There was something I was wondering about that I am not exactly sure how to do.

Suppose I take a normal circle on a plane and represent it by the parametric equations x=Cos[t] and y=Sin[t]. Now suppose I put it into 3d dimensional space and rotate it around anyway i want at different angles. What is an easy way of doing this?

I know there is rotation matrices that you can use, and I have used those before, but they seem to get really icky. The thing I was wondering about is if there is some way you can can do it with a vector. Like have a vector that points in the direction of the rotation you want and then have some way of transforming the initial equation so that its rotated in line with the vector or perhaps perpendicular to the vector.

So how do you do this?

Would it be better to use a position vector function r=fi+gj+hk? Where f,g,h are functions of t and i,j, and k are unit vectors
Then the initial equation looks like r=Cos[t]i+Sin[t]j and you can just rotate it somehow.

I think you'll have to go for the icky way. The best idea I can think of immediately (after not too much thought) is to change your reference base to one which is simplier. This will involve the very same icky rotation matrices however.

But once you get used to the icky matrices, they're not that icky anymore. Just try performing a rotation about two axis to see what I mean, it is just the product of two rotation matrices, which all have the familiar cos/sin/0/-sin/cos/0/0/0/1 structure.

Bye!

Crisp

Originally posted by cephas1012
Suppose I take a normal circle on a plane and represent it by the parametric equations x=Cos[t] and y=Sin[t]. Now suppose I put it into 3d dimensional space and rotate it around anyway i want at different angles. What is an easy way of doing this?


You are about to say it yourself...


I know there is rotation matrices that you can use, and I have used those before, but they seem to get really icky. The thing I was wondering about is if there is some way you can can do it with a vector.


The rotation matrices are the things you use with a vector. Take your initial position vector x. The one you specified would be written in column form as:


[cos(t)]
x= [sin(t)]
[ 0 ]


To transform to a rotated vector x', use the rotation matrices as appropriate. For reference, the rotation matrices are listed as equations 5, 6, and 7 on the following page:


Just remember: the order of the rotation matrices is from right to left, and they do not commute. For instance:

x’=R_x(a)R_y(b)x

means that first you do a rotation of b radians about the y axis, and then you do a rotation of a radians about the x axis. That is not the same as:

x’’=R_y(b)R_x(a)x,

which reverses the order of rotation.

Use two parameters.