# Parametric Equation (u,v,θ): Explained

• Logarythmic
In summary, the given equations represent a change in coordinates in \mathbb{R}^{3}, specifically from cylindrical coordinates to Cartesian coordinates. The applications defined by these equations may or may not be invertible, depending on the restrictions placed on the parameters. This is a parametric equation for a surface in \mathbb{R}^{3}, but in order to fully describe a specific surface, two parameters must be used instead of three.
Logarythmic
In coordinates $$(u,v,\theta)$$:

$$x = \sqrt{uv} \cos{\theta}, y=\sqrt{uv} \sin{\theta}, z = \frac{1}{2}(u-v)$$

What does this represent?

A change in coordinates on $\mathbb{R}^{3}$. You should check whether the applications thus defined are invertible or not.

Daniel.

That's not a part of my problem. This is a parametric equation for something, I'm just curious about what this something looks like...

It's a parametric equation for a change in coordinates in R^3. It should be an application of R^3 into itself, invertible and differentiable everywhere, i.e. diffeomorphism.

Daniel.

Yeah, but I mean

$$x = r \cos \theta, y = r \sin \theta, z = z$$

is an parametric equation for a cylinder. And my example is a parametric equation for..? For what?

Last edited:
Logarythmic said:
Yeah, but I mean

$$x = r \cos \theta, y = r \sin \theta, z = z$$

is an parametric equation for a cylinder. And my example is a parametric equation for..? For what?

No, they are not. Those are the equations for changing from cylindrical coordinates to Cartesian coordinates in R3, just as Dextercioby said. IF you put restrictions on them, such as $0\le \theta \le 2\pi$, $0\le r \le 1$, $0\le z\le 1$, then they are parametric equations describing a cylinder of radius 1, length 1. If you set $0\le \theta \le 2\pi$, [itexr = 1[/itex], $-\infty\le z\le\infty$, then you have parametric equations for the surface of an infinite cylinder.

The equations you give, both here and in your original post can take on any values for x, y, z because u, v, $\theta$ can have any values. If you want to describe a specific region in R3, then you must put restrictions on them. If you want to describe a surface then, since a surface is two-dimensional, you must have x, y, z given in terms of two parameters, not three.

## 1. What is a parametric equation and how is it different from a regular equation?

A parametric equation is a mathematical expression that defines the relationship between two or more variables in terms of a third variable. This third variable is usually referred to as a parameter, which can take on various values. Unlike a regular equation, which typically has a single dependent and independent variable, a parametric equation can have multiple dependent and independent variables.

## 2. What is the significance of the variables u, v, and θ in a parametric equation?

The variables u, v, and θ in a parametric equation represent the parameters that define the relationship between the dependent and independent variables. U and v are often used as the two independent variables, while θ is commonly used as the parameter that determines the shape or orientation of the graph.

## 3. How do you plot a graph using a parametric equation?

To plot a graph using a parametric equation, you first need to define the range of values for the parameter θ. Then, you can plug in different values of θ into the equation to calculate corresponding values for u and v. Plot these points on a coordinate system and connect them to create a graph.

## 4. What are the advantages of using parametric equations?

Parametric equations offer several advantages over regular equations, such as the ability to represent complex curves and surfaces that cannot be easily described by a single equation. They also allow for more flexibility in manipulating and transforming graphs, as the parameters can be easily adjusted to change the shape and orientation of the graph.

## 5. Can parametric equations be used in real-world applications?

Yes, parametric equations have numerous real-world applications in fields such as physics, engineering, and computer graphics. They are commonly used to describe the motion of objects in space, such as the trajectory of a projectile or the movement of a robot arm. They are also used in computer animation to create 3D models and special effects.

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