📄 homogenous coordinates.htm
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3c.org/TR/1999/REC-html401-19991224/loose.dtd">
<!-- saved from url=(0049)http://bishopw.loni.ucla.edu/AIR5/homogenous.html -->
<HTML><HEAD><TITLE>homogenous coordinates</TITLE>
<META http-equiv=Content-Type content="text/html; charset=windows-1252">
<META content="MSHTML 6.00.2800.1106" name=GENERATOR></HEAD>
<BODY bgColor=#ffffff>
<CENTER>
<H3>Homogenous Coordinates</H3></CENTER>
<UL>
<LI><A href="http://bishopw.loni.ucla.edu/AIR5/homogenous.html#vectors">4x1
homogenous coordinate vectors</A>
<LI><A href="http://bishopw.loni.ucla.edu/AIR5/homogenous.html#matrices">4x4
homogenous coordinate matices</A>
<LI><A
href="http://bishopw.loni.ucla.edu/AIR5/homogenous.html#translations">translations</A>
<LI><A
href="http://bishopw.loni.ucla.edu/AIR5/homogenous.html#rotations">rotations</A>
<LI><A
href="http://bishopw.loni.ucla.edu/AIR5/homogenous.html#rescaling">rescaling</A>
<LI><A
href="http://bishopw.loni.ucla.edu/AIR5/homogenous.html#perspective">perspective</A>
</LI></UL>
<P>Homogenous coordinates utilize a mathematical trick to embed
three-dimensional coordinates and transformations into a four-dimensional matrix
format. As a result, inversions or combinations of linear transformations are
simplified to inversion or multiplication of the corresponding matrices.
Homogenous coordinates also make it possible to define perspective
transformations.<BR></P>
<H4><A name=vectors></A>4x1 Homogenous Coordinate Vectors</H4>
<P>Instead of representing each point (x,y,z) in three-dimensional space with a
single three-dimensional vector:<BR><IMG height=49 alt=""
src="homogenous coordinates_files/vector3by1.gif" width=30
align=top><BR><BR>homogenous coordinates allow each point (x,y,z) to be
represented by any of an infinite number of four dimensional vectors:<BR><IMG
height=61 alt="" src="homogenous coordinates_files/vector4by1.gif" width=50
align=top><BR><BR>The three-dimensional vector corresponding to any
four-dimensional vector can be computed by dividing the first three elements by
the fourth, and a four-dimensional vector corresponding to any three-dimensional
vector can be created by simply adding a fourth element and setting it equal to
one.<BR><BR>Many textbooks define homogenous coordinates in such a way that
points are represented by 1x4 vectors:</P><PRE>[T*x T*y T*z T]</PRE>
<P>instead of 4x1 vectors. This definition is not used in the AIR package and
results in different 4x4 homogenous coordinate transformation matrices than
those described below.</P>
<HR>
<H4><A name=matrices></A>4x4 Homogenous Coordinate Transformation Matrices</H4>
<P>Homogenous coordinate transformation matrices operate on four-dimensional <A
href="http://bishopw.loni.ucla.edu/AIR5/homogenous.html#vectors">homogenous
coordinate vector</A> representations of traditional three-dimensional
coordinate locations. Any three-dimensional linear transformation (rotation,
translation, skew, perspective distortion) can be represented by a 4x4
homogenous coordinate transformation matrix. In fact, because of the redundant
representation of three space in a homogenous coordinate system, an infinite
number of different 4x4 homogenous coordinate transformation matrices are
available to perform any given linear transformation. This redundancy can be
eliminated to provide a unique representation by dividing all elements of a 4x4
homogenous transformation matrix by the last element (which will become equal to
one). This means that a 4x4 homogenous transformation matrix can incorporate as
many as 15 independent parameters. The generic format representation of a
homogenous transformation equation for mapping the three dimensional coordinate
(x,y,z) to the three-dimensional coordinate (x',y',z') is:<BR><IMG height=61
alt="" src="homogenous coordinates_files/homogenous.gif" width=260
align=top><BR></P>
<P>If any two matrices or vectors of this equation are known, the third matrix
(or vector) can be computed and then the redundant T element in the solution can
be eliminated by dividing all elements of the matrix by the last element.</P>
<P>Various <A
href="http://bishopw.loni.ucla.edu/AIR5/models.html">transformation models</A>
can be used to constrain the form of the matrix to tranformations with fewer
degrees of freedom.</P>
<P>In many textbooks, you will find homogenous tranformation matrices defined
such that 1x4 homogenous coordinate vectors are placed to the left of the 4x4
homogenous coordinate transformation matrix and multiplied. This format is not
supported in the AIR package and the difference accounts for the fact that
translations are represented in the fourth column and perspective distortions in
the fourth row of AIR homologous transformation matrices rather than vice
versa.</P>
<HR>
<H4><A name=translations></A>Translations</H4>
<P>Translations can be represented by the 4x4 homogenous coordinate
transformation matrix:<BR><IMG height=136 alt=""
src="homogenous coordinates_files/rigid_translations.gif" width=270
align=top></P>
<HR>
<H4><A name=rotations></A>Rotations</H4>
<P>A series of rotations (in the order [roll matrix]*[pitch matrix]*[yaw
matrix]) can be represented by the 4x4 homogenous coordinate transformation
matrix:<BR><IMG height=148 alt=""
src="homogenous coordinates_files/rigid_rotations.gif" width=475 align=top></P>
<HR>
<H4><A name=rescaling></A>Rescaling</H4>
<P>Rescaling along the major axes can be represented by the 4x4 homogenous
coordinate transformation matrix:<BR><IMG height=136 alt=""
src="homogenous coordinates_files/talamatrix.gif" width=350 align=top></P>
<HR>
<H4><A name=perspective></A>Perspective</H4>
<P>Perspective distortion is achieved by applying the 4x4 homogenous coordinate
transformation matrix:<BR><IMG height=136 alt=""
src="homogenous coordinates_files/perspective.gif" width=350 align=top></P>
<HR>
<A href="http://bishopw.loni.ucla.edu/AIR5/index.html"><IMG height=50 alt=""
src="homogenous coordinates_files/AIRlogo.gif" width=52
align=bottom></A>Modified: December 10, 2001<BR><BR>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -