关于用公式表示曲线所有切线平行的正确性确认

Hey there! Let's clear up this confusion for you right away.

First, let's get straight what it means for all tangent lines to a curve to be parallel: all tangent lines share the same fixed direction, which translates to the curve's tangent vector $\alpha'(s)$ being a constant (non-zero) vector. That's the core condition here.

Now, looking at your proposed formula:
$$\alpha(s)+\lambda(s)\alpha'(s)=k$$
where $k$ is a constant. This actually describes a different property entirely: all tangent lines to the curve pass through a single fixed point $k$, not that they're parallel. Think about it—for each value of $s$, you're saying some point along the tangent line (parameterized by $\lambda(s)$) lands exactly at the constant point $k$. That's the definition of a curve with concurrent tangents (all tangents meet at one spot), not parallel ones.

So to correctly formalize that all tangent lines are parallel, you should use one of these equivalent statements:

• $\alpha'(s) = c$ for all $s$, where $c$ is a non-zero constant vector (this fixes the tangent direction for the entire curve)
• $\alpha''(s) = 0$ for all $s$ (since the derivative of a constant vector is zero, this is just another way to state the tangent vector doesn't change)

Intuitively, a curve with all parallel tangents is just a straight line (or line segment), which makes sense—its tangent direction never shifts at all.

Hope that clears things up for you!

备注：内容来源于stack exchange，提问作者CharlesJA