Rolling ball rotation

Physical interpretation of rolling ball rotation

The rolling ball rotation ( "rotation of the rolling ball" even rollerball rotation ) is a human-computer interaction technology , which allows using a standard pointing device to perform three-dimensional rotations of an object in all directions. It is built into many 3D modeling and CAD tools .

During rolling ball rotation, a mouse button is pressed at any point, the mouse pointer is moved and the mouse button is released again. The rotation determined by this is that of a ball that rolls on a flat surface without sliding and that is moved from above with the flat of the hand in accordance with the movement of the mouse pointer.

The rolling ball rotation is context-free, which means that the rotation is completely defined by the displacement vector of the mouse pointer without earlier positions being included in the rotation.

implementation

The following rotation matrix expresses the rotation depending on the movement of the mouse pointer:

${\displaystyle {\begin{pmatrix}\cos \theta +(dy/dr)^{2}(1-\cos \theta )&-(dx/dr)(dy/dr)(1-\cos \theta )&(dx/dr)\sin \theta \\\\-(dx/dr)(dy/dr)(1-\cos \theta )&\cos \theta +(dx/dr)^{2}(1-\cos \theta )&(dy/dr)\sin \theta \\\\-(dx/dr)\sin \theta &-(dy/dr)\sin \theta &\cos \theta \end{pmatrix}}}$

${\displaystyle dx}$and indicate the movement of the mouse pointer in the X or Y direction in pixels . The other values ​​are defined as follows: ${\displaystyle dy}$

${\displaystyle dr={\sqrt {dx^{2}+dy^{2}}}}$

${\displaystyle \cos \theta ={\frac {R}{\sqrt {R^{2}+dr^{2}}}}}$

${\displaystyle \sin \theta ={\frac {dr}{\sqrt {R^{2}+dr^{2}}}}}$

${\displaystyle R}$is a freely selectable parameter that specifies the radius of the virtual rolling ball and thus determines the sensitivity of the rotation. A value on the order of 100 pixels is common.

The matrix is ​​defined so that the X, Y and Z axes result in a right-handed coordinate system . Your application assumes that the center of rotation is at the origin. If a rotation around another point is desired, a corresponding shift must be applied before and after . In addition, the rotation must be done all at once; successive rotations around the different axes do not give the same result.

Other methods of interaction

Further, context-sensitive interaction methods for controlling the rotation are the arcball method from Shoemake or Chens Virtual Sphere .

literature

• Andrew J. Hanson: The rolling ball. In David Kirk (ed.): Graphics Gems III , pp. 51-60, Academic Press, San Diego 1992, ISBN 0-12-409671-9 ( Online, Gzipped PostScript )
• Andrew J. Hanson: Rotations for N-Dimensional Graphics. In Alan W. Paeth (ed.): Graphics Gems V , pp. 55–64, Morgan Kaufmann, San Diego 1995, ISBN 0-12-543457-X ( Online, PDF )

Other methods:

1. ↑ Ken Shoemake: Arcball rotation control. In Paul Heckbert (ed.): Graphics Gems IV. Pp. 172-192. Academic Press, Boston 1994, ISBN 0-12-336155-9
2. ↑ Michael Chen et al: A Study in Interactive 3-D Rotation Using 2-D Control Devices. In SIGGRAPH 1988 Proceedings. Pp. 121-130. ACM, New York 1988, ISBN 0-89791-275-6 ( Online, PDF )