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c-13.5 - 13.5 Curvature Try to measure complexity of a...

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13.5 Curvature Try to measure complexity of a curve For a function y = f ( x ), we knew from Calculus I and II that if f ′′ ( x ) = 0 for all x , then f ( x ) = ax + b , i.e., the graph of f is a line; if f ′′ ( x ) > 0 for all x , then the graph of f is concave down; if f ′′ ( x ) < 0 for all x , then the graph of f is concave up. Consider any curve C given by a di±erentiable parametrized function v r ( t ) = x ( t ) v i + y ( t ) v j + z ( t ) v k for t [ a, b ]. We may try to use v r ′′ to measure complexity of the curve. Nevertheless, if we choose another parametrized function v φ , we should have v φ ′′ ( t ) n = v r ′′ ( t ) in general. Therefore we cannot use v r ′′ to measure complexity of the curve. To de²ne a quantity which measure complexity of a curve, we use the “best” (also unique) parametrized function v R ( s ) := v r ψ 1 ( s ), de²ned at the end of last section. DeFnition of curvature of a curve Let C be given by a parametrization: v r : [ a, b ] R 3 , t m→ v r ( t ) . Let v R = v r ψ 1 : [0 , L ( C ] R 3 , s m→ v r ψ 1 ( s ) 59
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be the induced new parametrization defned at the end oF last section, where Ψ( t ) = i t a b v r ( u ) b du . . Then we defne the curvature For C κ := v v v v d 2 v R ds 2 v v v v . (7) More precisely, κ ( v R ( s )) = b d 2 v R dt 2 b ( s ) For any s [0 , L ( C )]. Proposition 13.6 κ ( t ) = b v T ( t ) b b v r ( t ) b . (8) More precisely, κ ( v r ( t )) = b v T ( t ) b b v r ( t ) b for any t [ a, b ] .
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