- #1
Spinnor
Gold Member
- 2,234
- 433
Edit, the vector that rotates below might not rotate at all.
Please forgive any mistaken statements or sloppiness on my part below.
I think that by some measure a helicoid can be considered a smooth curved 2 dimensional surface except for a line of points?
Consider not the helicoid above but one that has an arbitrary orientation, rotation, and position in space. A pair of vectors and a line segment perpendicular to one of the vectors can encode all the information of such a helicoid with arbitrary position, orientation, and rotation? That would be 7 numbers? One vector represents the distance from some origin to a point on the axis of the helicoid, the other vector represents three pieces of information, the orientation of the axis of the helicoid, the pitch of the helicoid, and the handedness of the helicoid. The perpendicular line segment represents the rotational orientation of the helicoid.
So we have a smooth surface represented by a set of numbers. What I would like to do is take an object that could be represented by a set of numbers, which is easy to visualize, and associate if possible one or more smooth curved manifolds with that object, ideally this manifold's shape changes with time.
Consider the following simple object in 4D Minkowski space-time, a point which moves through space in a straight line at the speed of light. To our moving point attach a vector that moves along with our point and that rotates in the plane perpendicular to the line that is the path of our point through space. We need a number for the rate at which the vector rotates and we need to specify which way the vector rotates. This is the object that I would like to know what smooth curved manifolds can be associated with it, if any. There may be some small set of points where our curved manifold is not defined.
I may have not defined my problem as precisely as I should and will edit if any errors occur to me but I hope you got the idea.
Thanks for any help or suggestions.
Please forgive any mistaken statements or sloppiness on my part below.
I think that by some measure a helicoid can be considered a smooth curved 2 dimensional surface except for a line of points?
Consider not the helicoid above but one that has an arbitrary orientation, rotation, and position in space. A pair of vectors and a line segment perpendicular to one of the vectors can encode all the information of such a helicoid with arbitrary position, orientation, and rotation? That would be 7 numbers? One vector represents the distance from some origin to a point on the axis of the helicoid, the other vector represents three pieces of information, the orientation of the axis of the helicoid, the pitch of the helicoid, and the handedness of the helicoid. The perpendicular line segment represents the rotational orientation of the helicoid.
So we have a smooth surface represented by a set of numbers. What I would like to do is take an object that could be represented by a set of numbers, which is easy to visualize, and associate if possible one or more smooth curved manifolds with that object, ideally this manifold's shape changes with time.
Consider the following simple object in 4D Minkowski space-time, a point which moves through space in a straight line at the speed of light. To our moving point attach a vector that moves along with our point and that rotates in the plane perpendicular to the line that is the path of our point through space. We need a number for the rate at which the vector rotates and we need to specify which way the vector rotates. This is the object that I would like to know what smooth curved manifolds can be associated with it, if any. There may be some small set of points where our curved manifold is not defined.
I may have not defined my problem as precisely as I should and will edit if any errors occur to me but I hope you got the idea.
Thanks for any help or suggestions.
Last edited: