Hauptachse

Hauptachse is a German term that literally means main axis or principal axis. It is used in geometry, physics, and engineering to denote the primary direction or axis of symmetry of a figure or body. In many contexts, it refers to the axis along which a property is maximized or the axes that align with the natural symmetry of the object. The term is often paired with Nebenachsen (secondary axes) to describe orthogonal directions.

In two-dimensional geometry, the most common use is with ellipses. The Hauptachse of an ellipse is the major axis, the longest diameter that passes through the center and both foci. Perpendicular to it is the minor axis, the minor diameter. If the ellipse has semi-major axis a and semi-minor axis b, the length of the major axis is 2a and the distance to the foci is c = sqrt(a^2 - b^2). The foci lie on the Hauptachse.

In three-dimensional bodies, the term expands to principal axes. For objects such as ellipsoids or rigid bodies, the Hauptachsen are the three mutually perpendicular axes that align with the object's symmetry. In dynamics, these principal axes are the eigenvectors of the inertia tensor and define the directions in which the body’s rotation decouples, characterized by the principal moments of inertia I1, I2, and I3. Aligning coordinates with the Hauptachsen simplifies mathematical descriptions of rotation and deformation.

Summary: Hauptachse denotes the main directional axis of a figure or body, most commonly the major axis of an ellipse or the principal axes of a three-dimensional object, reflecting symmetry and simplifying analysis in geometry and mechanics.