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

Write short notes on rendering bi-cubic surface, Write short notes on rende...

Write short notes on rendering bi-cubic surface patches of constant u and v method?  The simple way is to draw the iso-parmetric lines of the surface. Discrete approximations t

Merits -phong shading or normal vector interpolation shading, Merits -Phong...

Merits -Phong shading or Normal Vector Interpolation Shading Hence by finding intensities at various points across the edge we determine the intensity that is varying across t

Plane equation - curves and surfaces, Plane Equation - Curves and Surfaces ...

Plane Equation - Curves and Surfaces Plane is a polygonal surface that bisects its environment in two halves. One is termed to as forward and another as backward half of som

Relationships between scaling and wavelet function spaces, QUESTION (a)...

QUESTION (a) Median filters do not cater for a dynamic range of pixels in a given area, S. Thus to ensure that no loss of image details occur in S, adaptive median filters coul

Video games, why do video game characters look better today?

why do video game characters look better today?

Transformation for 3-d reflection, Transformation for 3-D Reflection F...

Transformation for 3-D Reflection For 3-Dimensions reflections, we should to know the reference plane, which is a plane about that reflection is to be considered. Remember tha

Vanishing point - viewing transformations, Vanishing Point - Viewing Transf...

Vanishing Point - Viewing Transformations This point is that point at those parallel lines shows to converge and vanish. A practical illustration is a long straight railroad

Explain what you understand by corporate style guide, Question 1: (a) ...

Question 1: (a) Explain the term ‘logo' with the use of an example. (b) Explain in detail what three basic questions you need to ask yourself before creating a logo. (c) You

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