# Show reparameterized curvature equals curvature up to a sign

• MxwllsPersuasns
In summary, we used the general formula for curvature to show that the curvature of a reparameterised curve is equal to the curvature of the original curve, up to a sign determined by the orientation of the reparameterisation function.
MxwllsPersuasns

## Homework Statement

Let γ: I → ℝ2 be a smooth regular curve and let λ = γ ο φ with φ: Iλ → I be a reparameterisation of γ. Show, by using the general formula for curvature of a regular curve that κλ = ±κ ο Φ where the ± depends on whether φ is orientation preserving (+) or reversing (-).

## Homework Equations

κ = Det{γ',γ''}/{absval(γ')}3

## The Attempt at a Solution

So, the way I would approach this problem I guess is by first taking the curvature of γ...
- Though the problem I immediately run into, and did on another problem I just posted, is whether this should be done in general or for a specific case of γ and λ
+ i.e., I would parameterize γ with t perhaps by <acos(t), bsin(t)> then take φ to be something orientation reversing (maybe like φ = -t) and again as something orientation preserving (like maybe φ = t2?) and then show that the curvature of each λ± equals the curvature of the original curve (up to a sign) reparameterized with Φ.
+ Another question is, why are we using k composed with Φ for the reparameterised curvature when the reparamterization is done by composing with the function φ? Wouldn't the curvature then be something like κλ = ±κ ο φ?

Any help or guidance is extremely appreciated I'm having a bit of trouble grasping how to start and where to go with this.

Thanks!To start, we can use the given formula for curvature to find the curvature of γ:

κ = Det{γ',γ''}/{absval(γ')}3

Next, using the chain rule, we can find the curvature of λ:

κλ = Det{λ',λ''}/{absval(λ')}3

Now, we need to express λ in terms of γ and φ. Since λ = γ ο φ, we can substitute this into the above equation:

κλ = Det{(γ ο φ)',(γ ο φ)''}/{absval((γ ο φ)')}3

Using the chain rule again, we can expand (γ ο φ)' and (γ ο φ)'':

κλ = Det{(γ'φ' + γ''φ),(γ''φ' + γ'''φ + γ'φ'')}/{absval(γ'φ' + γ''φ)}3

Now, using the properties of determinants, we can simplify this expression to:

κλ = Det{γ',γ''}/{absval(γ')}3 * Det{φ',φ''}/{absval(φ')}3

Since κ = Det{γ',γ''}/{absval(γ')}3, we can substitute this into the above equation to get:

κλ = κ * Det{φ',φ''}/{absval(φ')}3

Now, we can consider the orientation of φ. If φ is orientation preserving, then φ' and φ'' will have the same sign. This means that Det{φ',φ''} will be positive. Similarly, if φ is orientation reversing, then φ' and φ'' will have opposite signs, and Det{φ',φ''} will be negative.

Therefore, if φ is orientation preserving, κλ = κ * Det{φ',φ''}/{absval(φ')}3 = κ * (+1) = κ. And if φ is orientation reversing, κλ = κ * Det{φ',φ''}/{absval(φ')}3 = κ * (-1) = -κ.

This shows that the curvature of the reparameterised curve λ is equal to the curvature of the original curve γ, up to a sign depending on the orientation of φ. This is what we wanted to prove.

Hope this helps! Let me know if you have any further questions.

## What does it mean for a show to be reparameterized?

Reparameterization of a show refers to a mathematical transformation that changes the parameterization of the show without altering its shape or structure. This can be done by changing the speed at which the show is performed or by changing the direction in which the show is executed.

## What is curvature and how is it related to reparameterization?

Curvature is a mathematical concept that measures the amount by which a geometric object deviates from being flat. In the context of reparameterization, it refers to the rate at which the show changes direction. Reparameterizing a show does not change its curvature, but changes in the direction or speed of the show can affect the perception of curvature.

## What does it mean for two curvatures to be equal up to a sign?

When two curvatures are equal up to a sign, it means that they have the same magnitude but opposite signs. In other words, they have the same absolute value but differ in direction. This is important in the context of reparameterization as it allows us to compare the curvatures of a show before and after it has been reparameterized.

## How does this concept apply to real-world scenarios?

The concept of reparameterized curvature being equal to curvature up to a sign is applicable in various real-world scenarios, such as analyzing the trajectory of a moving object, understanding the behavior of waves, and studying the curvature of surfaces. In these cases, reparameterization allows us to manipulate the speed or direction of the object or wave while still maintaining its overall curvature.

## What are the practical implications of this concept in scientific research?

The idea of reparameterized curvature being equal to curvature up to a sign is important in scientific research as it allows for a more accurate analysis of data. By taking into account the effects of reparameterization, scientists can better understand the underlying patterns and relationships in their data. This concept also has applications in fields such as computer graphics and robotics, where precise control of curvature is necessary.

### Similar threads

• Calculus and Beyond Homework Help
Replies
10
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K