# How do I find the Euclidean Coordinate Functions of a parametrized curve?

• MHB
• banananaz
In summary, the given curve is parametrized by t and has x-coordinate of cos(t), y-coordinate of t^2, and z-coordinate of 0. To find the euclidean coordinate functions, we can write the second equation as t= +/- sqrt(y) and then use the fact that cosine is an even function to get x= cos(sqrt(y)).
banananaz
I've been given a curve α parametrized by t :

α (t) = (cos(t), t^2, 0)

How would I go about finding the euclidean coordinate functions for this curve? I know how to find euclidean coord. fns. for a vector field, but I am a bit confused here.

I'm not certain what you mean by "Euclidean Coordinates". Perhaps you mean what I would call "Cartesian Coordinates"- a direct relation of x and y. The equations you have are x= cos(t), y= t^2, z= 0. The second equation can be written $t= \pm\sqrt{y}$ so we get two equations, $t= \sqrt{y}$ and $t= -\sqrt{y}$ and then $x= cos(\sqrt{y})$ and $x= cos(-\sqrt{y})$.

But since cosine is an "even function", $cos(-\theta)= cos(\theta)$, those both give $x= cos(\sqrt{y})$.

## 1. What is a parametrized curve?

A parametrized curve is a mathematical representation of a curve in which the coordinates of each point on the curve are expressed in terms of one or more parameters. These parameters can be thought of as variables that determine the position of a point on the curve.

## 2. How do I determine the Euclidean Coordinate Functions of a parametrized curve?

To find the Euclidean Coordinate Functions of a parametrized curve, you need to first express the coordinates of the curve in terms of the parameters. Then, you can use the Pythagorean theorem to find the Euclidean distance between two points on the curve. Finally, you can use this distance to determine the Euclidean Coordinate Functions.

## 3. What is the Pythagorean theorem?

The Pythagorean theorem is a mathematical principle that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

## 4. Can I use a parametrized curve to represent any type of curve?

Yes, a parametrized curve can represent any type of curve, including straight lines, circles, ellipses, and more complex curves.

## 5. What are some applications of parametrized curves?

Parametrized curves have many applications in mathematics, physics, and engineering. They are commonly used to model the motion of objects in space, such as the trajectory of a projectile or the path of a planet around a star. They are also used in computer graphics to create smooth and realistic curves for animations and simulations.

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