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)



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

Related Discussions:- Rotation about the origin - 2-d and 3-d transformations, Assignment Help, Ask Question on Rotation about the origin - 2-d and 3-d transformations, Get Answer, Expert's Help, Rotation about the origin - 2-d and 3-d transformations Discussions

Write discussion on Rotation about the origin - 2-d and 3-d transformations
Your posts are moderated
Related Questions
Line Drawing - Points and lines Line drawing is accomplished through computing the intermediate point coordinates along the line path between two given end points. Since, scre

pagemaker is a image editor

What is the difference between odd-even rule and non-zero winding number rule to identify interior regions of an object? Develop an algorithm for a recursive method for filling a 4

Boundary Fill Algorithm Boundary fill algorithm is suitable when the boundary has single color while flood fill algorithm is more suitable for filling regions which are defined

Q. Explain about Unified Memory Architecture? UMA signifies Unified Memory Architecture. It is an architecture that reduces the cost of PC construction.  In this a part of main

Orthographic and Oblique Projection - Viewing Transformation Orthographic projection is the easiest form of parallel projection that is commonly utilized for engineering drawi

The hand-held pointer and tablet in the form of a stylus i.e. pen or puck can function one or more of these three functions: (i)  For choosing positions on a drawing or on a men

What is an axonometric orthographic projection and cabinet projection? The orthographic projection can show more than one face of an object.  Such an orthographic projection i

What are the advantages of rendering by patch splitting?  i. It is fast-especially on workstations with a hardware polygon-rendering pipeline.  ii. Its speed can be varied b

Interlaced GIF: The conventional which is non-interlaced GIF graphic downloads one line of pixels at one time from top to bottom and browsers display all lines of the image as it