Rotation about the origin - 2-d and 3-d transformations
Specified a 2-D point P(x,y), which we need to rotate, along with respect to the origin O. The vector OP has a length 'r' and making a +ive or anticlockwise angle φ with respect to x-axis.
Suppose P' (x'y') be the outcome of rotation of point P by an angle θ regarding the origin that is demonstrated in Figure 3.
P(x,y) = P(r.cos φ,r.sin φ)
P'(x',y')=P[r.cos(φ+ θ),rsin(φ+ θ)]
The coordinates of P' are as:
x'=r.cos(θ+ φ)=r(cos θ cos φ -sin θ sin φ)
=x.cos θ -y.sin θ (where x=rcosφ and y=rsinφ)
As like;
y'= rsin(θ+ φ)=r(sinθ cosφ + cosθ.sinφ)
=xsinθ+ycosθ
Hence,
Hence, we have acquired the new coordinate of point P after the rotation. Within matrix form, the transformation relation among P' and P is specified by:
There is P'=P.R_{q} ---------(5)
Here P'and P represents object points in 2-Dimentional Euclidean system and R_{q} is transformation matrix for anti-clockwise Rotation.
In terms of Homogeneous Coordinates system, equation (5) becomes as
There is P'h=Ph.R_{q}, ---------(7)
Here P'h and Ph represent object points, after and before needed transformation, in Homogeneous Coordinates and R_{q} is termed as homogeneous transformation matrix for anticlockwise or =ive Rotation. Hence, P'h, the new coordinates of a transformed object, can be determined by multiplying previous object coordinate matrix, Ph, along with the transformation matrix for Rotation R_{q}.
Keep in mind that for clockwise rotation we have to put q = -q, hence the rotation matrix R_{q} , in Homogeneous Coordinates system, becomes: