Relation between 2-d euclidean system and homogeneous system, Computer Graphics

Assignment Help:

Relation between 2-D Euclidean system and Homogeneous coordinate system

Suppose that P(x,y) be any point in 2-D Euclidean system. In HCS, we add a third coordinate to the point. In place of (x,y), all points are represented via a triple (x,y,H) so H≠0; along with the condition as (x1,y1,H1)=(x2,y2,H2) ↔ x1/H1 = x2/H2 ; y1/H1 = y2/H2. In two dimensions the value of H is generally remained at 1 for simplicity. If we take H=0 now, then this presents point at infinity, which is generation of horizons.

The subsequent table demonstrates an association between 2-D Euclidean (Cartesian coordinate) system and Homogeneous coordinate system.

2-D Euclidian System                                        Homogeneous Coordinate System (HCS)

Any point (x,y)                         →                                             (x,y,1)

If (x,y,H) be any point in HCS(such that H≠0);

                                                                                    then (x,y,H)=(x/H,y/H,1), which is

(x/H,y/H)                        ←                                                                     (x,y,H)

Any one point (x,y) → (x+tx,y+ty) in 2-D Euclidian system. By using Homogeneous coordinate system, this translation transformation can be presented as (x,y,1) → (x+tx,y+ty,1). In two dimensions the value of H is generally maintained at 1 for simplicity. Here, we are capable to represent this translation transformation in matrix form as:

242_Relation between 2-D Euclidean (Cartesian) system and Homogeneous coordinate system 2.png

 (x',y',1)=(x,y,1)

P'h=Ph.Tv    

Here P'h and Ph   demonstrate here object points in Homogeneous Coordinates and Tv is termed as homogeneous transformation matrix for translation. Consequently, P'h, the new coordinates of a transformed object, can be determined by multiplying earlier object coordinate matrix, Ph, along with the transformation matrix for translation Tv.

The benefit of initiating the matrix form of translation is to simplify the operations on complicated objects which are, we can now build complicated transformations by multiplying the basic matrix transformations. Such process is termed as concatenation of matrices and the resulting matrix is frequently referred as the composite transformation matrix.


Related Discussions:- Relation between 2-d euclidean system and homogeneous system

Transformation for 3-d scaling, As we already seen that the scaling proces...

As we already seen that the scaling process is mainly utilized to change the size of an object. The scale factors find out whether the scaling is a magnification as s>1 or a red

Determine the transformation matrix for the reflection, Determine the trans...

Determine the transformation matrix for the reflection about the line y = x. Solution: The transformation for mirror reflection regarding to the line y = x, comprises the subs

Types of bitmap images, Types of Bitmap Images Bitmap images can includ...

Types of Bitmap Images Bitmap images can include any number of colours but we distinguish among four main categories as: 1)      Line-art: These are images that include

Raster & Vector display, what is refresh buffer/ identify the content and o...

what is refresh buffer/ identify the content and organisation of the refresh buffer for the case of raster display and vector display.

Medicine-applications for computer animation, Medicine: this is very tough...

Medicine: this is very tough for a doctor to get inside a living human body and to observe what is occurrence. Computer animation once again comes in very helpful. Every particula

Disadvantage of the raster scan display device, Disadvantage of the Raster ...

Disadvantage of the Raster Scan Display Device The major disadvantage of the raster scan is the jagged nature of the lines, happening from the information that the pixels are

Overstriking, why overstriking is harmful.justify

why overstriking is harmful.justify

Reflection, determine the tranformation matrix for reflection,computer grap...

determine the tranformation matrix for reflection,computer graphics

Trivial rejection case of cohen sutherland line clippings, Trivial Rejectio...

Trivial Rejection Case of cohen sutherland line clippings Case: it  is Trivial Rejection Case; if the logical intersection (AND) of the bit codes of the end points P, Q of

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