What is parametric representation and how is it used

[ArmChairPhysicist]

Ive been working through calculus this year and will be into next year, and as nearly every time I open my calculus book I have found something new and mysterious. This time it's something called parametric representation.

It isn't clearly explained what this means or how you go about converting between implicit/explicit and parametric forms.

An example of what my book is giving me.
X^2 +Y^2 = 9 is the equation of a circle with center at the origin and radius equal to 3.
If [theta] is the angle that the radius to the point (x,y) on the circle makes with the x axis, find the parametric representation of the circle in terms of [theta].
What rules govern this subject?
Why do I need it?
How and when do I use parametric representation?
Many thanks in advance.

 

[fresh_42]

Ive been working through calculus this year and will be into next year, and as nearly every time I open my calculus book I have found something new and mysterious. This time it's something called parametric representation.

It isn't clearly explained what this means or how you go about converting between implicit/explicit and parametric forms.

An example of what my book is giving me.
X^2 +Y^2 = 9 is the equation of a circle with center at the origin and radius equal to 3.
If [theta] is the angle that the radius to the point (x,y) on the circle makes with the x axis, find the parametric representation of the circle in terms of [theta].
What rules govern this subject?
Why do I need it?
How and when do I use parametric representation?
Many thanks in advance.

A parametrization of a curve $C$ (or a straight) by a parameter $t$ (loosely associated with "time") is a "walkthrough", a "path". You have to find a (continuous) function $p : I \longmapsto C$ where $I \subseteq \mathbb{R}$ is an interval, it's often $I = [0,1]\, , \,t=0$ the staring point and $t=1$ the end point of the path $p(t)$. This means, that to every point of the curve $C$ there is a point in time (or a few), where the path is at this point.

In the case of your example, $C$ is the circle and it isn't "time" here, but the angle $t=\theta$ that measures your path. For the interval you can take $I=[0,2\pi ]$ or $I=\mathbb{R}$ depending on how often you want to circle.

Why do you need it? Well, e.g. it's the general concept that defines a motion along a curve. You use it each time you drive your car. A path can divide regions into interior and exterior parts, can be used to speak of the length of a curve, and many more. Some are of mathematical interest like "path connected points", but many more of physical, as it is motion and basically defines what can be reached and what can not.

 

[Ray Vickson]

Ive been working through calculus this year and will be into next year, and as nearly every time I open my calculus book I have found something new and mysterious. This time it's something called parametric representation.

It isn't clearly explained what this means or how you go about converting between implicit/explicit and parametric forms.

An example of what my book is giving me.
X^2 +Y^2 = 9 is the equation of a circle with center at the origin and radius equal to 3.
If [theta] is the angle that the radius to the point (x,y) on the circle makes with the x axis, find the parametric representation of the circle in terms of [theta].
What rules govern this subject?
Why do I need it?
How and when do I use parametric representation?
Many thanks in advance.

If you do not understand something in your book, look in another book---or nowadays, go on-line and use Google. For example, if I Google the key words "parametric equation" I get immediate (free) access to hundreds of articles giving lots of examples. Your exact example is explained in detail in many of the articles.

 

[Eclair_de_XII]

X^2 +Y^2 = 9 is the equation of a circle with center at the origin and radius equal to 3.
If [theta] is the angle that the radius to the point (x,y) on the circle makes with the x axis, find the parametric representation of the circle in terms of [theta].

That sounds like Calculus III. If possible, I'd go one step further and look up "polar coordinates".

 

[fresh_42]

Here's a nice video. It is about something else, but beginning at 7:00 there is a nice visualization of a parametrization.
https://www.physicsforums.com/threa...in-calculus-on-manifolds.914809/#post-5763187

 

[ehild]

That sounds like Calculus III. If possible, I'd go one step further and look up "polar coordinates".

Ive been working through calculus this year and will be into next year, and as nearly every time I open my calculus book I have found something new and mysterious. This time it's something called parametric representation.

It isn't clearly explained what this means or how you go about converting between implicit/explicit and parametric forms.

An example of what my book is giving me.
X^2 +Y^2 = 9 is the equation of a circle with center at the origin and radius equal to 3.
If [theta] is the angle that the radius to the point (x,y) on the circle makes with the x axis, find the parametric representation of the circle in terms of [theta].

It means that you write the x and y coordinates in terms of r and theta. See figure.