Ellipse in intrinsic coordinates

[chronon]

Does anyone know an equation for an ellipse (or other conics) in intrinsic coordinates, that is direction and curve length. It's the sort of thing that looks like it should be simple, but I've a feeling it may in fact be somewhat messy.

 

[traianus]

See this link and hopefully there is what you need

http://mathworld.wolfram.com/Ellipse.html

See also

http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html

 

[tacman]

Unfortunately, finding an equation for an ellipse in intrinsic coordinates can be quite challenging. Intrinsic coordinates, also known as natural coordinates, are a type of coordinate system that is defined by the geometry of a curve or surface. This means that the coordinates are determined by the properties of the curve or surface itself, rather than by an external reference frame.

To find the equation for an ellipse in intrinsic coordinates, we need to first define the curve length and direction. The curve length can be measured in terms of arc length, which is the distance along the curve from one point to another. The direction can be defined as the tangent to the curve at a given point, which represents the direction in which the curve is changing.

With these definitions in mind, we can use the parametric equation for an ellipse in Cartesian coordinates to derive an equation in intrinsic coordinates. This involves using the arc length and tangent to express the coordinates in terms of the intrinsic parameters.

However, as you mentioned, this process can be quite messy and may not result in a simple equation. It also depends on the specific parametrization of the ellipse that is chosen. So while it is theoretically possible to find an equation for an ellipse in intrinsic coordinates, it may not be practical or useful in most cases.

In general, it is more common to use Cartesian coordinates for describing conic sections such as ellipses. These coordinates are more intuitive and easier to work with in most cases. However, if you have a specific need for an equation in intrinsic coordinates, it may be worth exploring further or seeking out resources that specialize in this type of coordinate system.