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In geometry, an axis-aligned object (axis-parallel, axis-oriented) is an object in n-dimensional space whose shape is aligned with the coordinate axes
Axis-aligned_object
Smallest box which encloses some set of points
coordinate for the points in S. Axis-aligned minimal bounding boxes are used as an approximate location of an object in question and as a very simple
Minimum_bounding_box
A rigid body with 3 distinct axes of inertia is unstable rotating about the middle axis
rotation of an object around its first and third principal axes is stable, whereas rotation around its second principal axis (or intermediate axis) is not.
Tennis_racket_theorem
Hypothetical Solar System planet
alignment also switched, from more aligned to anti-aligned with increasing semi-major axis, and from anti-aligned to aligned with increasing perihelion distance
Planet_Nine
Solar system objects beyond the other known trans-Neptunian objects
trans-Neptunian object (ETNO) orbits are suspected to be aligned with the hypothetical Planet Nine while the blue-colored ETNO orbits are anti-aligned. The highly
Extreme trans-Neptunian object
Extreme_trans-Neptunian_object
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
\end{aligned}}} Here I x x {\displaystyle I_{xx}} denotes the moment of inertia around the x {\displaystyle x} -axis when the objects are rotated
Moment_of_inertia
Movement of an object which leaves at least one point unchanged
perpendicular to that axis). Similarly, the rotation rate of an object in three dimensions at any instant is about some axis, although this axis may be changing
Rotation
vary with semi-major axis of the object and if the object is in resonance. At smaller semi-major axes the aligned and anti-aligned regions shrink and eventually
Effects of Planet Nine on trans-Neptunian objects
Effects_of_Planet_Nine_on_trans-Neptunian_objects
Type of motion
kinetic energy of the object, and for the forces on the parts of the object, are also simpler for rotation around a fixed axis, than for general rotational
Rotation_around_a_fixed_axis
Term in geometry; longest and shortest semidiameters of an ellipse
{\displaystyle {\begin{aligned}b&=a{\sqrt {1-e^{2}}},\\\ell &=a(1-e^{2}),\\a\ell &=b^{2}.\end{aligned}}} The semi-major axis of a hyperbola is, depending
Semi-major and semi-minor axes
Semi-major_and_semi-minor_axes
Polygon in which all angles are right
iso-oriented, axis-aligned, and axis-oriented polygons. These adjectives are less confusing when the polygons of this type are rectangles, and the term axis-aligned
Rectilinear_polygon
Major alliance of World War II
The Axis powers, originally called the Rome–Berlin Axis and later, the Rome–Berlin–Tokyo Axis, was the military coalition which initiated World War II
Axis_powers
Design technique
y = s z a z + c z {\displaystyle {\begin{aligned}b_{x}&=s_{x}a_{x}+c_{x}\\b_{y}&=s_{z}a_{z}+c_{z}\end{aligned}}} where the vector s is an arbitrary scale
3D_projection
Theorem in planar dynamics
of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular
Parallel_axis_theorem
Coordinate system using perpendicular axes
{\displaystyle {\begin{aligned}(+x,+y,+z)&&(-x,+y,+z)&&(+x,-y,+z)&&(+x,+y,-z)\\(+x,-y,-z)&&(-x,+y,-z)&&(-x,-y,+z)&&(-x,-y,-z)\end{aligned}}} The coordinates
Cartesian_coordinate_system
Optical imaging rule
image and object distances, so that m = − v u = − v − f f . {\displaystyle {\begin{aligned}m&=-{\frac {v}{u}}\\&=-{\frac {v-f}{f}}.\end{aligned}}} On the
Scheimpflug_principle
Loss of one degree of freedom in a three-dimensional, three-gimbal mechanism
accommodate rotation about one axis, leaving the suspended object effectively locked (i.e. unable to rotate) around that axis. The problem can be generalized
Gimbal_lock
Graph of space and time in special relativity
horizontal axis and time on the vertical axis. Additionally, the time and space units of measurement are chosen in such a way that an object moving at
Spacetime_diagram
Celestial coordinate system used to specify the positions of celestial objects
+ X η = y + Y ζ = z + Z {\displaystyle {\begin{aligned}\xi &=x+X\\\eta &=y+Y\\\zeta &=z+Z\end{aligned}}} Celestial coordinate system Planetary coordinate
Equatorial_coordinate_system
Mounting system for camera or telescope
