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In geometry, the attitude of an object such as a rigid body or a plane is its orientation relative to a set of reference axes.

Examples

Lattice plane

The attitude of a lattice plane is the orientation of the line normal to the plane, and is described by the plane's Miller indices. In three-space a family of planes (a series of parallel planes) can be denoted by its Miller indices (hkl), so the family of planes has an attitude common to all its constituent planes.

Rigid body

The attitude of a rigid body is its orientation as described, for example, by the orientation of a frame fixed in the body relative to a fixed reference frame. The attitude is described by attitude coordinates, and consists of at least three coordinates. One scheme for orienting a rigid body is based upon body-axes rotation; successive rotations three times about the axes of the body's fixed reference frame, thereby establishing the body's Euler angles. Another is based upon roll, pitch and yaw, although these terms also refer to incremental deviations from the nominal attitude.

Algebraic geometry in three-dimensional space

The various objects of algebraic geometry are charged with three attributes or features: attitude, orientation, and magnitude. For example, a vector has an attitude given by a straight line parallel to it, an orientation given by its sense (often indicated by an arrowhead) and a magnitude given by its length. Similarly, a bivector in three dimensions has an attitude given by the family of planes associated with it (possibly specified by the normal line common to these planes ), an orientation (sometimes denoted by a curved arrow in the plane) indicating a choice of sense of traversal of its boundary (its circulation), and a magnitude given by the area of the parallelogram defined by its two vectors.

Geological structures

Many features observed in geology are planes or lines, and their orientation is commonly referred to as their attitude. These attitudes are specified with two angles.

For a line, these angles are called the trend and the plunge. The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane.

For a plane, the two angles are called its strike (angle) and its dip (angle). A strike line is the intersection of a horizontal plane with the observed planar feature (and therefore a horizontal line), and the strike angle is the bearing of this line (that is, relative to geographic north or from magnetic north). The dip is the angle between a horizontal plane and the observed planar feature as observed in a third vertical plane perpendicular to the strike line.

Notes

1. ^ Robert J. Twiss, Eldridge M. Moores (1992). "§2.1 The orientation of structures". Structural geology (2nd ed.). Macmillan. p. 11. ISBN 0716722526. ...the attitude of a plane or a line — that is, its orientation in space — is fundamental to the description of structures.
2. ^ William Anthony Granville (1904). "§178 Normal line to a surface". Elements of the differential and integral calculus. Ginn & Company. p. 275.
3. Augustus Edward Hough Love (1892). A treatise on the mathematical theory of elasticity, Volume 1. Cambridge University Press. p. 79 ff.
4. Marcus Frederick Charles Ladd, Rex Alfred Palmer (2003). "§2.3 Families of planes and interplanar spacings". Structure determination by X-ray crystallography (4rth ed.). Springer. p. 62 ff. ISBN 0306474549.
5. Hanspeter Schaub, John L. Junkins (2003). "Rigid body kinematics". Analytical mechanics of space systems. American Institute of Aeronautics and Astronautics. p. 71. ISBN 1563475634.
6. ^ Jack B. Kuipers (2002). "Figure 4.7: Aircraft Euler angle sequence". Quaternions and rotation sequences: a primer with applications to orbits, aerospace, and virtual reality. Princeton University Press. p. 85. ISBN 0691102988.
7. Bong Wie (1998). "§5.2 Euler angles". Space vehicle dynamics and control. American Institute of Aeronautics and Astronautics. p. 310. ISBN 1563472619.
8. Lorenzo Sciavicco, Bruno Siciliano (2000). "§2.4.2 Roll-pitch-yaw angles". Modelling and control of robot manipulators (2nd ed.). Springer. p. 32. ISBN 1852332212.
9. Leo Dorst, Daniel Fontijne, Stephen Mann (2009). Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 32. ISBN 0123749425. The algebraic bivector is not specific on shape; geometrically it is an amount of oriented area in a specific plane, that's all.{{cite book}}: CS1 maint: multiple names: authors list (link)
10. B Jancewicz (1996). "Tables 28.1 & 28.2 in section 28.3: Forms and pseudoforms". In William Eric Baylis (ed.). Clifford (geometric) algebras with applications to physics, mathematics, and engineering. Springer. p. 397. ISBN 0817638687.
11. David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics (2nd ed.). Springer. p. 21. ISBN 0792353021.
12. Stephen Mark Rowland, Ernest M. Duebendorfer, Ilsa M. Schiefelbein (2007). "Attitudes of lines and planes". Structural analysis and synthesis: a laboratory course in structural geology (3rd ed.). Wiley-Blackwell,. p. 1 ff. ISBN 1405116528.{{cite book}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)

See also

• Aircraft principal axes
• Axis angle
• Orientation (geometry)
• Orientation (mathematics)
• Rotation matrix
• Tait-Bryan angles

• Geometry