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







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
Web Design and Editing To edit and make a website, the big three softwares are use: 1)   DreamWeaver (MacroMedia) 2)   Frontpage (MicroSoft) 3)   Go Live (Adobe) 4)

Bitmap Graphics: The information below illustrates bitmap data. Bitmap images are a set of bits that form an image. The image comprises a matrix of individual dots or pixels wh

Progressive Scan: Progressive or non-interlaced scanning is a process which displays, transmits or stores moving images wherein, the lines of all frame are drawn in order. It is i

What is  Raster Scan Display A raster scan display device using CRT on the other hand directs the electron beam across the screen, one row at a time from top to bottom. In a ra

Question 1: (a) Explain in detail what you understand by the term image compositing and where it is more often used? (b) You are given 3-5 images to make a photo montage/ima

Question (a)  List any four audio file formats you know. (b)  UTMDigitlab ltd, specialized in digitizing sound, converts an audio stream of the latest album of Shakira into

Sound and Audio: Sound is a mechanical energy disturbance which propagates by matter as a wave. Sound is characterized through the various properties that are: frequency, per

1. Implement proper exception handling mechanism for this application. 2. Display all data a. Search specific data (depending of the user selection, your application should e

LCD - Liquid Crystal Display A liquid crystal display consists of two glass plates each containing a light polarizer. One glass plate contains vertical polarizer and the ot

self test exercise 17 asked you to overload the operator >> and the operator Overload biinary operator + to add pairs according to the rule (a, b) + (c, d) = (a + c, b, + d) overl