instrument attached to it to stay fixed on any celestial object with diurnal motion by driving one axis at a constant speed. Such an arrangement is called a
Equatorial_mount
Type of motion which combines translation and rotation with respect to a surface
in the object axis is a line, while the trajectory of any point in the object rim is a cycloid. The velocity of any point in the rolling object is given
Rolling
Six points which determine imaging properties of an optical system
centred on the optical axis there will only pass rays that were emitted from the object at a sufficiently small angle from the optical axis. Using a sufficiently
Cardinal_point_(optics)
Mathematical construct in engineering
{\displaystyle J} (for an axis perpendicular to the plane). In both cases, it is calculated with a multiple integral over the object in question. Its dimension
Second_moment_of_area
Object movement along a circular path
In kinematics, circular motion is the motion of an object along a circular path. Examples of this include a stone tied to a string, a car moving around
Circular_motion
Variance in a celestial body's axis of rotation over time
a phenomenon which causes the orientation of the axis of rotation of a spinning astronomical object to vary over time. It is caused by the gravitational
Astronomical_nutation
Lens with a thickness that is negligible
{\begin{aligned}e&\approx nr_{2}\\&=n(i-r_{1})\\&\approx n(i-{\frac {i}{n}})\end{aligned}}} If the incoming ray is parallel to the optical axis and distance
Thin_lens
Conserved physical quantity; rotational analogue of linear momentum
{\begin{aligned}\mathbf {r} _{i}&=\mathbf {v} _{i}=\mathbf {0} ,\\\mathbf {r} &=\mathbf {R} ,\\\mathbf {v} &=\mathbf {V} ,\\m&=M,\end{aligned}}} ∑ i r
Angular_momentum
Periodic change in the direction of a rotation axis
symmetry axis. When an object is not perfectly rigid, inelastic dissipation will tend to damp torque-free precession, and the rotation axis will align itself
Precession
Method for visually representing three-dimensional objects
with the camera aligned parallel to the floor and aligned to the coordinate axes, it is first rotated horizontally (around the vertical axis) by ±45°, then
Isometric_projection
Type of inertial force
appears to act on all objects when viewed in a rotating frame of reference. It appears to be directed perpendicularly from the axis of rotation of the frame
Centrifugal_force
Celestial coordinate system used to describe Solar System objects
z = r sin b {\displaystyle {\begin{aligned}x&=r\cos b\cos \ell \\y&=r\cos b\sin \ell \\z&=r\sin b\end{aligned}}} [ x equatorial y equatorial z equatorial
Ecliptic_coordinate_system
Process of enlarging the apparent size of something
subtended by an object (w.r.t the optical axis) and ε {\textstyle \varepsilon } is the angle subtended by its image (also w.r.t the optical axis) made by an
Magnification
Straight figure with zero width and depth
{\displaystyle {\begin{aligned}X&=(x_{1},x_{2},\dots ,x_{n}),\\Y&=(y_{1},y_{2},\dots ,y_{n}),\\Z&=(z_{1},z_{2},\dots ,z_{n}),\end{aligned}}} are collinear if
Line_(geometry)
Representation of a curve by a function of a parameter
parametrisation) of the object. For example, the equations x = cos t y = sin t {\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}} form a parametric
Parametric_equation
Number of rotations per unit time
lowercase Greek nu, and also n), is the frequency of rotation of an object around an axis. Its SI unit is the reciprocal seconds (s−1); other common units
Rotational_frequency
Optical device which transmits and refracts light
material), u {\textstyle u} is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the
Lens
Position of something in relation to its surroundings
similar method, called axis–angle representation, describes a rotation or orientation using a unit vector aligned with the rotation axis, and a separate value
Orientation_(geometry)
Geometric objects with a common centre
quadrics. Geometric objects are coaxial if they share the same axis (line of symmetry). Geometric objects with a well-defined axis include circles (any
Concentric_objects
Speed and direction of a motion
{u}})+a^{2}t^{2}\end{aligned}}} ( 2 a ) ⋅ x = ( 2 a ) ⋅ ( u t + 1 2 a t 2 ) = 2 t ( a ⋅ u ) + a 2 t 2 = v 2 − u 2 {\displaystyle {\begin{aligned}(2{\boldsymbol
Velocity
Movement of an object which leaves one point unchanged
cos θ . {\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}} The vectors [ x y ] {\displaystyle
2D_rotation
Geometric civil engineering calculation technique
{MPa}}\\\end{aligned}}} σ 2 = σ a v g − R = − 30 MPa {\displaystyle {\begin{aligned}\sigma _{2}&=\sigma _{\mathrm {avg} }-R\\&=-30{\textrm {MPa}}\\\end{aligned}}}
Mohr's_circle
Device to facilitate robotic insertion into holes with tight clearance
peg over the hole, and then push the peg along the axis of the hole. If the peg is perfectly aligned and centered, it would then slide into the hole. However
Remote_center_compliance
Coordinates comprising two distances and an angle
ρ sin φ z = z {\displaystyle {\begin{aligned}x&=\rho \cos \varphi \\y&=\rho \sin \varphi \\z&=z\end{aligned}}} in one direction, and ρ = x 2 + y 2 φ
Cylindrical_coordinate_system
Design rule for optical systems
optical system in order for it to produce sharp images of off-axis as well as on-axis objects. It was formulated by Ernst Abbe in the context of microscopes
Abbe_sine_condition
Interface in computer science, human-computer interaction, and interaction design
three arrows translates the object along the appropriate axis. The axes may be aligned with the world-space axes, the object-space axes, or some other space
Direct_manipulation_interface
Time-telling device
solid object. In some cases, the sundials are formed as hollows in a solid object, e.g., a cylindrical hollow aligned with the Earth's rotational axis (in
Sundial
Visible star that is nearly aligned with Earth's axis of rotation
A pole star is a visible star that is approximately aligned with the axis of rotation of an astronomical body; that is, a star whose apparent position
Pole_star
Curved path of an object around a point
an object under the influence of an attracting force. Alternatively, it is known as an orbital revolution, because it is a rotation around an axis external
Orbit
Measure of how closely the shape of an object approaches that of a circle
points on the object, but is a separate precision bearing usually on the measuring instrument. The axis of the object or part of the object to be measured
Roundness
Coordinates comprising a distance and two angles
{\displaystyle {\begin{aligned}x&=r\sin \theta \,\cos \varphi ,\\y&=r\sin \theta \,\sin \varphi ,\\z&=r\cos \theta .\end{aligned}}} Cylindrical coordinates
Spherical_coordinate_system
Closed volume that completely contains the union of a set of objects
object. In most applications the axis of the cylinder is aligned with the vertical direction of the scene. Cylinders are appropriate for 3-D objects that
Bounding_volume
Correspondence between quaternions and 3D rotations
+\alpha }{2}}\right)\mathbf {B} \times \mathbf {A} \end{aligned}}} finally normalizing the rotation axis: D 2 sin 1 2 γ {\textstyle {\frac {\mathbf {D} }{2\sin
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Optical aberration
i + o , {\displaystyle {\begin{aligned}s&={\frac {R_{2}+R_{1}}{R_{2}-R_{1}}}\\[8pt]p&={\frac {i-o}{i+o}},\end{aligned}}} one can write the longitudinal
Spherical_aberration
Description of the orientation of a rigid body
the object. As an example, consider a top. If we define the Z axis to be the symmetry axis of the top, then the top spinning around its own axis of symmetry
Euler_angles
Method of orienting telescopes and other celestial observation devices
the act of aligning the rotational axis of a telescope's equatorial mount or a sundial's gnomon with a celestial pole to parallel Earth's axis. The method
Polar_alignment
Rate of change of velocity
{\displaystyle {\begin{aligned}a_{x}&={\frac {dv_{x}}{dt}}={\frac {d^{2}x}{dt^{2}}},\\a_{y}&={\frac {dv_{y}}{dt}}={\frac {d^{2}y}{dt^{2}}}.\end{aligned}}} The two-dimensional
Acceleration
Directional planes
vertical can be drawn from "up" to "down" (or down to up), such as the y-axis in the Cartesian coordinate system. The word horizontal is derived from the
Vertical_and_horizontal
superset, BSP trees). Finally, BIH is axis-aligned as are its ancestors. Although a more general non-axis-aligned implementation of the BIH should be possible
Bounding_interval_hierarchy
Geometric method for visualizing a rotating rigid body
intermediate axis is not aligned with the angular momentum, then the podhode will be a curbe going from one end of the intermediate axis to the other
Poinsot's_ellipsoid
3D model's surface projected to a 2D image
P} to the sphere's origin. Assuming that the sphere's poles are aligned with the Y axis, UV coordinates in the range [ 0 , 1 ] {\displaystyle [0,1]} can
UV_mapping
Coordinate frames
J2000 or EME2000. The x-axis is aligned with the mean vernal equinox. The z-axis is aligned with the Earth's rotation axis (or equivalently, the celestial
Earth-centered_inertial
Diagnostic instrument used to assess the extent and axis of astigmatism
4 apertures – As there are two prisms, each aligned perpendicular to the other, the major and minor axis powers can be measured independently without
Keratometer
Small patches of nebulosity associated with newly born stars
Herbig–Haro objects are commonly found in star-forming regions, and several are often seen around a single star, aligned with its rotational axis. Most of
Herbig–Haro_object
frame, with X-axis aligned to the south, where the local meridian intersects the horizon, Y-axis toward the eastern horizon, and Z-axis toward the zenith
Parallactic_angle
Process in computer graphics programming
performance or accuracy outcomes. One common hit-test algorithm for axis aligned bounding boxes. A key idea is that the box being tested must be either
Hit-testing
Type of telescope mount and related software
coordinates. To track the object so that it stays in the eyepiece despite Earth's rotation, only the right-ascension axis is moved. Smart telescopes
GoTo_(telescopes)
Frame-dependent apparent force in Physics
apparent change in the velocity of an object when its position changes, putting it nearer to or further from the axis of rotation (the change in ∑ x j d
Fictitious_force
Astronomy device for a telescope
used on telescopes equipped with an equatorial mount to find celestial objects by their equatorial coordinates, often used in star charts and ephemerides
Setting_circles
Apparent force in a rotating reference frame
proportional to the object's speed in the rotating frame (more precisely, to the component of its velocity that is perpendicular to the axis of rotation). The
Coriolis_force
{\begin{aligned}\epsilon &={\frac {v^{2}}{2}}-{\frac {\mu }{r}}=-{\frac {1}{2}}{\frac {\mu ^{2}}{h^{2}}}\left(1-e^{2}\right)=-{\frac {\mu }{2a}}\end{aligned}}}
Specific_mechanical_energy
Estimation of orbits of objects
orbiting objects. As time goes by, the actual path of an orbiting object tends to diverge from the predicted path (especially if the object is subject
Orbit_determination
Celestial coordinate system in spherical coordinates, with the Sun as its center
velocities of galactic objects. In these systems the xyz-axes are designated UVW, but the definitions vary by author. In one system, the U axis is directed toward
Galactic_coordinate_system
Family of linear transformations
{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}} where (t, x, y, z)
Lorentz_transformation
Energy of a moving physical body
an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is
Kinetic_energy
Surface formed by rotating an ellipse
{\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}} All modern geodetic ellipsoids are defined by the semi-major axis plus either
Spheroid
Type of synthetic microstructure
vertically aligned carbon nanotube arrays (VANTAs) are a unique microstructure consisting of carbon nanotubes oriented with their longitudinal axis perpendicular
Vertically aligned carbon nanotube arrays
Vertically_aligned_carbon_nanotube_arrays
Quantum state of a system
object such as an electron, atom, or molecule, having a quantum spin S = 1. It has three allowed values of the spin's projection along a given axis mS
Triplet_state
Limiting case which is different from the rest of the class
lower-dimensional hyperrectangle, all the way down to a point if the sides aligned with every axis have length zero. A convex polygon is degenerate if at least two
Degeneracy_(mathematics)
Constraint equations of a mechanical system
t\end{aligned}}.} In programming, this is often performed with the algorithm, x ← x + v t + 1 2 a t 2 v ← v + a t . {\displaystyle {\begin{aligned}\mathbf
Kinematics_equations
Method for aligning firearm sights
Boresighting is a method of visually pre-aligning a firearm barrel's bore axis with the target, in order to more easily zero the gunsight (optical or
Boresight_(firearm)
Graphics structure
corresponding data objects very tightly. One of the most commonly used bounding volumes is an axis-aligned minimum bounding box. The axis-aligned minimum bounding
Bounding_volume_hierarchy
Surface created by rotating a curve about an axis
dt\end{aligned}}} where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar
Surface_of_revolution
Physical phenomenon
following is valid: objects that are at rest on the surface of the Earth, co-rotating with the Earth, are circling the Earth's axis, so they are in centripetal
Eötvös_effect
Distant body in the outer Solar System
its semi-major axis to the 3/2 power, times a constant 1.7×1016.(see equation 4) If an object's Λ is greater than 1, then that object will eventually
Sedna_(dwarf_planet)
Standard color space with color-opponent values
8840 {\displaystyle {\begin{aligned}X_{\mathrm {n} }&=95.0489,\\Y_{\mathrm {n} }&=100,\\Z_{\mathrm {n} }&=108.8840\end{aligned}}} For illuminant D50, which
CIELAB_color_space
Classical mechanics rule
of inertia of a rigid object is unchanged when the object is stretched parallel to an axis of rotation that is a principal axis, provided that the distribution
Stretch_rule
Device that allows one to sight a distant object
cylinder (G), the alidade was aligned with the position of the star. This could not be used with a closely located object. A star, being so far away as
Alidade
Measure of compression between circle to ellipse or sphere to an ellipsoid of revolution
n , n = f 2 − f . {\displaystyle {\begin{aligned}f={\frac {2n}{1+n}},\\[5mu]n={\frac {f}{2-f}}.\end{aligned}}} The flattenings are related to other parameters
Flattening
Equations that describe the behavior of a physical system
unit vector in the direction of the axis of rotation, and θ is the angle the object turns through about the axis. The following relation holds for a point-like
Equations_of_motion
Group of rotations in 3 dimensions
2 , {\displaystyle {\begin{aligned}q&=w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,\\1&=w^{2}+x^{2}+y^{2}+z^{2},\end{aligned}}} is mapped to the rotation
3D_rotation_group
Defining the orbit of an object in space
ascending node, is one of the orbital elements used to specify the orbit of an object in space. Denoted with the symbol Ω, it is the angle from a specified reference
Longitude of the ascending node
Longitude_of_the_ascending_node
Spiral with constant distance from itself
Archimedean spirals. Suppose a point object moves in the Cartesian system with a constant velocity v directed parallel to the x-axis, with respect to the xy-plane
Archimedean_spiral
Equation for the velocity of a body in viscous fluid
z–axis is through the centre of the sphere and aligned with the mean flow direction, while r is the radius as measured perpendicular to the z–axis. The
Stokes's_law
Sednoid
2023 KQ14, informally nicknamed Ammonite, is a trans-Neptunian object (TNO) orbiting the Sun on an extremely wide elliptical orbit. It was discovered by
2023_KQ14
Turning force around an axis
body, a torque can be thought of as a twist applied to an object with respect to a chosen axis. For example, when driving a screw, a screwdriver applies
Torque
Visual illusion of 3D scene
intelligible icons which represent objects or concepts. In this autostereogram, patterns become smaller and smaller down the y-axis, until they look like random
Autostereogram
Type of technical drawing
cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off at an angle (typically
Oblique_projection
Plumbing tool
at a stop, the jaws will be perpendicular to the shaft and thus aligned to the object. When the shaft is held in a vertical orientation, the jaws are
Basin_wrench
Deviation from rectilinear projection (optics)
the image sharpness without changing an object shape or structure in the image (e.g., a straight line in an object is still a straight line in the image
Distortion_(optics)
Topics referred to by the same term
its orientation aligned with its direction of movement Toe (automotive), the symmetric angle that each wheel makes with the long axis of a vehicle Video
Tracking
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