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27-450: An orbital pole is either point at the ends of the orbital normal , an imaginary line segment that runs through a focus of an orbit (of a revolving body like a planet , moon or satellite ) and is perpendicular (or normal ) to the orbital plane . Projected onto the celestial sphere , orbital poles are similar in concept to celestial poles , but are based on the body's orbit instead of its equator . The north orbital pole of

54-497: A closed line segment as above, and an open line segment as a subset L that can be parametrized as for some vectors u , v ∈ V . {\displaystyle \mathbf {u} ,\mathbf {v} \in V.} Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points. In geometry , one might define point B to be between two other points A and C , if

81-420: A line , the object is a secant line . The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow"). More generally, a chord is a line segment joining two points on any curve , for instance, on an ellipse . A chord that passes through a circle's center point is the circle's diameter . Among properties of chords of a circle are the following: The midpoints of

108-624: A circuit around the nearer ecliptic pole every 25,800 years. As of 1 January 2000, the positions of the ecliptic poles expressed in equatorial coordinates , as a consequence of Earth's axial tilt , are the following: The North Ecliptic Pole is located near the Cat's Eye Nebula and the South Ecliptic Pole is located near the Large Magellanic Cloud . It is impossible anywhere on Earth for either ecliptic pole to be at

135-429: A directed line segment semi-infinitely produces a directed half-line and infinitely in both directions produces a directed line . This suggestion has been absorbed into mathematical physics through the concept of a Euclidean vector . The collection of all directed line segments is usually reduced by making equipollent any pair having the same length and orientation. This application of an equivalence relation

162-407: A more extensive table of chords in his book on astronomy , giving the value of the chord for angles ranging from ⁠ 1 / 2 ⁠ to 180 degrees by increments of ⁠ 1 / 2 ⁠ degree. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the integer part. The chord function is defined geometrically as shown in

189-428: A revolving body is defined by the right-hand rule . If the fingers of the right hand are curved along the direction of orbital motion , with the thumb extended and oriented to be parallel to the orbital axis , then the direction the thumb points is defined to be the orbital north. The poles of Earth's orbit are referred to as the ecliptic poles . For the remaining planets, the orbital pole in ecliptic coordinates

216-411: A set of parallel chords of a conic are collinear ( midpoint theorem for conics ). Chords were used extensively in the early development of trigonometry . The first known trigonometric table, compiled by Hipparchus in the 2nd century BC, is no longer extant but tabulated the value of the chord function for every ⁠7 + 1 / 2 ⁠ degrees . In the 2nd century AD, Ptolemy compiled

243-416: A triangle include those connecting various triangle centers to each other, most notably the incenter , the circumcenter , the nine-point center , the centroid and the orthocenter . In addition to the sides and diagonals of a quadrilateral , some important segments are the two bimedians (connecting the midpoints of opposite sides) and the four maltitudes (each perpendicularly connecting one side to

270-598: A two-dimensional simplex is a triangle. This article incorporates material from Line segment on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . Chord (geometry) A chord (from the Latin chorda , meaning " bowstring ") of a circle is a straight line segment whose endpoints both lie on a circular arc . If a chord were to be extended infinitely on both directions into

297-460: Is a subset of V , then L is a line segment if L can be parameterized as for some vectors u , v ∈ V {\displaystyle \mathbf {u} ,\mathbf {v} \in V} where v is nonzero. The endpoints of L are then the vectors u and u + v . Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define

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324-414: Is called the major axis , and a segment from the midpoint of the major axis (the ellipse's center) to either endpoint of the major axis is called a semi-major axis . Similarly, the shortest diameter of an ellipse is called the minor axis , and the segment from its midpoint (the ellipse's center) to either of its endpoints is called a semi-minor axis . The chords of an ellipse which are perpendicular to

351-490: Is given by the longitude of the ascending node ( ☊ ) and inclination ( i ): ℓ = ☊ − 90° , b = 90° − i . In the following table, the planetary orbit poles are given in both celestial coordinates and the ecliptic coordinates for the Earth. When an artificial satellite orbits close to another large body, it can only maintain continuous observations in areas near its orbital poles. The continuous viewing zone (CVZ) of

