How do curves consist of points? According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in other words there is nothing, then how is it possible to draw any curve? 
The thing I could imagine is that maybe a point is not as Euclid thought. I mean a point can be thought of as a small line segment whose length is approaching zero but never becomes exactly zero. In this way, we can say that curves consist of points (the line segment with length approaching zero). But then it breaks the fact as given by Euclid that a point has no dimensions.  Please help me to get out of this dilemma.

A curve is completely determined by two facts:


*

*Knowledge of all of the points lying on the curve

*Knowledge that the curve is drawn on the Euclidean plane


When it's said that a curve is made out of points, one really means to include in the latter fact too, or something similar (e.g. a topology or a metric on the collection of points).
There are more sophisticated geometric techniques (e.g. tangent spaces, halos, germs, stalks) that probe the "infinitesimal" shape of the curve at the point; 
For example, studying the tangent space to the curve would indeed allow you to say that, at each point where the curve is smooth, it consists of an infinitesimal line.
But just to reinforce my initial point, the shape of that infinitesimal line can be ascertained simply by knowing the curve is being drawn on the plane along with which 'nearby' points lie on the curve.
