Finding the parametric equation of a curve

In summary, the conversation discusses how to parameterize the part of a curve that allows an equilateral triangle, with a height of 3R, to roll from one vertex to the next while its center maintains a constant height. The solution involves defining a curve with a parameter t, using the sine rule to find the length of the curve, and then using the arc length equation to find the tangential and normal unit vectors. The conversation ends with a helpful hint to find the solution.
  • #1
Westlife
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2

Homework Statement


Parameterize the part of the curve which allows an equilateral triangle, with the height 3R, to roll from one vertex to the next one, while its center travels at a constant height.

Homework Equations


I will include some pictures to show what I'm doing

The Attempt at a Solution


Hey I'm just checking soo far if I have the right idea.

let ##f(t)=(x(t),y(t))## be our curve

Here I defined the parameter ##t## to be the angel between the height and the point at which the triangle intersects the curve .
20181210_155949.jpg


I said that the length of ##f## has to be equal to the length of the side from the vertex to the point of intersection and using the sine rule I got.
20181210_160135.jpg


so the length of the curve is ##l=\frac{2R\sin(t)}{\sin(\frac{2\pi}{3}-t)}##
so from the arc length equation we know that ##\sqrt{x'^2+y'^2}=l##
the tangential unit vector of the curve is ##(x',y')/l## and the normal unit vector is ##(-y',x')/l##
Here i defined the curve of the center of the triangle to be ##S(t)=(m(t),n(t))## where
$$n(t)=y(t)+\lambda y'/l +Rx'/l=2R\quad \lambda=\frac{R\sin(60-t)}{sin(30+t)}$$
Where I got lambda from using the sine rule again.
And now I'm lost. I don't know how to continue from here
 

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  • #2
Hello Westlife,

Here is a nice hint ...
 
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  • #3
Thank you I was able to find the solutin once I saw the video
 

Related to Finding the parametric equation of a curve

1. What is a parametric equation of a curve?

A parametric equation of a curve is a set of equations that describe the x and y coordinates of a point on a curve in terms of a third variable, typically referred to as the parameter. This allows for a more flexible and general way of representing curves compared to traditional equations like y = f(x).

2. Why is it important to find the parametric equation of a curve?

Finding the parametric equation of a curve allows for a more precise and efficient way of representing and analyzing curves. It also enables us to better understand the behavior and properties of the curve, such as its rate of change and direction of movement.

3. How do you find the parametric equation of a curve?

The process for finding the parametric equation of a curve involves determining the x and y coordinates of a point on the curve in terms of a parameter, and then eliminating the parameter to obtain a single equation with x and y. This can be done through various methods such as using trigonometric functions, vector operations, or calculus techniques.

4. What are some real-life applications of parametric equations?

Parametric equations are commonly used in fields such as physics, engineering, and computer graphics to model and analyze various phenomena. They are also used in designing curves and shapes in industries like architecture and manufacturing.

5. Are there any limitations or drawbacks to using parametric equations?

One limitation of parametric equations is that they can sometimes be more complex and difficult to work with compared to traditional equations. Additionally, parametric equations may not always provide a clear visual representation of the curve, making it challenging to interpret the curve's shape and behavior.

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