378-417: Is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. The segment addition postulate can be used to add congruent segment or segments with equal lengths, and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a degenerate case of an ellipse , in which the semiminor axis goes to zero,

405-407: Is the following collection of points: In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an isometry of a line (used as a coordinate system). Segments play an important role in other theories. For example, in a convex set , the segment that joins any two points of the set is contained in the set. This

432-584: The Hubble Space Telescope lies inside roughly 24° of Hubble's orbital poles, which precess around the Earth's axis every 56 days. The ecliptic is the plane on which Earth orbits the Sun . The ecliptic poles are the two points where the ecliptic axis, the imaginary line perpendicular to the ecliptic, intersects the celestial sphere . The two ecliptic poles are mapped below. Due to axial precession , either celestial pole completes

459-470: The Pythagorean theorem to calculate the chord length: The last step uses the half-angle formula . Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably, a great deal was known about them. In the table below (where c is the chord length, and D

486-439: The foci go to the endpoints, and the eccentricity goes to one. A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a radial elliptic trajectory . In addition to appearing as

513-419: The perpendicular bisectors of the sides (perpendicularly connecting the midpoint of a side to one of the other sides), and the internal angle bisectors (each connecting a vertex to the opposite side). In each case, there are various equalities relating these segment lengths to others (discussed in the articles on the various types of segment), as well as various inequalities . Other segments of interest in

540-671: The zenith in the night sky . By definition, the ecliptic poles are located 90° from the Sun's position . Therefore, whenever and wherever either ecliptic pole is directly overhead, the Sun must be on the horizon . The ecliptic poles can contact the zenith only within the Arctic and Antarctic circles. The galactic coordinates of the North ecliptic pole can be calculated as ℓ = 96.38° , b = 29.81° (see celestial coordinate system ). Line segment Examples of line segments include

567-485: The distance | AB | added to the distance | BC | is equal to the distance | AC | . Thus in ⁠ R 2 , {\displaystyle \mathbb {R} ^{2},} ⁠ the line segment with endpoints A = ( a x , a y ) {\displaystyle A=(a_{x},a_{y})} and C = ( c x , c y ) {\displaystyle C=(c_{x},c_{y})}

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594-410: The edges and diagonals of polygons and polyhedra , line segments also appear in numerous other locations relative to other geometric shapes . Some very frequently considered segments in a triangle to include the three altitudes (each perpendicularly connecting a side or its extension to the opposite vertex ), the three medians (each connecting a side's midpoint to the opposite vertex),

621-425: The major axis and pass through one of its foci are called the latera recta of the ellipse. The interfocal segment connects the two foci. When a line segment is given an orientation ( direction ) it is called a directed line segment or oriented line segment . It suggests a translation or displacement (perhaps caused by a force ). The magnitude and direction are indicative of a potential change. Extending

648-406: The midpoint of the opposite side). Any straight line segment connecting two points on a circle or ellipse is called a chord . Any chord in a circle which has no longer chord is called a diameter , and any segment connecting the circle's center (the midpoint of a diameter) to a point on the circle is called a radius . In an ellipse, the longest chord, which is also the longest diameter ,

675-416: The picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle . The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be ( cos θ , sin θ ), and then using

702-570: The sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron , the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal . When the end points both lie on a curve (such as a circle ), a line segment is called a chord (of that curve). If V is a vector space over ⁠ R {\displaystyle \mathbb {R} } ⁠ or ⁠ C , {\displaystyle \mathbb {C} ,} ⁠ and L

729-421: Was introduced by Giusto Bellavitis in 1835. Analogous to straight line segments above, one can also define arcs as segments of a curve . In one-dimensional space, a ball is a line segment. An oriented plane segment or bivector generalizes the directed line segment. Beyond Euclidean geometry, geodesic segments play the role of line segments. A line segment is a one-dimensional simplex ;

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