what is the relationship between cartesian equation and parametric equation?

Jul 2015
82
1
malaysia
hmm can anyone give a guidance on what is the relationship between cartesian equation and parametric equation.

My ans is cartesian equation is an equation which is y=x^2 while parametric equation is an equation with parameter which is t as a link to form a cartesian equation, for example y=sint, x = cos t, through basic identity of trigonometry, sint^2 +cost^2 =1, y^2+x^2=1. therefore this show that parametric equation is the linker to form an equation from parameter given.

but the question is that, do my answer actually explain the relationship between cartesian equation and parametric equation?
do I need to add on more information on the relationship between cartesian equation and parametric equation?
thanks :) ur comment i will appreciate them :)
 

romsek

MHF Helper
Nov 2013
6,744
3,037
California
I don't know that there is any magical relationship.

As you noted in a Cartesian equation you can write

$y=f(x)$

where as in a parametric equation you write

$\{x,y\} = \{x(t), y(t)\}$

to achieve the same functional mapping.

There will be plenty of parametric equations that don't have Cartesian equivalents given the usual vocabulary of functions.

You can always find a simple parametric equivalent of a Cartesian equation. $\{x,y\} = \{t,f(t)\}$ where $f$ is such that $y=f(x)$
 
Sep 2012
1,061
434
Washington DC USA
There are actually 3 different things here: graphs of functions, "Cartesian Equations", and parametric equations.

The distinction between graphs of functions and "Cartesian Equations" (to use your terminology) is best seen by considering a circle.

The equation \(\displaystyle x^2 + y^2 = 1\) is satisfied by a set of points (x, y) in the plane that form the unit circle at the origin. But it's not the graph of any function y = f(x).
(Would f(0) be 1, or -1? It can't be both for a function f. This is the origin of the so-called "vertical line rule", which the circle fails.)​

So by "Cartesian Equations", I assuming you're including things like \(\displaystyle x^2 + y^2 = 1\), which is not the graph of any function. (It's the union of the graphs of 2 different functions, \(\displaystyle g_1(x) = \sqrt{1 - x^2}\) and \(\displaystyle g_2(x) = - \sqrt{1 - x^2}\).)

The following are the 3 types of sets in the plane it seems you want to consider:

1. Graphs of Functions: \(\displaystyle \{ (x, f(x)) \in \mathbb{R}^2 \ | \ x \in D_f \}\) (where \(\displaystyle D_f \subset \mathbb{R}\) is the function's domain.)

2. "Cartesian Equations": \(\displaystyle \{ (x, y) \in \mathbb{R}^2 \ | \ F(x, y) = 0 \}\)

3. Parametric Equations: \(\displaystyle \{ Q(t) = (x(t), y(t)) \in \mathbb{R}^2 \ | \ t \in D \}\), where \(\displaystyle Q : D (\subset \mathbb{R}) \rightarrow \mathbb{R}^2\).

The differences? Well, every graph of a function y = f(x) could be represented by a "Cartesian Equation" or a parametric equation.
Ex: \(\displaystyle F(x, y) = y - f(x)\) when x is in f's domain, and define \(\displaystyle F(x, y)\) however you like, other than zero, otherwise. And for parametric, set \(\displaystyle x(t) = t, y(t) = f(x(t))\), where the t-domain is f's domain.​

However, not every "Cartesian Equation" can be continuously parameterized.
\(\displaystyle F(x, y) = xy - x^3 = x(y - x^2)\), whose graph is the union of a parabola and the x-axis.​

For practical purposes, the same is true in reverse (the formal claim would be a mess, and likely be false), meaning that not every parametric equation has the same image as a "Cartesian Equation"
Ex: Try to find F such that \(\displaystyle F(x(t), y(t)) = 0 \ \forall \ t\) where \(\displaystyle y(t) = t^{35} + 4 t^4 - 3 e^{\sin(t)}, x(t) = 2t^{10} - t \tan(t) - e^t\cos(\ln(t))\).​

Note that the set of points in the plane via "Cartesian Equation" is given by a function inverse at a point (it's the set \(\displaystyle F^{-1}(0)\)), while in the parametric case it's the image set of a function (the function Q).

