Tangent to a 2-Dimensional curve, in 3 Dimensions

Question

For line to be tangent to a given curve, they should pass through a common point and both should have same slope at that point. But how do we compare slope in 3D?

E.g.=> For a circle in 2-Dimensions, we know which lines will be it's tangents. But what if I ask, which lines will be tangent to it in 3-Dimensions? Will the answer be same for both cases, or will many others lines, lying in some other planes, also be included?

All I could think of is that, we need to find tangents in planes, and in every plane the curve will be set of points, except the one in which the curve is a circle. And because slope of a point is not defined, there will be no new tangents added when we go from 2D to 3D.

Answer 1

Formally, a curve in $\mathbb{R}^3$ is a continuous map $\gamma: [0,1] \rightarrow \mathbb{R}^3$ defined by $t \mapsto (\gamma_1(t), \gamma_2(t), \gamma_3(t))$. Then one can define the tangent vector to this curve at a point $t \in [0,1]$ to be

$$ \gamma'(t) = \lim_{h \rightarrow 0, h \neq 0} \frac{\gamma(t+h) - \gamma(t)}{h} = (\gamma_1'(t), \gamma_2'(t), \gamma_3'(t))$$

For a curve in $\mathbb{R}^2$ this amounts to the usual idea of a derivative (slope to a curve):

For curves in $\mathbb{R}^3$ however, we get the following: (here the red vector is the tangent)

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Also, if you were to consider not curves but surfaces in $\mathbb{R}^3$ a simple "line" would not be sufficient as a tangent. You will need a plane. For a much more general treatise of these things have a look here.

Answer 2

You can define slope at a point of a curve just as you did in calculus 1 - take the limit of approximating secant lines. The only difference this time is that the limiting object will be a line pointing in, well, any direction. So, at this point, it is best to use vectors.

Two curves in space will be tangent to eachother at a point if they both pass through this point and if they both have the same tangent vectors at this point.

Now, with a little bit of imagination, you can see why you will have new curves which are tangent to the circle beyond the curves which are coplanar with the circle.