# Conversions between intrinsic/extrinsic coordinates

• Jhenrique
Thank you!In summary, the conversion between Cartesian coordinates and intrinsic coordinates can be found in textbooks and online resources. In the 3D case, the conversion involves using the Frenet-Serret formulas and can be written in terms of arc length, curvature, and torsion.
Jhenrique
I noticed that in wiki there is the follows conversion:

that is the conversion beetwen the cartesian coordinates and the intrinsic (arc length, curvature) "coordinate". In this case, the system is 2D.

There is the 3D case too:

However, is missing the (x, y, z) coordinates in terms of (s, κ, τ). Where I can find this conversion?

Hello,

Thank you for bringing this up. The conversion between Cartesian coordinates and intrinsic coordinates can be found in many textbooks and online resources. In the 3D case, the conversion can be written as:

x = s cos(τ)cos(κ)
y = s sin(τ)cos(κ)
z = s sin(κ)

Where s is the arc length, κ is the curvature, and τ is the torsion. This conversion is derived from the Frenet-Serret formulas, which describe the relationship between the tangent, normal, and binormal vectors of a curve in 3D space.

I hope this helps and please let me know if you have any further questions.

## 1. What are intrinsic and extrinsic coordinates?

Intrinsic coordinates refer to the coordinates of a point or object within a specific frame of reference, such as the coordinates of a point on a curve. Extrinsic coordinates, on the other hand, are the coordinates of a point or object in relation to a fixed external frame of reference, such as the coordinates of a point on a map.

## 2. How are intrinsic and extrinsic coordinates related?

Intrinsic and extrinsic coordinates are related through a mathematical transformation. This transformation allows us to convert coordinates from one frame of reference to another, making it possible to describe the same point or object using different sets of coordinates.

## 3. What is the purpose of converting between intrinsic and extrinsic coordinates?

The purpose of converting between intrinsic and extrinsic coordinates is to describe the same point or object using different frames of reference. This can be useful in various fields, such as mathematics, physics, and computer graphics.

## 4. What are some common techniques used for converting between intrinsic and extrinsic coordinates?

Some common techniques used for converting between intrinsic and extrinsic coordinates include rotation, translation, scaling, and reflection. Other techniques may also be used depending on the specific problem or application.

## 5. Are there any challenges or limitations when converting between intrinsic and extrinsic coordinates?

Yes, there can be challenges and limitations when converting between intrinsic and extrinsic coordinates. For example, if the transformation is not well-defined or if the coordinate systems are incompatible, the conversion may not be accurate. Additionally, some transformations may result in loss of information or introduce errors in the coordinates.

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