# tangent lines to curves

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• Oct 18th 2011, 06:04 AM
Sogan
tangent lines to curves
Hello,

i consider a (smooth) curve in euclidean space $c:I->\mathbb{R}^3$
and i want to prove:
if all the tangent lines to the curve meet up in a single point, then the curve must be a straight line.

Can you please help me to proove this fact?
Lets say, the point where all the tangent lines meet is $P\in\mathbb{R}^3.$
Then all the tangent lines look like this:
$c(t_0)+t*(P-c(t_0)).$

But how can i show that c'(t) is constant?

Regards
• Oct 18th 2011, 10:17 AM
xxp9
Re: tangent lines to curves
Let s be the unit parameter, c(s)-P = f(s)c'(s), f is some unknow function of s. So
c'(s) = f'(s)c'(s) + f(s)c''(s)
That is (1-f'(s))c'(s) = f(s)c''(s)
since c'(s) is unit, their inner product <c'(s), c''(s)>=0, so we must have
1-f'(s)=0, that is f(s)=f(0)+s. f(s) is not 0 when s is not -f(0).
So c''(s)=0 when s is not -f(0)