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Concept in geometry
Look up subtend in Wiktionary, the free dictionary. In geometry, an angle subtended (from Latin for "stretched under") by a line segment at an arbitrary
Subtended_angle
Angle between two radii of a circle
distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured
Central_angle
Measure in 3-dimensional geometry
viewing direction. The solid angle of a sphere measured from any point in its interior is 4π sr. The solid angle subtended at the center of a cube by one
Solid_angle
Angle formed in the interior of a circle
inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a
Inscribed_angle
SI derived unit of angle
many areas of mathematics. It is defined such that one radian is the angle subtended at the center of a plane circle by an arc that is equal in length to
Radian
Figure formed by two rays meeting at a common point
Irrational angle Phase (waves) Protractor Solid angle Spherical angle Subtended angle Tangential angle Transcendent angle Trisection Zenith angle This approach
Angle
SI derived unit of solid angle
measure a solid angle of any projected shape. The solid angle subtended is the same as that of a cone with the same projected area. A solid angle of one steradian
Steradian
Angle created by applying the golden ratio to a circle
{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}} The golden angle is then the angle subtended by the smaller arc of length b. It measures approximately
Golden_angle
inscribed angle theorem, the central angle subtended by the chord A C ¯ {\displaystyle {\overline {AC}}} at the circle's center is twice the angle ∠ A D C
List of trigonometric identities
List_of_trigonometric_identities
Unit of length in astronomy
one au (astronomical unit, the average Earth–Sun distance) and the subtended angle of the vertex opposite that leg measures one arcsecond (1⁄3600 of a
Parsec
Clarity of vision
30 seconds of arc, corresponding to the foveal cone diameter or the angle subtended at the nodal point of the eye. To get reception from each cone, as
Visual_acuity
Simplification of the basic trigonometric functions
or angle subtended by the image of a distant object is often only a few arcseconds (denoted by the symbol ″), so it is well suited to the small angle approximation
Small-angle_approximation
Units for measuring angles
by simple trigonometry, it can be derived that the angle θ {\displaystyle \theta } subtended by an object of diameter or length d {\displaystyle
Minute_and_second_of_arc
90° angle (π/2 radians)
right angle. Two application examples in which the right angle and the Thales' theorem are included (see animations). The solid angle subtended by an
Right_angle
Optical Phenomenon
given area element is reduced by the cosine of the emission angle, the solid angle, subtended by surface visible to the viewer, is reduced by the very same
Lambert's_cosine_law
Optical illusion
lens of the eye. The larger the angle an object subtends, the larger the apparent size of the object. The subtended angle increases as the object moves
Forced_perspective
Unit of plane angle where a full circle equals 1
(symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center.
Turn_(angle)
Polygonal curve made from right triangles
{\displaystyle \tan \varphi _{n}={\frac {1}{\sqrt {n}}}} Therefore, the subtended angle φ n {\displaystyle \varphi _{n}} of the next triangle n {\displaystyle
Spiral_of_Theodorus
US 1960s-era study into nuclear weapons to create beams of plasma
created thrust. Moving the plate closer to the bomb increased the subtended angle that was captured, and thus efficiency, but at the cost of greatly
Casaba-Howitzer
Angle an object subtends at the eye
Visual angle is the angle a viewed object subtends at the eye, usually stated in degrees of arc. It also is called the object's angular size. The diagram
Visual_angle
Angular extent of given scene imaged by camera
view. It is important to distinguish the angle of view from the angle of coverage, which describes the angle at which the lens projects the image circle
Angle_of_view_(photography)
Displacement measured angle-wise when a body is showing circular or rotational motion
minimum and maximum angles of rotation admited; this interval delimits a circular arc, spans a flat pencil of lines, and subtends a given angular separation
Angular_displacement
Functions of an angle
describing angle measures. When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends
Trigonometric_functions
Process of enlarging the apparent size of something
\varepsilon _{0}} is the angle subtended by an object (w.r.t the optical axis) and ε {\textstyle \varepsilon } is the angle subtended by its image (also w
Magnification
Famous mathematical optimization problem
far from the wall should the viewer stand in order to maximize the angle subtended by the painting and whose vertex is at the viewer's eye? If the viewer
Regiomontanus' angle maximization problem
Regiomontanus'_angle_maximization_problem
Concept of computer visual displays
Figure 2) a viewing cone of 70°–90° subtended angle means (for a rotationally symmetric viewing cone) a maximum angle of inclination of 35°-45°. Fig. 5
Viewing_cone
Equation to derive time of sunset and sunrise
neglects the influence of atmospheric refraction and the non-zero angle subtended by the solar disc — i.e. the apparent diameter of the sun. The times
Sunrise_equation
Angle between the vertical axis and the steering axis of a steered wheel, in side view
constructors refer to the angle subtended by the mechanical trail at the wheel center as caster. Bicycle and motorcycle dynamics Camber angle Toe (automotive)
Caster_angle
Argument of the hyperbolic functions
if two chords P0P1 and P0P2 subtend angles L1 and L2 at the centre of a circle, their sum L1 + L2 is the angle subtended by a chord P0Q, where P0Q is
Hyperbolic_angle
Triangle containing a 90-degree angle
"In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Three integers
Right_triangle
Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle
, and using (1) yields Ptolemy's theorem. Let the inscribed angles subtended by A B {\displaystyle AB} , B C {\displaystyle BC} and C D {\displaystyle
Ptolemy's_theorem
Quaternion of norm 1 (unit quaternion)
3-dimensional rotation defined by the versor has the angle two times the arc's subtended angle, and preserves the same plane. It is a rotation about
Versor
Eye chart
the distance from eye to chart, and θ {\displaystyle \theta } is the angle subtended by the optotype, which is 5 arcminutes as specified by Snellen. Another
Snellen_chart
Archimedean solid with 38 faces
\quad \arccos \left({\tfrac {1}{3}}(1-2t)\right).} A snub cube has a subtended angle by an edge from the center arccos ( ( t 2 − 2 ) − 1 ) . {\displaystyle
Snub_cube
Unit of measurement of an angle, equal to 1/400th of a circle
Earth subtends an angle of one centigon at the centre of the Earth. However the gon is rarely used. Angular frequency – Rate of change of angle Milliradian –
Gradian
Fluctuations in the density of the normal matter of the universe
function of redshift ( z ) {\displaystyle (z)} . By measuring the subtended angle, Δ θ {\displaystyle \Delta \theta } , of the ruler of length Δ χ {\displaystyle
Baryon_acoustic_oscillations
Atmospheric optical phenomenon
range of incident angles, we can find the angle of the outgoing ray with respect to the vertical—which in fact is the angle subtended at the eye of the
22°_halo
Part of a circle between two points
the minor arc, subtends a central angle that is less than π radians (180 degrees); and the other arc, the major arc, subtends an angle greater than π
Circular_arc
Map projection system
projections are conformal, maps in UTM coordinates do not distort subtended angles or local shapes, and scale distortion is isotropic. Distortion at a
Universal Transverse Mercator coordinate system
Universal_Transverse_Mercator_coordinate_system
1 minus the cosine of an angle
vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ) is the distance AC (half of the chord). On the
Versine
Simple curve of Euclidean geometry
said to subtend the angle, known as the central angle, at the centre of the circle. One radian is the measure of the central angle subtended by a circular
Circle
Spherical triangle used in astronavigation
measuring the angle of the celestial body in the sky, the observer can get the local hour angle (LHA) of X, which is the angle subtended at P between Z
Navigational_triangle
List of constellations by area
are ranked by the solid angle that they subtend in the sky, measured in square degrees and millisteradians. These solid angles depend on arbitrary boundaries
IAU designated constellations by solid angle
IAU_designated_constellations_by_solid_angle
In human visual perception, the visual angle, denoted θ, subtended by a viewed object sometimes looks larger or smaller than its actual value. One approach
Perceived_visual_angle
Spherical triangle with three right angles
Bézier triangle. The solid angle subtended by a spherical octant is π/2 steradian or one-eighth of a spat, the solid angle of a full sphere. Hemisphere
Octant_of_a_sphere
Angular measurement, thousandth of a radian
given angle (angular distance or angular diameter) increases with viewing distance to the target. For instance the same angle of 0.1 mrad will subtend 10 mm
Milliradian
Unit of angle in solid geometry
the OEIS). Thus 1 sp is the solid angle subtended by a complete sphere at its center. Turn (angle) — the plane angle counterpart of the spat, equivalent
Spat_(angular_unit)
Long narrow depression bordering the Aegean Sea to the south
base line, the subduction line, and a chord across the arc of the subtended angle. Currently the vertex opposite the base line does not extend as far
Hellenic_Trench
Property of all triangles on a Euclidean plane
sphere, a, b, and c are the angles at the center of the sphere subtended by those arcs, in radians. Let A, B, and C be the angles opposite those respective
Law_of_sines
Theory in aviation
unguided projectiles are used. At the vast engagement distances, the subtended angle of the target would be minuscule, and the projectile flight time to
Big_sky_theory
Hunting strategy by some predators
likely to engage in pursuit when prey come within a subtended angle of around 1-2 degrees. Angles greater than this are outside of a dragonflies visual
Pursuit_predation
Physical quantity in radiometry
at that surface from a specified angle of view. In this case, the solid angle of interest is the solid angle subtended by the optical system's entrance
Radiance
Spherical collection of stars
radius (rt). The radius can be expressed as a physical distance or as a subtended angle in the sky. Considering a radius around the core, the surface luminosity
Globular_cluster
Polyhedron with four faces
which the solid angles subtended by the faces are equal, having a common value of π {\displaystyle \pi } sr, and at which the angles subtended by opposite
Tetrahedron
Perceived variation in the Moon's size
angular size when at the horizon than when above. The angle that the diameter of the full Moon subtends at an observer's eye can be measured directly with
Moon_illusion
Using measures of converging rays to improve fixed points for mapping
Snell in 1615–17, who showed how a point could be located from the angles subtended from three known points, but measured at the new unknown point rather
Triangulation_(surveying)
Angle between the two sightlines or two objects as viewed from an observer
the angle between the orientation of two straight lines, rays, or vectors – typically in three-dimensional space – or the central angle subtended by the
Angular_distance
Radio wave detection and transmission system defined by its separation
transmit and receive antennas, but if the angle subtended between transmitter, target and receiver (the bistatic angle) is close to zero, then they would still
Bistatic_radar
edges, as they perform better at oblique incidence angles. This enabled the subtended (flare) angle of the feed horn to be increased, and hence the length
Transmitarray_antenna
Distance measuring technique
difference in position of a star as seen from the Earth and Sun, i.e. the angle subtended at a star by the mean radius of the Earth's orbit around the Sun. The
Parallax_in_astronomy
Geometry of figures on the surface of a sphere
to the vertices of the triangle (on the unit sphere). The arc BC subtends an angle of magnitude a at the centre and therefore OB→ · OC→ = cos a. Introduce
Spherical_trigonometry
Spectroscopy technique
at a detector lying at an angle θ and subtending a solid angle ΔΩ is defined as the intensity of light per unit solid angle per unit incident intensity
Multiangle_light_scattering
Constant equal to twice pi
clearer or pedagogically preferable if formulas and quantities (including angles) were described in terms of τ instead of π. A constant named tau is implemented
Tau_(mathematics)
Angle in certain right triangles in the hyperbolic plane
semicircles, making a right angle with the unit semicircle and a variable angle Φ with Q. The angle at the center of Q subtended by the radius to (0, y) is
Angle_of_parallelism
Generalization of Pythagorean theorem
obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice
Law_of_cosines
Diagrammatic representation of Sun's position over a period of time
on the diagram, but it should be expressed in terms of the angle that would be subtended at an observer on the ground by the corresponding distance in
Analemma
Geometric line segment whose endpoints lie on a circular arc
center if and only if their lengths are equal. Equal chords are subtended by equal angles from the center of the circle. A chord that passes through the
Chord_(geometry)
Tool for angle measurement
which is basically a quintant: the frame is a sector of a circle subtending an angle of 76° at the pivot of the index arm. The necessity for the doubled
Sextant
Trap used to catch insects
aids, and that the placement of a closer-than-the-moon light causes subtended angles of light at the insect's eye to alter so rapidly that it has to fly
Moth_trap
Time of day when the sun appears above the horizon
for calculation, neglecting atmospheric refraction and the non-zero angle subtended by the solar disc. Neglecting the effects of refraction and the Sun's
Sunrise
Photometric measure
source inside the solid angle dΩΣ; dΩS is the infinitesimal solid angle subtended by dΣ as seen from dS; and θS is the angle between the normal nS to
Luminance
Measure of the "spread" of light in an optical system
solid angle the source subtends as seen from the pupil. These definitions must be applied for infinitesimally small "elements" of area and solid angle, which
Etendue
Formula for the great-circle distance between two points on a sphere
unit sphere, the lengths a, b, and c are simply equal to the angles (in radians) subtended by those sides from the center of the sphere (for a non-unit
Haversine_formula
Mathematical relation in spherical triangles
proportional to the measures of the central angles they subtend. The angles between sides are dihedral angles between the planes containing them. In the
Spherical_law_of_cosines
Intersection of a sphere and cone emanating from its center
is the solid angle of the spherical sector in steradians, the SI unit of solid angle. One steradian is defined as the solid angle subtended by a cap area
Spherical_sector
chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected
History_of_trigonometry
Pattern of birefringence colors
power objective lens is typically used. This both maximises the solid angle subtended by the lens, and hence the angular variation of the light intercepted
Conoscopic interference pattern
Conoscopic_interference_pattern
Relation between sides of a right triangle
Elements: Book VI, Proposition VI 31: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described
Pythagorean_theorem
Surveying instrument for angle measurements
for angle measurements. It consists of a semicircular limb divided into 180 degrees and sometimes subdivided into minutes. The limb is subtended by the
Graphometer
following quantities are considered: length, area, volume, plane angle, solid angle, mass, density, time, frequency, velocity, volumetric flow rate, acceleration
List_of_conversion_factors
the solid angle subtended by the infinitesimal 'target' (or 'aperture') area d A a {\displaystyle dA_{a}} θ {\displaystyle \theta } is the angle between
Intensity_(heat_transfer)
Relationship between two lines that meet at a right angle
geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity
Perpendicular
Intensity of electromagnetic radiation
to the solid angle into which the light is emitted. When calculating radiance received by a detector, Ω refers to the solid angle subtended by the source
Radiant_intensity
Measure of how strongly an optical system converges or diverges light
film or sensor format is known as a normal lens; its angle of view is similar to the angle subtended by a large-enough print viewed at a typical viewing
Focal_length
Any of the five regular polyhedra
is a regular q-gon. The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4π steradians)
Platonic_solid
Longest side of a right-angled triangle, the side opposite of the right angle
equivalent Latin verb, subtend, is still used in English in a more general sense to refer to any line or curve stretched across an angle. The Pythagorean theorem
Hypotenuse
Archaic rapid surveying method
surveyors. The horizontal distance S is inferred from the vertical angle subtended between two well-defined points on the staff and the known distance
Tacheometry
Periodic comet with 6 year orbit
discovery, these shed fragments were spread out along the orbital path subtending an angle of 27 arcminutes from the comet's surviving head. The comet has many
57P/du_Toit–Neujmin–Delporte
Tetrahedron where all three face angles at one vertex are right angles
T_{0}={\frac {xyz}{2h}}.} The solid angle at the right-angled vertex, from which the opposite face (the base) subtends an octant, has measure π/2 steradians
Trirectangular_tetrahedron
Geographic coordinate specifying north-south position
through the centre and the latitude (ϕ) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned
Latitude
Television placement for immersive experience
distance places the viewer where the horizontal angle subtended by the screen is the same as the horizontal angle captured by the camera. If this is the case
Optimum_HDTV_viewing_distance
Recreational mathematics planar boundary and area problem
{1}{2}}r^{2}\theta } if the angle is in radians. Imagine an infinitely thin circular sector from o {\displaystyle o} to p {\displaystyle p} subtended by an infinitely
Goat_grazing_problem
Area bounded by a circular arc and a straight line
arc which forms part of the perimeter of the segment, θ the central angle subtending the arc in radians, c the chord length, s the arc length, h the sagitta
Circular_segment
How large a sphere or circle appears
units of angle) describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in
Angular_diameter
Mathematical model of the physical space
triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47) Books
Euclidean_geometry
Meteorological phenomenon
perceived angle that the rainbow subtends as follows. Given a spherical raindrop, and defining the perceived angle of the rainbow as 2φ, and the angle of the
Rainbow
Far-field diffraction
Most of the diffracted light falls between the first minima. The angle, α, subtended by these two minima is given by: α ≈ 2 λ W {\displaystyle \alpha
Fraunhofer_diffraction
Width of an electromagnetic beam
refer to the angular diameter (i.e., angular width), which is the angle subtended by the beam at the source. The angular width is also called the beam
Beam_diameter
Plane curve
arc length of a portion of the circumference, as a function of the angle subtended (or x coordinates of any two points on the upper half of the ellipse)
Ellipse
Aberration inherent to certain optical designs or due to imperfection in the lens
appear as wedge-shaped. The further off-axis (or the greater the angle subtended by the point with the optical axis), the worse this effect is. This causes
Coma_(optics)
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