Also, if you don't put some minimal restrictions - especially continuity - on the possible functions F and Q for "Cartesian Equations" and parametric equations respectively, then ANY subset of the plane comes from such an F or Q.
Ex: For F, you could use 1 minus the indicator function, meaning F of a point is 0 if the point is in the given set, and 1 otherwise. For Q, one could exploit a set bijection between \(\displaystyle \mathbb{R}\) and \(\displaystyle \mathbb{R}^2\).​

Even requiring continuity, there are very counter-intuitive examples possible under these definitions.
Ex: F(x,y) := 0 is the "Cartesian Equation" whose graph is the entire plane. Likewise, parametricly, there are continuous space-filling curves (like the Peano curve: https://en.wikipedia.org/wiki/Peano_curve)... counter-intuitive indeed!​

A set of points in a plane is not a function! A corresponding function is uniquely recoverable from that only in the first case, where you're told that the set of points is the graph of a function. In the other two cases, "Cartesian coordinates" and parametric equations, there are multiple functions F and Q, respectively, which could produce those sets.
Ex: For the "Cartesian Equation" case, if F defines the set in the plane, then kF defines the same set for any non-zero constant k. For parametric, consider the line y = x via \(\displaystyle x(t) =t, y(t) = t\), OR \(\displaystyle x(t) =t^3, y(t) = t^3\).​

The functions F and Q contain more information than you get from just the set of points in the plane given by \(\displaystyle F^{-1}(0)\) and \(\displaystyle \text{Image}(Q)\) respectively!

The "best" form in general is the parametric form, because it contains the most information, is the most flexible, is the most direct (it's just a function \(\displaystyle \mathbb{R} \rightarrow \mathbb{R}^2\)) and is the most useful for modelling.

Parametric Equations common use of t is, of course, for "time". So you think of parametric equations as describing a point moving in the plane in time - which clearly contains vastly more information than just the set that moving point eventually covers. And even more valuable than all that additional information is that the description of a point moving through time is EXACTLY what's needed to model most dynamic real world scenarios.

At the level of generality of your question - asking for "the relationship" between them - that's about all I can think to say.
 
Last edited:
Jul 2015
82
1
malaysia
There are actually 3 different things here: graphs of functions, "Cartesian Equations", and parametric equations.

The distinction between graphs of functions and "Cartesian Equations" (to use your terminology) is best seen by considering a circle.

The equation \(\displaystyle x^2 + y^2 = 1\) is satisfied by a set of points (x, y) in the plane that form the unit circle at the origin. But it's not the graph of any function y = f(x).
(Would f(0) be 1, or -1? It can't be both for a function f. This is the origin of the so-called "vertical line rule", which the circle fails.)​

So by "Cartesian Equations", I assuming you're including things like \(\displaystyle x^2 + y^2 = 1\), which is not the graph of any function. (It's the union of the graphs of 2 different functions, \(\displaystyle g_1(x) = \sqrt{1 - x^2}\) and \(\displaystyle g_2(x) = - \sqrt{1 - x^2}\).)

The following are the 3 types of sets in the plane it seems you want to consider:

1. Graphs of Functions: \(\displaystyle \{ (x, f(x)) \in \mathbb{R}^2 \ | \ x \in D_f \}\) (where \(\displaystyle D_f \subset \mathbb{R}\) is the function's domain.)

2. "Cartesian Equations": \(\displaystyle \{ (x, y) \in \mathbb{R}^2 \ | \ F(x, y) = 0 \}\)

3. Parametric Equations: \(\displaystyle \{ Q(t) = (x(t), y(t)) \in \mathbb{R}^2 \ | \ t \in D \}\), where \(\displaystyle Q : D (\subset \mathbb{R}) \rightarrow \mathbb{R}^2\).

The differences? Well, every graph of a function y = f(x) could be represented by a "Cartesian Equation" or a parametric equation.
Ex: \(\displaystyle F(x, y) = y - f(x)\) when x is in f's domain, and define \(\displaystyle F(x, y)\) however you like, other than zero, otherwise. And for parametric, set \(\displaystyle x(t) = t, y(t) = f(x(t))\), where the t-domain is f's domain.​

However, not every "Cartesian Equation" can be continuously parameterized.
\(\displaystyle F(x, y) = xy - x^3 = x(y - x^2)\), whose graph is the union of a parabola and the x-axis.​

For practical purposes, the same is true in reverse (the formal claim would be a mess, and likely be false), meaning that not every parametric equation has the same image as a "Cartesian Equation"
Ex: Try to find F such that \(\displaystyle F(x(t), y(t)) = 0 \ \forall \ t\) where \(\displaystyle y(t) = t^{35} + 4 t^4 - 3 e^{\sin(t)}, x(t) = 2t^{10} - t \tan(t) - e^t\cos(\ln(t))\).​

Note that the set of points in the plane via "Cartesian Equation" is given by a function inverse at a point (it's the set \(\displaystyle F^{-1}(0)\)), while in the parametric case it's the image set of a function (the function Q).

Also, if you don't put some minimal restrictions - especially continuity - on the possible functions F and Q for "Cartesian Equations" and parametric equations respectively, then ANY subset of the plane comes from such an F or Q.
Ex: For F, you could use 1 minus the indicator function, meaning F of a point is 0 if the point is in the given set, and 1 otherwise. For Q, one could exploit a set bijection between \(\displaystyle \mathbb{R}\) and \(\displaystyle \mathbb{R}^2\).​

Even requiring continuity, there are very counter-intuitive examples possible under these definitions.
Ex: F(x,y) := 0 is the "Cartesian Equation" whose graph is the entire plane. Likewise, parametricly, there are continuous space-filling curves (like the Peano curve: https://en.wikipedia.org/wiki/Peano_curve)... counter-intuitive indeed!​

A set of points in a plane is not a function! A corresponding function is uniquely recoverable from that only in the first case, where you're told that the set of points is the graph of a function. In the other two cases, "Cartesian coordinates" and parametric equations, there are multiple functions F and Q, respectively, which could produce those sets.
Ex: For the "Cartesian Equation" case, if F defines the set in the plane, then kF defines the same set for any non-zero constant k. For parametric, consider the line y = x via \(\displaystyle x(t) =t, y(t) = t\), OR \(\displaystyle x(t) =t^3, y(t) = t^3\).​

The functions F and Q contain more information than you get from just the set of points in the plane given by \(\displaystyle F^{-1}(0)\) and \(\displaystyle \text{Image}(Q)\) respectively!

The "best" form in general is the parametric form, because it contains the most information, is the most flexible, is the most direct (it's just a function \(\displaystyle \mathbb{R} \rightarrow \mathbb{R}^2\)) and is the most useful for modelling.

Parametric Equations common use of t is, of course, for "time". So you think of parametric equations as describing a point moving in the plane in time - which clearly contains vastly more information than just the set that moving point eventually covers. And even more valuable than all that additional information is that the description of a point moving through time is EXACTLY what's needed to model most dynamic real world scenarios.

At the level of generality of your question - asking for "the relationship" between them - that's about all I can think to say.
what if i change it to relation between cartesian equation and parametric equation? is there any relation between these two?
 
Jul 2015
82
1
malaysia
what if i change it to relation between cartesian equation and parametric equation? is there any relation between these two?
 
Sep 2012
1,061
434
Washington DC USA
At this level of generality, I think I've pointed out most of the obvious similarities and differences. (FYI: "Cartesian Equation" isn't particularly standard terminology - I was just following your lead.)