How to say if a curve is flat?

Question

Given a curve $\omega$ : $$ \left\{ \begin{array}{c} X=t-1 \\ Y=t^2-2t \\ Z=t^2-3t+1 \end{array} \right. $$ Which way can i use to prove if such a curve like $\omega$ is flat ? And what does it means geometrically?

Answer 1

I'm guessing "flat" means that the curve lies on a plane.

HINT: A set of points lies on a plane if and only if the coordinates $(x, y, z)$ of such points satisfy some linear relation of the kind: $$ Ax + By + Cz + D = 0 $$ Where $A, B, C, D$ are constants. Can you show this is true for your curve?

Answer 2

Note that $$ x-y+z = (t-1)-(t^2-2t)+(t^2-3t+1) =0$$

Thus the points $(x,y,z)$ are on the plane $$x-y+z=0$$ for all values of $t$

That means your curve is on a flat plane, so it is flat.