Cylindrical coordinates allow points to be specified using two linear distances and one angle. These three coordinates are shown on the diagram as:
Cylindrical Coordinates (r − θ − z) Polar coordinates can be extended to three dimensions in a very straightforward manner. We simply add the z coordinate, which is then treated in a cartesian like manner. Every point in space is determined by the r and θ coordinates of its projection in the xy plane, and its z coordinate. 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the position of each point on the plane is determined by an angle and a distance. The distance is usually denoted rand the angle is usually denoted.
Cylinder aligned with the z axisCylindrical coordinates to Cartesian coordinates
Cartesian coordinates to Cylindrical coordinates
Cylinder aligned with the arbitrary axisCylindrical coordinates to Cartesian coordinates
We want to rotate the above so that the h axis is aligned with the arbitrary axis (Ax, Ay, Az) in other words we want to lookat the point (Ax, Ay, Az) see lookat
Multiplying gives:
Cartesian coordinates to Cylindrical coordinates
In order to do this we need to invert the above matrix, since this is orthogonal we can invert by transposing as follows:
Multiplying gives,
So in terms of θ, r , h
Using Tensors
This is more advanced stuff and only really needed if you need to do physics or advanced geometry in curvilinear coordinates.
On the curvilinear coordinates page we saw that the expression of coordinates as a linear equation:
ei Ai = e1 A1 + e2 A2 + e3 A3
can be modified for curvilinear coordinates where either the basis or the components depend on the location:
e(x,y,z)i Ai = e(x,y,z)1 A1 + e(x,y,z)2 A2 + e(x,y,z)3 A3…
or
ei A(x,y,z)i = e1 A(x,y,z)1 + e2 A(x,y,z)2 + e3 A(x,y,z)3…
where:
We have already looked at the situation where the terms are expressed as a function of a global linear coordinate system. Now lets look at the situation where the linear basis is a function of position. There are two ways to do this:
Covariant Axis
where:
where:
Jacobean
Inverse Jacobean
Contravariant Axis
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