Parametric Equation and Euclidean Distance

In summary: From $(x_1, y_1)$ and $(x_2, y_2)$, you can find the equation of the line by using the Pythagorean theorem. The equation of the line is $x= a_2+ b_1$
  • #1
Aleister911
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  • #2
Okay, what do you understand of this problem and what have you done on it? The first part asks for parametric equations for the line from $(x_1, y_1)$ to $(x_2, y_2)$. Do you know what "parametric equation" are? Do you know how to find parametric equations for a line?
 
  • #3
I do understand that to find the parametric equation we focus on our vector equation which needs to be first computed (r=r0+tv). However there are no given values from each point to find r0 and tv to formulate the vector equation. I'm really confused with the overall idea of the question above. I kindly ask if anyone could help me solve this question.
 
  • #4
You are told that the line passes through the points $(a_1, b_1)$ and $(a_2, b_2$.
A straight line can be written with parametric equations $x= r_1+ tv_1$ and and $y= r_2+ tv_2$. There are many different parametric equations describing the same line. We can arbitrarily take t= 0 at $(a_1, b_1)$ and t= 1 at $(a_2, b_2)$.

you need four equations to find the values of $r_1$, $v_1$, $r_2$, and $v_2$. Those four equation are:
$a_1= r_1+ 0v_1= r_1$
$b_1= r_2+ 0v_2= r_2$
$a_2= r_1+ 1v_1= r_1+ v_1$ and
$b_2= r_2+ 1v_2= r_2+ v_2$.

From the first two equations, $r_1= a_1$ and $r_2= a_2$. The next two equations give
$a_2= r_1+ v_1= a_1+ v_1$ so $v_1= a_2- a_1$ and $b_2= r_2+ v_2= a_2+ v_2$ so $v_2= b_2- a_2$

The parametric equations for the line are
$x= a_2+ (b_1- a_1)t$
$y= a_2+ (b_2- a_2)t$
 
  • #5
Thank you for the respond. I do appreciate. My question is that are we going to use the parametric equation to solve for the euclidean distance from the robot to the line segment, using the euclidean formula D = sqrt[(x2-x1)^2+(y2-y1)^2]. If so then how can we determine the value of x1,x2,y1,y2 ??
 
  • #6
Take any two points on the line. From a practical point of view, the farther the two points are, the more accurate the calculation will be so I would recommend the two endpoints of line, $(a_1, b_1)$ and $(a_2, b_2)$ in your picture.
 
  • #7
Correct me if I'm wrong, but since we have 2 points, is it really mean that we should have two euclidean distance? if so then, can you help me solve this and optimise the distance as well?

Kind suggestion
 
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  • #8
No, you were not asked for the distance to the two endpoints. You were asked to find the distance from the point (x, y) to the line. That is the distance from (x, y) to the point at the foot of the perpendicular,
 
  • #9
Okay, However from the Euclidean distance formula we have x2, x1 and y2,y1. What value are we going to used? is it the x,y values from the parametric equation?
I'm really confused now, can you help me solve this?
 
  • #10
You use all the values! You are given two points, $(x_1, y_1)$ and $(x_2, y_2)$. Do you understand what $(x_1, y_1)$ and $(x_2, y_2)$ mean? Whoever gave you this problem expects you to be able to find the equation of the line between them. Can you do that?
 

1. What is a parametric equation?

A parametric equation is a mathematical representation of a curve or surface in terms of one or more parameters. It allows us to describe the coordinates of points on the curve or surface in terms of these parameters, rather than in terms of x and y coordinates.

2. How is a parametric equation different from a standard equation?

A standard equation is typically in the form y = f(x), where x and y are both variables. In a parametric equation, the coordinates of a point are expressed in terms of one or more parameters, rather than just x and y. This allows for more flexibility in describing complex curves and surfaces.

3. What is Euclidean distance?

Euclidean distance is the shortest distance between two points in a straight line. It is calculated using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

4. How is Euclidean distance related to parametric equations?

In parametric equations, the coordinates of a point are expressed in terms of parameters. By plugging in different values for these parameters, we can find the coordinates of different points on a curve or surface. The Euclidean distance between any two points on the curve or surface can then be calculated using their coordinates in the parametric equation.

5. Why is Euclidean distance important in science?

Euclidean distance is important in science because it allows us to measure the distance between two points in a simple and precise way. It is used in various fields such as physics, engineering, and computer science to calculate distances, velocities, and other important quantities. It also has applications in data analysis and machine learning, where it is used to measure the similarity between data points.

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