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curvature is define as how quickly/ abruptly a curve changes with respect to its arc length.

okay so the normal vecor (N = T ') is the change in the tangent vector of a curve with respect to some parameter t.

but conceptually i don't get the difference. if the parameter t = Const. nothing moves (everything is still) and so the space curve doesn't get to traverse freely through space. but when it does get to traverse, it traverses the path of the function it is set by and so, wouldn't [itex]\frac{dT}{dS}[/itex] [itex]\alpha[/itex] [itex]\frac{dT}{dt}[/itex] (variable 'S' = arc length)? i mean, when there is a sharp turn, the T(t) value with respect to the length of the curve will be high and also.. the change in the tangent vector with respect to the parameter will be high. i just don't see the difference between the two. they're both quantities involving the change in the tangent vector. and S is always proportional to t. except.. i understand dT/dt is. but i don't understand the difference between dT/dt and dT/dS. i already looked up curvature in my book, notes, and google, i still don't get the difference though.