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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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1  
A function of the form $f(x)=a_0+a_1x +a_2 x^2$ is already called "parabola". Why should we call it "nonlinear straight line of order 2"? –  Siminore Aug 30 '12 at 12:22
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This is close to incomprehensible. Can you work a little more to explain what your question is? –  rschwieb Aug 30 '12 at 12:24

2 Answers 2

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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Yes, reparameterize is a choice. Please don't mind one more stupid question. Are the lines in $P_0+tV_0=C(t)$ and yours in $C(s)$ cubic in $t$ the same? E.g. At t=0.5, so $s=0.5+0.5^2+0.5^3=0.875, C(t=0.5) = C(s=0.875)$? –  l l Aug 30 '12 at 14:01
    
Sorry, Ignore the above. I should ask about the derivative –  l l Aug 30 '12 at 14:05

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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Yes, thats true. Then how about the straightness? A straight line must be linear? –  l l Aug 30 '12 at 12:24
    
@ll Let's suppose we are working in the $x-y$ plane, just so we're on the same page. The answer is "every nonvertical straight line is the graph of a linear function of $x$, and conversely every graph of a linear function (in $x$) is a nonvertical straight line. –  rschwieb Aug 30 '12 at 12:27
    
Yes, sir. You are right. There are also non-Euclidean geometry. Because I limited myself on the parameterization, so I was confused on the derivative/continuity. If consider the whole set of points, "straight" should be definited by "the direction of shortest path". –  l l Aug 30 '12 at 14:19
    
@ll From the above answer, I can now see what you were getting at with the parameterization :) –  rschwieb Aug 30 '12 at 14:37
    
Um... so the derivative for a "straight" line need not to be constant depends on its parameterization? –  l l Aug 31 '12 at 4:32

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