# Is this a valid parametrisation of a curve?

• rickonstark999
In summary: But @rickonstark999, you do understand that your first post above is correct and your parameterization ##x=t^2,~y=4t## with ##t## both positive and negative, ie ##-\infty < t < \infty## is a correct parameterization of your curve, right? You don't need any ##\pm## sign.Yes, I understand that my parameterization is correct.Yes, I understand that my parameterization is correct.
rickonstark999
Member warned that the homework template is required
Hi, so I am learning how to parametrise curves.

For the curve y^2=16x, I have said let x=t^2. Then, we can say y^2=16t^2, so that we can take the root of this and get y=4t. What I wanted to ask was do we have to say "plus or minus" in front of the 4t, or do we just leave it as positive to get a valid parametrisation?

Thanks.

rickonstark999 said:
Hi, so I am learning how to parametrise curves.

For the curve y^2=16x, I have said let x=t^2. Then, we can say y^2=16t^2, so that we can take the root of this and get y=4t. What I wanted to ask was do we have to say "plus or minus" in front of the 4t, or do we just leave it as positive to get a valid parametrisation?

Thanks.
The question to ask is what values can ##t## take?

rickonstark999 said:
Hi, so I am learning how to parametrise curves.

For the curve y^2=16x, I have said let x=t^2. Then, we can say y^2=16t^2, so that we can take the root of this and get y=4t. What I wanted to ask was do we have to say "plus or minus" in front of the 4t, or do we just leave it as positive to get a valid parametrisation?

Thanks.

Is ##y## allowed to be ##<0## along the curve? Is ##t < 0## allowed?

Ray Vickson said:
Is ##y## allowed to be ##<0## along the curve? Is ##t < 0## allowed?

I suppose so? The question doesn't specify so I am not sure.

rickonstark999 said:
I suppose so? The question doesn't specify so I am not sure.

How would you draw a graph of the relation ##x = \frac{1}{16} y^2##?

Ray Vickson said:
How would you draw a graph of the relation ##x = \frac{1}{16} y^2##?

You would have a sideways parabola where y can be any real number, but x cannot be below 0 because you are squaring.

rickonstark999 said:
You would have a sideways parabola where y can be any real number, but x cannot be below 0 because you are squaring.

OK, so now what does that tell you about ##t## in your parametrization?

Ray Vickson said:
OK, so now what does that tell you about ##t## in your parametrization?

From my parametrisations, t can be both positive and negative. Would you say that is correct?

rickonstark999 said:
Then, we can say y^2=16t^2, so that we can take the root of this and get y=4t.
No, that isn't right. If ##y^2 = 16t^2##, then ##y = \pm 4t##.

Mark44 said:
No, that isn't right. If ##y^2 = 16t^2##, then ##y = \pm 4t##.

Thank you. That is what I was asking in the sentence after that.

rickonstark999 said:
From my parametrisations, t can be both positive and negative. Would you say that is correct?

Mark44 said:
No, that isn't right. If ##y^2 = 16t^2##, then ##y = \pm 4t##.

rickonstark999 said:
Thank you. That is what I was asking in the sentence after that.

But @rickonstark999, you do understand that your first post above is correct and your parameterization ##x=t^2,~y=4t## with ##t## both positive and negative, ie ##-\infty < t < \infty## is a correct parameterization of your curve, right? You don't need any ##\pm## sign.

PeroK

## 1. What is a parametrisation of a curve?

A parametrisation of a curve is a way of representing a curve using a set of equations or parameters. It allows us to describe the movement and shape of a curve in a systematic way.

## 2. Why is it important to have a valid parametrisation of a curve?

A valid parametrisation of a curve is important because it helps us to understand and analyze the behaviour of the curve. It also allows us to perform calculations and make predictions about the curve's properties.

## 3. How can I determine if a parametrisation is valid?

A parametrisation is considered valid if it meets certain criteria, such as being continuous, differentiable, and covering the entire curve. To determine if a parametrisation is valid, you can check if it satisfies these conditions.

## 4. What are some common mistakes when parametrising a curve?

Some common mistakes when parametrising a curve include not covering the entire curve, having discontinuous or non-differentiable parts, and using incorrect equations or parameters. It is important to double check the parametrisation to ensure it accurately represents the curve.

## 5. Can a curve have multiple valid parametrisations?

Yes, a curve can have multiple valid parametrisations. However, some parametrisations may be more useful or convenient for certain applications than others. It is important to choose a parametrisation that best suits the purpose and analysis of the curve.

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