Question

How do you find the parametric equations for a line segment?

Answer 1

Hint: If there are functions $ f $ and $ g $ with a common domain $ T $ , the equations $ x = f\left( t \right) $ and $ y = g(t) $ , for $ t $ in $ T $ , are parametric equations of the curve consisting of all points $ \left( {f\left( t \right),g\left( t \right)} \right) $ , for $ t $ in $ T $ . The variable $ t $ is the parameter. A parameter is a third, independent variable. A line segment is a section or a part of a line that allows polygons to be created, slopes to be determined as well as measured. Its length is finite and it has two endpoints which helps to evaluate it. Here, in this question we will try to understand how to find parametric equations for a line segment.

Complete step-by-step answer:
As we know a line segment can be determined by two endpoints so, let the two endpoints be $ \left( {{x_0},{y_0}} \right) $ and $ \left( {{x_1},{y_1}} \right) $ . The line segment between $ \left( {{x_0},{y_0}} \right) $ and $ \left( {{x_1},{y_1}} \right) $ can be written as:
 $
   \Rightarrow x\left( t \right) = \left( {1 - t} \right){x_0} + t{x_1} \\
   \Rightarrow y(t) = \left( {1 - t} \right){y_0} + t{y_1} \;
  $
where $ 0 \leqslant t \leqslant 1 $
We know that the direction vector from $ \left( {{x_0},{y_0}} \right) $ to $ \left( {{x_1},{y_1}} \right) $ is
 $
   \Rightarrow \vec v = \left( {{x_1},{y_1}} \right) - \left( {{x_0},{y_0}} \right) \\
   \Rightarrow \vec v = \left( {{x_1} - {x_0},{y_1} - {y_0}} \right) \;
  $
By adding a scalar multiple of $ \vec v $ to the point $ \left( {{x_0},{y_0}} \right) $ , we can easily find any point $ \left( {x,y} \right) $ on the line segment. So now we have,
 $ \left( {x,y} \right) = \left( {{x_0},{y_0}} \right) + t\left( {{x_1} - {x_0},{y_1} - {y_o}} \right) $
Now, after simplifying the above equation we get,
 $ \left( {x,y} \right) = \left( {\left( {1 - t} \right){x_0} + t{x_1},\left( {1 - t} \right){y_0} + t{y_1}} \right) $ , where $ 0 \leqslant t \leqslant 1 $
So, the correct answer is “$\left( {\left( {1 - t} \right){x_0} + t{x_1},\left( {1 - t} \right){y_0} + t{y_1}} \right) $ , where $ 0 \leqslant t \leqslant 1 $”.

Note: There are two ways to form a parametric equation either from a cartesian equation or by using vectors. A vector is a line segment which has a direction or in geometric terms a vector can be defined as the difference between two points in the plane. Parametric equations are often used to simulate motion and in computer graphics to design a variety of figures.