Mathematics (Coordinate geometry to be more precise) offers us the equations of ellipse, straight line, circle, hyperbola etc.
Why doesn't it give us the equation of a line segment, not a straight line, but a line segment that has its both ends terminating at certain positions?
For example, if x becomes invalid after a certain value, it terminates at that value. Similarly if we enter that kind of equation of a line segment into a graphing calculator, the graph of the line segment should be simply invalid after the certain terminating points. (Let AB be a line segment. I am talking about the equation that represents AB, and the equation is invalid or gives no result for coordinates exceeding A and B. Or in other words, for points lying outside AB, the equation doesn't give a valid result.)
When I thought about this, all i could think was that it could be same as the equation of an ellipse with the length of its semi minor axis equal to zero.
But that raised the problem of the second term in the ellipse's general form becoming undefined.
So,
The question is:
Why doesn't this kind of equation of a line segment exist? And if it does, what is its general form?
Best Answer
One useful sort of equations for a line segment are the "parametric equations". Something like this: $$ \begin{cases}x = 5t+1,\\ y = -2t+4,\end{cases}\qquad 0 \le t \le 1 . $$