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From almost everywhere, a straight line is linear defined as $C(t)=P_0+tV_0$. And I am wonder what does a straight line but is not linear by the means of the parameter $t$. For example $C(t)=P_0+tV_0+t^2V_0+t^3V_0$.

So would it means a straight line is not really needed to be linear? Would that be the case that the definition for a straight line is: For any two points $p=C(a),q=C(b)$,

(1):$C'(a)\times C'(b)=0$

(2):$C'(a)\bullet C'(b)>0$

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    This is close to i$n$com$p$rehe$n$sible. Can you work a little more to explain what your question is?2012-08-30

2 Answers 2

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Your $C(t)=P_0+tV_0+t^2V_0+t^3V_0=P_0+sV_0$ where $s=t^3+t^2+t$ defines a straight line, the line through $P_0$ in the direction of $V_0$. The parameter $s$ has a nonlinear expression in terms of the parameter $t$. So what? I don't see what the difficulty is.

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    Sorry, Ignore the above. I should ask about the derivative2012-08-30
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A "linear function" refers to functions $f(t)$ having the properties $f(a+b)=f(a)+f(b)$ and $f(\lambda t)=\lambda f(t)$.

If you allow higher powers of $t$ than just degree 1, then you will usually fail to have these properties. "Straight lines" are usually considered to be the graphs of linear functions.

Of course, there is a more advanced notion of "straight line" from differential geometry that is more along the lines of "geodesic". I think they defining feature there is that "the second derivative is zero".

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    Um... so the derivative for a "straight" line need not to be constant depends on its parameterization?2012-08-31