If you have a curve in parametric form, and you have a point on that curve, how would you calculate the directional vector that has its origin in that point and its direction in the direction of the curve?
How to calculate the direction of a curve in a certain point
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$\begingroup$
derivatives
partial-derivative
curves
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0the derivative $(x'(t),y'(t))$ – 2017-01-14
1 Answers
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If $\overline{x}(t) = [x(t), y(t), z(t)]$ is the curve and the point is at $t_0$, then the direction vector is $[x'(t_0); y'(t_0); z'(t_0)]/\|\overline{x}(t_0)\|$, where $ \|\overline{x}(t_0)\| = \sqrt{x'(t_0)^2 + y'(t_0)^2 + z'(t_0)^2}$.
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0just a small note, in the off-chance the length of the vector is $0$, then one might need to use other methods or just accept there is no direction as the parameterization stopped at that point and hovered there. – 2017-01-14
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0I agree. If possible one should re-parameterize the curve in this case. – 2017-01-15