Math Equation for Backward Going Curve?

[pairofstrings]

TL;DR Summary

I have a curve that goes backwards.

Hello.
I have a curve, I want to write mathematical statements that describes all the features of the curve. For example: how do I write math statement that describes its curvature...
Is it possible to write equation for curves that goes backwards?
Thanks.

 

[jedishrfu]

Its not a function. You need to break it up into two distinct functions. From there you can compute what you need excluding the endpoints of the functions.

 

[fresh_42]

Since there are values of $x$ that have two values of $y$, and values of $y$ that have two values of $x$, it isn't a function. But it is a curve, so you can write an equation: $f\, : \,[0,1]\longrightarrow \mathbb{R}^2\, , \,f(t)=(x(t),y(t))$ that maps the way you go along the curve: $t=0$ at the beginning and $t=1$ at the end of the path. If it has no ends, then we need not an interval but the real axis for $t$ instead.

Crucial in any case is, that you have to describe the points of your plot somehow. Or did you only draw a curve?

 

[PeroK]

Summary:: I have a curve that goes backwards.

Hello.
I have a curve, I want to write mathematical statements that describes all the features of the curve. For example: how do I write math statement that describes its curvature...
Is it possible to write equation for curves that goes backwards?
Thanks.

You need to parameterise the curve, where $x(u)$ and $y(u)$ are functions of a parameter $u$.

PS or $t$ would do just as well as the parameter!

 

[FactChecker]

For this example, it might be possible to transform to an axis system where the curve is a parabola.
Some data points would be needed, like the point where it crosses the x-axis. I can not be more specific off the top of my head.

 

[robphy]

As others have said above, write the curve parametrically.
Then calculate the curvature using
https://en.wikipedia.org/wiki/Curvature#In_terms_of_a_general_parametrization

If you don't have a parametrization,
you can try https://en.wikipedia.org/wiki/Bézier_curve

 

[mathman]

any quadraic curve can be represented in the form $ax^2+by^2+cx+dy+f=0$.

 

[pairofstrings]

Crucial in any case is, that you have to describe the points of your plot somehow. Or did you only draw a curve?

Can you please give me an example that could have backward going curve?

Thanks..

 

[PeroK]

Can you please give me an example that could have backward going curve?

Thanks..

A simple example is $y = 1, x = -t$, where $t \in [0, 1]$.

Another example is an anticlockwise circular path: $x = \cos t, y = \sin t$

 

[robphy]

Implementing what I mentioned earlier...

Using someone's Desmos file (found by Googling) https://www.desmos.com/calculator/cahqdxeshd
for tuning a Bezier curve [see also https://pomax.github.io/bezierinfo/ ], I took your plot and manually fitted your curve using the 4 control points.
See https://www.desmos.com/calculator/nobtxod6mk

$\begin{align*} C_{simple} &=\left(\left( -x_{0}+3x_{1}-3x_{2}+x_{3}\right)t^{3}+\left(3x_{0}-6x_{1}+3x_{2}\right)t^{2}+\left(-3x_{0}+3x_{1}\right)t+x_{0}, \right. \\ &\qquad\left. \left( -y_{0}+3y_{1}-3y_{2}+y_{3}\right)t^{3}+\left(3y_{0}-6y_{1}+3y_{2}\right)t^{2}+\left(-3y_{0}+3y_{1}\right)t+y_{0}\right) \end{align*}$
where $(x_0,y_0)=(-0.64,0.37)$, etc...

Now you can use this parametrized curve in
https://en.wikipedia.org/wiki/Curvature#In_terms_of_a_general_parametrization
to find its curvature function,
which can be evaluated at any value of t.

 

[pairofstrings]

Its not a function. You need to break it up into two distinct functions. From there you can compute what you need excluding the endpoints of the functions.

There is a need is to have two distinct functions, so, these functions are going to be (called) piecewise-defined functions?

 

[Mark44]

There is a need is to have two distinct functions, so, these functions are going to be (called) piecewise-defined functions?

No. A piecewise-defined function can have different formulas on different parts of its domain, but it can't have two or more function values for a given input value.

 

[pairofstrings]

A piecewise-defined function can have different formulas on different parts of its domain, but it can't have two or more function values for a given input value.

Can you please give me an example?
Thanks.

 

[Mark44]

Can you please give me an example?

You can easily look up many examples on your own. Just do a search for "piecewise-defined function."

 

[jedishrfu]

Check here for examples and a discussion of piece wise functions

https://www.storyofmathematics.com/piecewise-functions

 

[pairofstrings]

"Piecewise functions are defined by different functions throughout the different intervals of the domain."
What is meaning of "domain" in the above statement?
I am confused between intervals and domain.

"A piecewise function is a function that is defined by different formulas or functions for each given interval. It’s also in the name: piece. The function is defined by pieces of functions for each part of the domain."
What is the meaning of "..each part of the domain"?

Is it this that is being referred to:

From the above piecewise-defined function:
$2x$ is a function $f(x)$, and "for $x > 0$" is a part of the domain?

 

[jedishrfu]

In the above example, the domain of x for the purposes of this piecewise function is divided into three parts and x>0 is one of them.

 

[DaveE]

$2x$ is a function $f(x)$, and "for $x > 0$" is a part of the domain?

$x > 0$ is the domain of the function $2x$ in this example. It is part of the domain of $f(x)$, whose domain also includes $x = 0$ and $x < 0$. $2x$ is not a function of $f$ ($2x$ is a function of $x$); I don't understand why they said that.