Explain the proof of meusniers theorem by andrew pressley
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The proof is short, and begins as follows:
Assume that γθγθ is a unit-speed parametrization of the curve of intersection of ΠθΠθ and SS. Then at pp, γ′θ=±vγθ′=±v, so γ′′θγθ″ is perpendicular to vv and is parallel to Πθ–––––––––––––––––––and is parallel to Πθ_.
My question is about the underlined statement. How do we know that γ′′θγθ″ lies on ΠθΠθ? Looking at the diagram given, it seems obvious, in the sense that γ′′θγθ″ is parallel to the principle normal to γθγθ, which, along with γ′θγθ′ forms the basis for the oscullating plane, which also "looks" like the plane ΠθΠθ, but other than that heuristic argument I can't formalize why γ′′θγθ″ should lie on ΠθΠθ. I'm sure its easy
Assume that γθγθ is a unit-speed parametrization of the curve of intersection of ΠθΠθ and SS. Then at pp, γ′θ=±vγθ′=±v, so γ′′θγθ″ is perpendicular to vv and is parallel to Πθ–––––––––––––––––––and is parallel to Πθ_.
My question is about the underlined statement. How do we know that γ′′θγθ″ lies on ΠθΠθ? Looking at the diagram given, it seems obvious, in the sense that γ′′θγθ″ is parallel to the principle normal to γθγθ, which, along with γ′θγθ′ forms the basis for the oscullating plane, which also "looks" like the plane ΠθΠθ, but other than that heuristic argument I can't formalize why γ′′θγθ″ should lie on ΠθΠθ. I'm sure its easy
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