Coordinate systems parameterized by pseudo arc-length

In summary, the conversation discusses defining a coordinate system for a circular arc that makes a specified angle ##\alpha## with a 90 degree wedge. A parametrization using trigonometric functions is suggested, with an example given for ##\alpha = {\pi\over 2}##. The conversation also considers the case of a non-90 degree angle and discusses the use of trigonometry to determine the domain of the parameter ##s##.
  • #1
member 428835
Hi PF!

Can anyone help me define a coordinate system for a circular arc that makes a specified angle ##\alpha## with a 90 degree wedge? See picture titled Geo.

As an example, a circular arc can be parameterized over a straight line by ##s##, making angle ##\alpha##, via $$\vec T = \left\langle \frac{\sin(s)}{\sin\alpha}, \frac{\cos (s) - \cos \alpha }{\sin\alpha} \right\rangle$$
 

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  • #2
Hi,
joshmccraney said:
define a coordinate system
are you defining a cordinate system, or are you searching for a parametrization ?

I like to keep things simple and pick an easy example, e.g. ##\alpha = {\pi\over 2}##, Now $$\vec T = (\sin s, \cos s) $$ is a quarter circle with radius 1 and centered at the origin. For ##s\in [0, {\pi\over 2}]## oriented clockwise.

But it's a parametrization in a cartesian coordinate system.
 
  • #3
BvU said:
Hi,
are you defining a cordinate system, or are you searching for a parametrization ?
Sorry, a parametrization!

BvU said:
I like to keep things simple and pick an easy example, e.g. ##\alpha = {\pi\over 2}##, Now $$\vec T = (\sin s, \cos s) $$ is a quarter circle with radius 1 and centered at the origin. For ##s\in [0, {\pi\over 2}]## oriented clockwise.

But it's a parametrization in a cartesian coordinate system.
Yep, this is simple, but what if I want ##\alpha \neq 90^\circ##?
 
  • #4
Simplest case: center of circle moves along ##y=x##. (If you want to restrict to a single ##\alpha##)
And you have to do some trig to find the domain of ##s##.
 

Related to Coordinate systems parameterized by pseudo arc-length

1. What is a coordinate system parameterized by pseudo arc-length?

A coordinate system parameterized by pseudo arc-length is a mathematical concept used in geometry and cartography to map points on a curved surface onto a flat plane. This type of coordinate system uses a parameter called pseudo arc-length, which is a measure of distance along a curve, to define the location of a point.

2. How is pseudo arc-length different from regular arc-length?

Pseudo arc-length is a measure of distance along a curve that is defined by a mathematical function, while regular arc-length is a physical distance along a curve. Pseudo arc-length is a normalized parameter that can be used to map points on any curve, while regular arc-length is specific to a particular curve.

3. What are the advantages of using a coordinate system parameterized by pseudo arc-length?

One advantage of using a coordinate system parameterized by pseudo arc-length is that it allows for a more accurate representation of curved surfaces on a flat plane. This can be useful in fields such as cartography, where accurate maps of curved surfaces are needed. Additionally, this type of coordinate system is more flexible and can be applied to a wide range of curves.

4. How is a coordinate system parameterized by pseudo arc-length calculated?

The calculation of a coordinate system parameterized by pseudo arc-length depends on the specific curve being mapped. In general, it involves finding the mathematical function that defines the curve, and then using this function to calculate the pseudo arc-length for each point on the curve. This can be done using mathematical techniques such as integration.

5. What are some real-world applications of coordinate systems parameterized by pseudo arc-length?

Coordinate systems parameterized by pseudo arc-length have many real-world applications, including mapping curved surfaces in cartography, creating accurate 3D models of objects in computer graphics, and analyzing data from curved surfaces in scientific research. They are also used in fields such as engineering, architecture, and geography to accurately represent curved surfaces in 2D or 3D space.

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