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左右手坐标系的转换.zip （1个子文件）

Conversion of Left-Handed Coordinates to Right-Handed Coordinates.pdf 198KB

Conversion of Left-Handed Coordinates to

Right-Handed Coordinates

David Eberly

Geometric Tools, LLC

http://www.geometrictools.com/

Created: December 20, 2003

Last Modiﬁed: March 1, 2008

Contents

1 Right-Handed Coordinate Systems 2

2 Left-Handed Coordinate Systems 3

2.1 Heading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Order of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Conversion from Left-Handedness to Right-Handedness 7

3.1 Conversion of Points and Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Conversion of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Conversion of Aﬃne Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Conversion of Cameras and Lights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

This document describes the process by which you convert data in a left-handed coordinate system to

that in a right-handed coordinate system. The data includes vertex locations, translational and rotational

transformations, camera location and orientation, and light location and orientation.

1 Right-Handed Coordinate Systems

A right-handed coordinate system is described as +X

in the left direction, +Y

in the up direction, and +Z

in the view direction. Figure 1.1 shows a typical representation.

Figure 1.1 A right-handed coordinate system. The coordinate names are subscripted with r to

stress the fact these are for the right-handed system. The +Z

axis is intend to have direction into

the plane of the page.

The origin of the system is (0, 0, 0). A point (x, y, z) is located relative to the origin by moving x units to

the left (when x > 0) or to the right (when x < 0), y units up (when y > 0) or down (when y < 0), and z

units away (when z > 0) or towards (when z < 0).

The placement of the axes in Figure 1.1 is based on wanting the Y

-axis to point upward and the Z

-axis to

point into the plane of the page. To make sure the coordinate system is right-handed, the only choice for

the X

-axis is to point to the left. The direction of X

is actually not relevant on its own. You may choose

to point to the right, Y

to point up, and Z

to point out of the plane of the page. Or choose X

to point

to the right, Y

to point down, and Z

to point into the plane of the page. For that matter, you may rotate

any of these conﬁgurations to any other conﬁguration you like, but you still have a right-handed system.

Choose any of the coordinate directions you wish to be the up vector, choose any remaining direction to be

the view direction. You have only one choice for the third vector. If this is the world coordinate system you

are choosing for an application, you need only position and orient the camera in such a way that your view

of the world is what you expect! You can do this regardless of your convention for what is the up direction

and what is the view direction.

In this document, the rotation conventions for right-handed coordinates are that positive angles are associated

with counterclockwise rotations when viewing the system along the negative direction of the axis of rotation.

The rotation about the x-axis is

(θ) =







1 0 0

0 cos(θ) − sin(θ)

0 sin(θ) cos(θ)







(1)

The rotation about the y-axis is

(θ) =







cos(θ) 0 sin(θ)

0 1 0

− sin(θ) 0 cos(θ)







(2)

The rotation about the z-axis is

(θ) =







cos(θ) − sin(θ) 0

sin(θ) cos(θ) 0

0 0 1







(3)

2 Left-Handed Coordinate Systems

A left-handed coordinate system is described as +X

in the right direction, +Y

in the up direction, and +Z

in the view direction. The prototypical modeling package that uses left-handed coordinates for all objects is

LightWave. The discussion here is eﬀectively a description of LightWave’s conventions. Figure 2.1 shows a

typical representation.

Figure 2.1 A left-handed coordinate system. The coordinate names are subscripted with ` to stress

the fact these are for the left-handed system. The +Z

axis is intend to have direction into the plane

of the page.

The origin of the system is (0, 0, 0). A point (x, y, z) is located relative to the origin by moving x units to

the right (when x > 0) or to the left (when x < 0), y units up (when y > 0) or down (when y < 0), and z

units away (when z > 0) or towards (when z < 0).

Figure 2.1 also shows how coordinate axis rotations are deﬁned in this left-handed system. The direction of

the X

axis is (1, 0, 0), the direction of the Y

axis is (0, 1, 0), and the direction of the Z

axis is (0, 0, 1). We

may conveniently store the direction vectors as the columns of the identity matrix,

I =







1 0 0

0 1 0

0 0 1







Notice that the ﬁrst column is the right direction, the second column is the up direction, and the third

column is the view direction.

2.1 Heading

A rotation about the Y

-axis represents heading. In Figure 2.1, the direction shown by the arrow labeled

“heading” corresponds to a positive angle. If you are looking in the −Y

direction, a positive angle is

associated with a clockwise rotation. Figure 2.2 shows a view in the −Y

direction.

Figure 2.2 A view in the −Y

direction. A positive angle for heading is associated with a clockwise

rotation in the plane of X

and Z

The heading angle is θ

and the ﬁgure illustrates a heading that is a positive angle (clockwise rotation).

The Z

axis is rotated to the Z

axis, the latter axis having direction (s

, 0, c

) where s

= sin(θ

) and

= cos(θ

). The X

axis is rotated to the X

axis, the latter axis having direction (c

, 0, −s

). The Y

axis is unchanged by the rotation, so the rotated axis Y

is the same as the original one. Just as the identity

matrix columns are the original directions in the order right, up, and view, the rotated axes may be listed

as the columns of another matrix in the same order. This matrix is the heading matrix

(θ

) =







0 s

0 1 0

−s

0 c







Observe that the ﬁrst column is the direction of the X

axis, the second column is the direction of the Y

(and Y

) axis, and the third column is the direction of the Z

axis.

The heading matrix was introduced as a convenient way to store the directions of the rotated coordinate

axes. The matrix has double duty in that it may also be used to show how points (x, y, z) in the original

coordinate system are rotated to points (x

, y

, z

). Speciﬁcally,



















0 s

0 1 0

−s

0 c

























x + s

−s

x + c







(4)

The standard rules of matrix multiplication have been applied.

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