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| The velocity vector is the derivative of the vector valued function r(t). |
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| The speed is the magnitude of the velocity vector. To obtain speed, square the components of the velocity vector and then take the square root. |
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| Acceleration is the second derivative of the vector valued function r(t), or the first derivative of the velocity vector. The acceleration vector can be written in components and in terms of curvature and speed. |
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| Let C be a smooth curve represented by r on an open interval I. The unit tangent vector T(t) is the velocity vector divided by the speed. (r'(t) cannot be zero) |
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| Let C be a smooth curve represented by r on an open interval I. The principal unit vector N(t) is the derivative on the unit tangent vector divided by its magnitude. (T'(t) cannot be zero |
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| The tangential component can be written as 3 different formulas: The derivative of speed; a dot t; and v dot a divided by speed. |
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| The normal component of acceleration can be written as 4 formulas: a dot N; the magnitude of v cross a divided by speed; the square root of ||a|| - at^2; and the curvature times the speed squared. Note: an is >= 0. |
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Definition
Curvature if s is the parameter.
Note: ||r'(s)||=1 |
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| Curvature if t is the parameter. |
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| Alternate formula for curvature. |
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