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

Assignment Help:

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


Related Discussions:- Rotation about the origin - 2-d and 3-d transformations

perform a perspective projection on the plane, Consider at line segment AB...

Consider at line segment AB in the Figure k, parallel to the z-axis along with end points A (3, 2, 4) and B (3, 2, 8). Perform a perspective projection on the z = 0 plane from the

Will the shape of the resulting b-spline curve change, If the spacing betwe...

If the spacing between the knot sequence is uniformly doubled, will the shape of the resulting B-spline curve change?  Justify your answer.

3d studio max -softwares for computer animation, 3DStudio Max -Softwares fo...

3DStudio Max -Softwares for computer animation The successor to 3-DStudio 3.0, 3-DStudio Max runs in WindowsNT. This is completely object-oriented, featuring new enhancements a

Area-subdivision method, Area-Subdivision method This method is a ty...

Area-Subdivision method This method is a type of an image-space method although uses object-space operations re-ordering or sorting of surfaces as per to depth. Area sub-div

Concept for eliminating hidden lines, Concept for Eliminating Hidden Lines,...

Concept for Eliminating Hidden Lines, Surfaces or Edges To exemplify the concept for eliminating hidden-lines, surfaces or edges, see a classic wire frame model of a cube as i

Operation reseach, #question. Steps involved in the solution of operation r...

#question. Steps involved in the solution of operation research problem

What happens while two polygons have similar z value , What happens while t...

What happens while two polygons have similar z value and the z-buffer algorithm is utilized? Solution : z-buffer algorithms, varies colors at a pixel if z(x,y)

What is the use of projection reference point, What is the use of Projectio...

What is the use of Projection reference point?  In Perspective projection, the object positions are transformed to the view plane with these converged projection line and the p

Animation and games - cad and cam, Animation and Games - CAD and CAM I...

Animation and Games - CAD and CAM In our childhood, we all have seen that the flip books of cricketers that came free with some soft drink and where some pictures of similar p

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd