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44-471: The selenographic coordinates of this range are 19°54′N 9°12′E  /  19.9°N 9.2°E  / 19.9; 9.2 , and the length is 560 km. The tallest peaks in this range climb as high as 2.4 km. This range is named after Haemus Mons , an old Thracian name of the Balkan Mountains . It appeared on the map of Moon due to Johannes Hevelius . But he assigned this name (in

88-421: A commutative ring . The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa ; this is the basis of analytic geometry . The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line . In this system, an arbitrary point O (the origin ) is chosen on a given line. The coordinate of

132-439: A right-handed or a left-handed system. Another common coordinate system for the plane is the polar coordinate system . A point is chosen as the pole and a ray from this point is taken as the polar axis . For a given angle θ , there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from

176-412: A rotation vector to understand how rotation occurs. It reveals a connection between the current and the magnetic field lines in the magnetic field that the current created. Ampère was inspired by fellow physicist Hans Christian Ørsted , who observed that needles swirled when in the proximity of an electric current -carrying wire and concluded that electricity could create magnetic fields . This rule

220-409: A 180° rotation around the remaining axis, also preserving the handedness. These operations can be composed to give repeated changes of handedness. (If the axes do not have a positive or negative direction, then handedness has no meaning.) In mathematics, a rotating body is commonly represented by a pseudovector along the axis of rotation . The length of the vector gives the speed of rotation and

264-483: A mathematical system for representing three-dimensional rotations, is often attributed with the introduction of this convention. In the context of quaternions, the Hamiltonian product of two vector quaternions yields a quaternion comprising both scalar and vector components. Josiah Willard Gibbs recognized that treating these components separately, as dot and cross product, simplifies vector formalism. Following

308-475: A point P is defined as the signed distance from O to P , where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system . In the plane , two perpendicular lines are chosen and

352-476: A point varies while the other coordinates are held constant, then the resulting curve is called a coordinate curve . If a coordinate curve is a straight line , it is called a coordinate line . A coordinate system for which some coordinate curves are not lines is called a curvilinear coordinate system . Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. A coordinate line with all other constant coordinates equal to zero

396-441: A position along the positive z -axis, the ¼ turn from the positive x- to the positive y- axis is counter-clockwise . For left-handed coordinates, the above description of the axes is the same, except using the left hand; and the ¼ turn is clockwise . Interchanging the labels of any two axes reverses the handedness. Reversing the direction of one axis (or three axes) also reverses the handedness. Reversing two axes amounts to

440-410: A positively charged particle moving to the north, in a region where the magnetic field points west, the resultant force points up. The right-hand rule has widespread use in physics . A list of physical quantities whose directions are related by the right-hand rule is given below. (Some of these are related only indirectly to cross products, and use the second form.) Unlike most mathematical concepts,

484-404: A second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix , which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of

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528-415: A substantial debate, the mainstream shifted from Hamilton's quaternionic system to Gibbs' three-vectors system. This transition led to the prevalent adoption of the right-hand rule in the contemporary contexts. The cross product of vectors a → {\displaystyle {\vec {a}}} and b → {\displaystyle {\vec {b}}}  is

572-402: A vector perpendicular to the plane spanned by a → {\displaystyle {\vec {a}}} and b → {\displaystyle {\vec {b}}}  with the direction given by the right-hand rule : If you put the index of your right hand on a → {\displaystyle {\vec {a}}}  and

616-405: Is a homeomorphism from an open subset of a space X to an open subset of R . It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an atlas covering the space. A space equipped with such an atlas is called a manifold and additional structure can be defined on a manifold if the structure

660-414: Is called a coordinate axis , an oriented line used for assigning coordinates. In a Cartesian coordinate system , all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwise orthogonal . A polar coordinate system is a curvilinear system where coordinate curves are lines or circles . However, one of the coordinate curves

704-417: Is consistent where the coordinate maps overlap. For example, a differentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function. In geometry and kinematics , coordinate systems are used to describe the (linear) position of points and the angular position of axes, planes, and rigid bodies . In the latter case, the orientation of

748-501: Is described by coordinate transformations , which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates ( x ,  y ) and polar coordinates ( r ,  θ ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x  =  r  cos θ and y  =  r  sin θ . With every bijection from

792-428: Is one where only the ratios of the coordinates are significant and not the actual values. Some other common coordinate systems are the following: There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length . These include: Coordinates systems are often used to specify the position of a point, but they may also be used to specify

836-418: Is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero. Many curves can occur as coordinate curves. For example, the coordinate curves of parabolic coordinates are parabolas . In three-dimensional space, if one coordinate is held constant and

880-523: Is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be dualistic . Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the principle of duality . There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems

924-426: Is used in two different applications of Ampère's circuital law : The cross product of two vectors is often taken in physics and engineering. For example, as discussed above, the force exerted on a moving charged particle when moving in a magnetic field B is given by the magnetic term of Lorentz force: The direction of the cross product may be found by application of the right-hand rule as follows: For example, for

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968-401: The ( n − 1) -dimensional spaces resulting from fixing a single coordinate of an n -dimensional coordinate system. The concept of a coordinate map , or coordinate chart is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map

1012-452: The cylindrical coordinate system , a z -coordinate with the same meaning as in Cartesian coordinates is added to the r and θ polar coordinates giving a triple ( r ,  θ ,  z ). Spherical coordinates take this a step further by converting the pair of cylindrical coordinates ( r ,  z ) to polar coordinates ( ρ ,  φ ) giving a triple ( ρ ,  θ ,  φ ). A point in

1056-442: The middle finger on b → {\displaystyle {\vec {b}}} , then the thumb points in the direction of a → × b → {\displaystyle {\vec {a}}\times {\vec {b}}} . The right-hand rule in physics was introduced in the late 19th century by John Fleming in his book Magnets and Electric Currents. Fleming described

1100-428: The position of the points or other geometric elements on a manifold such as Euclidean space . The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x -coordinate". The coordinates are taken to be real numbers in elementary mathematics , but may be complex numbers or elements of a more abstract system such as

1144-539: The right-hand rule is a convention and a mnemonic , utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors , as well as to establish the direction of the force on a current-carrying conductor in a magnetic field . The various right- and left-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. This can be seen by holding your hands together with palms up and fingers curled. If

1188-409: The z -axis. The threads of a screw are helical and therefore screws can be right- or left-handed. To properly fasten or unfasten a screw, one applies the above rules: if a screw is right-handed, pointing one's right thumb in the direction of the hole and turning in the direction of the right hand's curled fingers (i.e. clockwise) will fasten the screw, while pointing away from the hole and turning in

1232-498: The Rimae Sulpicius Gallus. Several small lakes, formed from basaltic lava , lie along the southwest face of the range. From the northwest to the southeast these are Lacus Odii , Lacus Doloris , Lacus Gaudii , and Lacus Hiemalis . The Lacus Lenitatis lies farther to the south. Coordinate In geometry , a coordinate system is a system that uses one or more numbers , or coordinates , to uniquely determine

1276-542: The axes of the local system; they are the tips of three unit vectors aligned with those axes. The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems. Starting with the Greeks of the Hellenistic period , a variety of coordinate systems have been developed based on the types above, including: Right-hand rule In mathematics and physics ,

1320-432: The coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three mutually orthogonal planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n -dimensional Euclidean space. Depending on the direction and order of the coordinate axes , the three-dimensional system may be

1364-410: The curl of the fingers represents a movement from the first or x-axis to the second or y-axis, then the third or z-axis can point along either right thumb or left thumb. The right-hand rule dates back to the 19th century when it was implemented as a way for identifying the positive direction of coordinate axes in three dimensions. William Rowan Hamilton , recognized for his development of quaternions ,

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1408-456: The direction of n̂ and the fingers curl along the orientation of the bounding curve C . Ampère's right-hand grip rule, also called the right-hand screw rule , coffee-mug rule or the corkscrew-rule; is used either when a vector (such as the Euler vector ) must be defined to represent the rotation of a body, a magnetic field, or a fluid, or vice versa, when it is necessary to define

1452-405: The direction of the axis gives the direction of rotation according to the right-hand rule: right fingers curled in the direction of rotation and the right thumb pointing in the positive direction of the axis. This allows some simple calculations using the vector cross-product. No part of the body is moving in the direction of the axis arrow. If the thumb is pointing north, Earth rotates according to

1496-406: The electromotive force set up in it will be indicated by the direction in which the middle finger points." For right-handed coordinates, if the thumb of a person's right hand points along the z -axis in the positive direction (third coordinate vector), then the fingers curl from the positive x -axis (first coordinate vector) toward the positive y -axis (second coordinate vector). When viewed at

1540-399: The form Mons Æmus ) to another feature – remains of the rim of crater Alexander , located on the other side of Mare Serenitatis. Later the name moved to the subject of this article. The same name, but with reversed order of words – Haemus Montes – belongs to one of mountain systems on Io . Several rille systems lie along the eastern side of this range. The eastern end of the range forms

1584-421: The new direction (i.e. counterclockwise) will unfasten the screw. In vector calculus , it is necessary to relate a normal vector of a surface to the boundary curve of the surface. Given a surface S with a specified normal direction n̂ (a choice of "upward direction" with respect to S ), the boundary curve C around S is defined to be positively oriented provided that the right thumb points in

1628-400: The orientation of the induced electromotive force by referencing the motion of the conductor and the direction of the magnetic field in the following depiction: “If a conductor, represented by the middle finger, be moved in a field of magnetic flux , the direction of which is represented by the direction of the forefinger , the direction of this motion, being in the direction of the thumb, then

1672-448: The origin is r for given number r . For a given pair of coordinates ( r ,  θ ) there is a single point, but any point is represented by many pairs of coordinates. For example, ( r ,  θ ), ( r ,  θ +2 π ) and (− r ,  θ + π ) are all polar coordinates for the same point. The pole is represented by (0, θ ) for any value of θ . There are two common methods for extending the polar coordinate system to three dimensions. In

1716-453: The other two are allowed to vary, then the resulting surface is called a coordinate surface . For example, the coordinate surfaces obtained by holding ρ constant in the spherical coordinate system are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak of coordinate planes . Similarly, coordinate hypersurfaces are

1760-422: The plane may be represented in homogeneous coordinates by a triple ( x ,  y ,  z ) where x / z and y / z are the Cartesian coordinates of the point. This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity . In general, a homogeneous coordinate system

1804-549: The position of more complex figures such as lines, planes, circles or spheres . For example, Plücker coordinates are used to determine the position of a line in space. When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term line coordinates is used for any coordinate system that specifies the position of a line. It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this

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1848-449: The right-hand rule ( prograde motion ). This causes the Sun, Moon, and stars to appear to revolve westward according to the left-hand rule. A helix is a curved line formed by a point rotating around a center while the center moves up or down the z -axis. Helices are either right or left handed with curled fingers giving the direction of rotation and thumb giving the direction of advance along

1892-399: The space to itself two coordinate transformations can be associated: For example, in 1D , if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more. Given a coordinate system, if one of the coordinates of

1936-469: The western terminus of a rille system designated Rimae Plinius. 100 km farther to the west the craters Menelaus and Auwers are embedded within the range, and to their northeast are the Rimae Menelaus. Where the mountain range curves up to the northwest, the cup-shaped crater Sulpicius Gallus lies nearby on the lunar mare . Just to the northwest of this crater, and paralleling the mountains, are

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