What is an equation of a "line segment"?

Question

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?

Answer 1

One useful sort of equations for a line segment are the "parametric equations". Something like this: $$ \begin{cases}x = 5t+1,\\ y = -2t+4,\end{cases}\qquad 0 \le t \le 1 . $$

Answer 2

Technically,

$$y=0\,\sqrt{x(1-x)}$$ is the equation of the line segment from $(0,0)$ to $(1,0)$.

Admittedly, this makes little sense. Instead you use an implicit or parametric equation with restrictions on the variables by means of inequalities.

Answer 3

Not sure if I understood you properly.Take it as comment if you like.

The line definition is inescapable. Only if within the segment interval we can have real values.

Plot for example the circle $ y= \sqrt{ (x-1)(5-x)} $

It has center on x-axis $(3,0) $ with radius $2$, the points $(1,0),(5,0)$ are ends of segment $AB$ which is the diameter of circle.This is the real, valid interval.

If on x-axis $x$ is situated between endpoints of segments $A$ or $B$ the y-axis value can be read off.

Else if $x$ is outside this segment interval, $y$ is imaginary.

Answer 4

Here is one reason why an equation for a line segment is not something commonly seen:

Planar curves given by equations are really altitude lines of two-input functions. For instance, the circle $x^2+y^2=1$ is really the collection of points at height $1$ in the graph of the function $f(x,y)=x^2+y^2$.

These curves thus separate the plane into regions where the function value is smaller than some constant and regions where the function value is larger. For instance, in the circle example above, on the inside of the circle the value of the function is less than $1$, and on the outside it's less than $1$.

So making an equation that gives a curve which doesn't separate the plane into regions is a somewhat unnatural thing to do.