle. A unit speed parametrisation of a circle in R³ may be written as y: R → R³ where y(s) = c + r cos * (²³) · +r sin V₁ + r sin ¹ ( ² ) ₁ V2 where r > 0 is a constant real number, c, V₁, V2 are constant vectors and vi. Vj = dij. Let 3: IR³ be a unit speed curve with K(0) > 0. Show that there is precisely one unit speed circle y that agrees with 3 to second order at (0), i.e. such that B(0) = y(0), 8(0) = (0) and (0) = 7(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 ß at B(0).
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