- #1
dipole
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(Note that the title of this thread might be incorrect - I'm just drawing on the vocabulary people use when discussing Lagrangian Mechanics...)
Hi, I'm trying to set up a coordinate system to represent points in space where one of the coordinates is the distance along a parametric curve, one is the shortest distance from a point to the curve, and one is an angle defined relative to some direction.
I have some curve in space defined by:
x = x(t)
y = y(t)
z = z(t)
and I want to define a coordinate system (s, \rho, \omega) relative to this curve.
\rho is defined by the shortest distance from a point p' = (x',y',z') (I will use primes to denote points not on the curve) in space to the curve (assume this is unique).
s is defined by the distance along the curve, starting from some initial point \left (x(t_0), y(t_0), z(t_0) \right) to the point \left (x(t), y(t), z(t) \right) such that
d \left (x', y', z', x(t), y(t), z(t) \right ) = \rho
where d is the euclidean distance.
In other words, say I have some point in space p' = (x',y',z'), then \rho is the length of the smallest line segment between p' and some point on the curve p(t) = (x(t), y(t), z(t)). I want to represent the vector from p(t) to p' by an orthogonal coordinate system that is attached to the curve.
Let \hat{s}(t) be the unit tangent vector to the curve at the point p(t). This defines a plane where the point p' lies on a circle of radius \rho in the plane. The only other thing I need to uniquely define the point p' in this plane is some reference direction to measure the angle at which the point p' lies on the circle. Call this direction \hat{\omega}(t).
Here's the kicker though - I want \hat{\omega}(t) to be defined in such a way that when the curve is a straight line, and in the z-direction, that the coordinate system becomes plain old cylindiral coordinates, and \hat{\omega}(t) = \hat{x} . However, when the curve is not a straight line, then \hat{\omega} should rotate with the curve in such a way that it is always orthogonal to \hat{s}(t), and locally if \hat{s}(t) \simeq \hat{z} then \hat{\omega}(t) \simeq \hat{x}.
My question is how to define \hat{\omega}(t) in such a way that satisfies these constraints.
Does this make sense? I hope at least the idea of what I want to do is clear. I further hope this problem is not ill-posed. Any help would be appreciated and please let me know if something is unclear.
Hi, I'm trying to set up a coordinate system to represent points in space where one of the coordinates is the distance along a parametric curve, one is the shortest distance from a point to the curve, and one is an angle defined relative to some direction.
I have some curve in space defined by:
x = x(t)
y = y(t)
z = z(t)
and I want to define a coordinate system (s, \rho, \omega) relative to this curve.
\rho is defined by the shortest distance from a point p' = (x',y',z') (I will use primes to denote points not on the curve) in space to the curve (assume this is unique).
s is defined by the distance along the curve, starting from some initial point \left (x(t_0), y(t_0), z(t_0) \right) to the point \left (x(t), y(t), z(t) \right) such that
d \left (x', y', z', x(t), y(t), z(t) \right ) = \rho
where d is the euclidean distance.
In other words, say I have some point in space p' = (x',y',z'), then \rho is the length of the smallest line segment between p' and some point on the curve p(t) = (x(t), y(t), z(t)). I want to represent the vector from p(t) to p' by an orthogonal coordinate system that is attached to the curve.
Let \hat{s}(t) be the unit tangent vector to the curve at the point p(t). This defines a plane where the point p' lies on a circle of radius \rho in the plane. The only other thing I need to uniquely define the point p' in this plane is some reference direction to measure the angle at which the point p' lies on the circle. Call this direction \hat{\omega}(t).
Here's the kicker though - I want \hat{\omega}(t) to be defined in such a way that when the curve is a straight line, and in the z-direction, that the coordinate system becomes plain old cylindiral coordinates, and \hat{\omega}(t) = \hat{x} . However, when the curve is not a straight line, then \hat{\omega} should rotate with the curve in such a way that it is always orthogonal to \hat{s}(t), and locally if \hat{s}(t) \simeq \hat{z} then \hat{\omega}(t) \simeq \hat{x}.
My question is how to define \hat{\omega}(t) in such a way that satisfies these constraints.
Does this make sense? I hope at least the idea of what I want to do is clear. I further hope this problem is not ill-posed. Any help would be appreciated and please let me know if something is unclear.