Rotation about the origin - 2-d and 3-d transformations, Computer Graphics

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.

1337_Rotation about the origin - 2-d and 3-d transformations.png

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,

1628_Rotation about the origin - 2-d and 3-d transformations 1.png

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:

346_Rotation about the origin - 2-d and 3-d transformations 2.png

There is P'=P.Rq                                               ---------(5)

Here P'and P represents object points in 2-Dimentional Euclidean system and Rq is transformation matrix for anti-clockwise Rotation.

In terms of Homogeneous Coordinates system, equation (5) becomes as

2409_Rotation about the origin - 2-d and 3-d transformations 3.png

There is P'h=Ph.Rq,                                                     ---------(7)

Here P'h and Ph   represent object points, after and before needed transformation, in Homogeneous Coordinates and Rq 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 Rq.

Keep in mind that for clockwise rotation we have to put q = -q, hence the rotation matrix Rq , in Homogeneous Coordinates system, becomes:

1007_Rotation about the origin - 2-d and 3-d transformations 4.png

Posted Date: 4/3/2013 5:18:10 AM | Location : United States







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