Visualizing Quaternions: Quaternion Maps

Brief Introduction Keyboard Shortcuts Mouse Interface

1. Quaternions Describe Orientations in 3D

In short: Quaternion Maps are the visualization of the space of orientations in three dimensional space.

An orientation is characterized by a direction vector (n) and an angle-of-rotation (θ) around this direction. Typically, the term 'orientation' is substituted by the term 'orientation-frame',  wherein, the frame is a triad of three mutually-perpendicular vectors (nx, ny, nz), with one axis aligned along the direction vector.

Quaternion maps are essentially visualizations of one or more quaternions. Quaternions are four-dimensional vectors, and unit-quaternions represent the orientation-frames as mentioned above. This mapping provides interesting and surprisingly crisp explanations for some intriguing phenomena as described in the book Visualizing Quaternions

2. Quaternion Maps

Quaternion maps are obtained by connecting a set of quaternions to form a smooth curve. Quaternion maps can be open or closed quaternion-curves. Since the configuration of the quaternion map signifies successive orientations of a frame in 3D space, a closed quaternion-curve indicates that the starting and end frames have the same orientation in 3D. An open quaternion-curve may or may not mean imply the same. This is because of the mapping between orientations in 3D and quaternions. The mapping is a 1:2-mapping, which implies that each 3D orientation is represented by a pair of quaternions (with opposite sign). This property is referred to as the 'double cover'. If the end quaternions in an open quaternion-curve form such a pair, the orientations in 3D will be identical.

The application demonstrates orientations of frames that sweep closed, knotted curves such as the torus knots. A highly interesting problem in itself is how the frames are assigned to a given curve. Three framing methods are exposed in this application, viz. the Frenet-Serret, Parallel Transport and Geodesic Reference.

Please visit the book website for a detailed description: www.visualizingquaternions.com.

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Keyboard Shortcuts...

Global options
 
operation key
Quit ESCAPE
Help "h"
Reset View "r"
 
toggle feature key
Axes "a"
Curve "c"
Curve-tube Edges "c"
Curve-tube Faces "f"
Frames "Shift+f"
Gap "g"
Overall help "h"
Screen-help text "Shift+h"
Labels "l"
Palette "p"
Quaternion Map Color options
operation key
Reflect Frenet-Serret map "p"
Quaternion Rings "Shift+r"
Quaternion-map Mesh "m"
Quaternion-map Surface "Shift+m"
Unit sphere "s"
   
Quaternion Map options
 
operation key
Show component XYZ "w"
Show component WYZ "x"
Show component WXZ "y"
Show component WXY "z"

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Mouse Interface...

Operation Button + Action
Rotate Model Left + drag
Scale Model Right + drag vertically

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Graphics and Visualization Lab,
Indiana University Bloomington,
Computer Science Department
Feb 2006