Rotation Matrix

Discussion in 'Math' started by Key, Apr 13, 2014.

1. Key Thread Starter New Member

Jun 21, 2013
14
2

Am I correct to assume in rotation matrix we compare the rotation of coordinate system B to A ? For example in row 1 column 2, we compare the coordinate B X-axis to coordinate system A Y-axis ?
And the formula on the right is when we rotate the X and Y axis of coordinate system B counterclockwise and we leave axis Z untouched ?
That is where we get our corner alpha ? and why zaxis value is 1 for parallel and the others are 0 for being crossed ?

Could someone poke me in the right direction if I am wrong here ?

And excuse the MSPaint, didnt have anything better at hand.

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2. Key Thread Starter New Member

Jun 21, 2013
14
2
if I happen to be right, could someone please explain to me how the sines cosines are gotten, in this more simpler 2 axis coordinate system.

Thank you

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3. MrChips Moderator

Oct 2, 2009
18,168
5,706
Go back to basic trigonometry.

Suppose you have a point (x,y) in X-Y Cartesian coordinates.
What is the position of the point in polar coordinates, (r,θ)?

x = r cos θ
y = r sin θ

Now rotate the X,Y coordinate system by α-degrees in a counter clockwise direction.

The position (x,y) in the new system is r, (θ-α)
The new (x,y) coordinates are
x = r cos(θ - α)
y = r sin(θ - α)

Expand these two equations:

x = r ( cos θ cos α + sin θ sin α )
y = r ( sin θ cos α - cos θ sin α )

See if you can come up with the rotational matrix for 2-D rotation.

4. Key Thread Starter New Member

Jun 21, 2013
14
2
why is it θ-α ? why isnt it
x=r*cos(θ-a)
y=r*sin(θ+a)