homogeneous coordinates


What are homogeneous coordinates?

Homogeneous coordinates are coordinates used to represent the cartesian coordinate (x,y) as three - element vector.We represent each cartesian coordinate position (x,y) with the homogeneous coordinate triple (h.x, h.y, h) where h is any non - zero value.We can set h = 1.

Hence each two dimensional position (x,y) is represented as (x,y,1) in homogeneous coordinates.


Why do you need homogeneous coordinates?

Many graphics applications involve sequences of geometric transformations.The sequence of basic transformations can be expressed in matrix form as

P' = M1.P + M2

where M1 is a 2x2 matrix containing multiplicative terms and

M2 is a column matrix containing translational terms.

We can combine the multiplicative and translational terms in to a single matrix representation by expanding 2x2 matrices to 3x3 matrices.

To express any two dimensional transformation as a matrix multiplication, we represent each cartesian coordinate position with homogeneous coordinates.

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Geometric Transformations by Mrs.S.Sathya

  • Introduction

    Geometric tranformations are the trasformations used to alter or manipulate pictures in a graphics package.Changes in orientation,size … Read more »

  • Basic Transformations

    There are three basic transformations from which any other transformation can be obtained.They change the orientation,size and shape… Read more »

  • Translation

    Translation is a transformation that repositions and object along a straight line path from one coordinate location to another. … Read more »

  • Rotation

    Rotation of an object is repositioning it along a circular path in the xy plane.To rotate, we need to specify a rotation angle θ and the rotation point(xr,yr) about which the object is rotated. … Read more »

  • Scaling

    A Scaling transformation alters the size of an object.Scaling is carried out by applying scaling factors sx,sy for xand y direction … Read more »

  • Homogeneous Coordinates

    Homogeneous coordinates are coordinates used to represent the cartesian coordinate (x,y) as three - element vector.We represent each cartesian … Read more »

  • Matrix Representations

    For translation, the transformation equvations are written as … Read more »

  • Other Transformations

    Other transformations inculde Reflection and Shear. Reflections : Reflections are transformations that rotate an object 180" about a reflection axis. … Read more »