a curve does not need to be injective?

Question

In diff. Geometry, curve is a differentiable mapping from an open interval to 3 dimensional euclidean space. Doesn't it need to be injective?

If it is not, then there might be a two different tangent vector at a point in the euclidean space...which make frenet analysis impossible at the point.

Any help will be appreciated!

score 3 · Accepted Answer · answered Jun 10 '15 at 09:51

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You need to realise that a curve is the function $\gamma \colon [a,b]\to \mathbb R^3$, not its image. Even if the image looks like a figure eight (for instance), the function, if differentiable, will still have a tangent at every point, it's just that it will have a tangent at every point in $[a,b]$.

answered Jun 10 '15 at 09:51

Ittay Weiss

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Oh...okay. Then in your 8 example, at the center of 8, the curve has two different frenet apparatus? – Mathcho Jun 10 '15 at 09:57
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yes @Mathcho, exactly. – Ittay Weiss Jun 10 '15 at 10:46
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Thank you so much! I really appreciate it – Mathcho Jun 10 '15 at 11:00

a curve does not need to be injective?

1 Answers1