In mathematics , a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point .
31-511: Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space . For any pair of distinct non- antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of
62-408: A {\displaystyle x=a} to x = b {\displaystyle x=b} is The area of the sector formed by an arc and the center of a circle (bounded by the arc and the two radii drawn to its endpoints) is The area A has the same proportion to the circle area as the angle θ to a full circle: We can cancel π on both sides: By multiplying both sides by r , we get
93-433: A functional is a certain type of function . The exact definition of the term varies depending on the subfield (and sometimes even the author). This article is mainly concerned with the second concept, which arose in the early 18th century as part of the calculus of variations . The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name linear form . The third concept
124-402: A circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc . The length (more precisely, arc length ) of an arc of a circle with radius r and subtending an angle θ (measured in radians) with
155-501: A diameter of the sphere, and therefore every great circle is concentric with the sphere and shares the same radius . Any other circle of the sphere is called a small circle , and is the intersection of the sphere with a plane not passing through its center. Small circles are the spherical-geometry analog of circles in Euclidean space. Every circle in Euclidean 3-space is a great circle of exactly one sphere. The disk bounded by
186-439: A function along all great circles of the sphere. Circular arc A circular arc is the arc of a circle between a pair of distinct points . If the two points are not directly opposite each other, one of these arcs, the minor arc , subtends an angle at the center of the circle that is less than π radians (180 degrees ); and the other arc, the major arc , subtends an angle greater than π radians. The arc of
217-392: A functional's value can be computed for small segments of the input curve and then summed to find the total value, the functional is called local. Otherwise it is called non-local. For example: F ( y ) = ∫ x 0 x 1 y ( x ) d x {\displaystyle F(y)=\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x}
248-417: A great circle is called a great disk : it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n -sphere are the intersection of the n -sphere with 2-planes that pass through the origin in the Euclidean space R . Half of a great circle may be called a great semicircle (e.g., as in parts of a meridian in astronomy ). To prove that
279-471: A meridian of the sphere. In a Cartesian coordinate system , this is which is a plane through the origin, i.e., the center of the sphere. Some examples of great circles on the celestial sphere include the celestial horizon , the celestial equator , and the ecliptic . Great circles are also used as rather accurate approximations of geodesics on the Earth 's surface for air or sea navigation (although it
310-415: A special class of functionals. They map a function f {\displaystyle f} into a real number, provided that H {\displaystyle H} is real-valued. Examples include Given an inner product space X , {\displaystyle X,} and a fixed vector x → ∈ X , {\displaystyle {\vec {x}}\in X,}
341-483: Is a functional of the curve given by According to the Euler–Lagrange equation , S [ γ ] {\displaystyle S[\gamma ]} is minimized if and only if where C {\displaystyle C} is a t {\displaystyle t} -independent constant, and From the first equation of these two, it can be obtained that Integrating both sides and considering
SECTION 10
#1732775873768372-612: Is a parameter . Provided that f {\displaystyle f} is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals . Integrals such as f ↦ I [ f ] = ∫ Ω H ( f ( x ) , f ′ ( x ) , … ) μ ( d x ) {\displaystyle f\mapsto I[f]=\int _{\Omega }H(f(x),f'(x),\ldots )\;\mu (\mathrm {d} x)} form
403-438: Is a function, where x 0 {\displaystyle x_{0}} is an argument of a function f . {\displaystyle f.} At the same time, the mapping of a function to the value of the function at a point f ↦ f ( x 0 ) {\displaystyle f\mapsto f(x_{0})} is a functional ; here, x 0 {\displaystyle x_{0}}
434-408: Is detailed in the computer science article on higher-order functions . In the case where the space X {\displaystyle X} is a space of functions, the functional is a "function of a function", and some older authors actually define the term "functional" to mean "function of a function". However, the fact that X {\displaystyle X} is a space of functions
465-430: Is local while F ( y ) = ∫ x 0 x 1 y ( x ) d x ∫ x 0 x 1 ( 1 + [ y ( x ) ] 2 ) d x {\displaystyle F(y)={\frac {\int _{x_{0}}^{x_{1}}y(x)\;\mathrm {d} x}{\int _{x_{0}}^{x_{1}}(1+[y(x)]^{2})\;\mathrm {d} x}}}
496-530: Is non-local. This occurs commonly when integrals occur separately in the numerator and denominator of an equation such as in calculations of center of mass. The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation F = G {\displaystyle F=G} between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, like when it
527-450: Is not a perfect sphere ), as well as on spheroidal celestial bodies . The equator of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the land and water hemispheres . A great circle divides the earth into two hemispheres and if a great circle passes through a point it must pass through its antipodal point . The Funk transform integrates
558-596: Is not mathematically essential, so this older definition is no longer prevalent. The term originates from the calculus of variations , where one searches for a function that minimizes (or maximizes) a given functional. A particularly important application in physics is search for a state of a system that minimizes (or maximizes) the action , or in other words the time integral of the Lagrangian . The mapping x 0 ↦ f ( x 0 ) {\displaystyle x_{0}\mapsto f(x_{0})}
589-483: Is said that an additive map f {\displaystyle f} is one satisfying Cauchy's functional equation : f ( x + y ) = f ( x ) + f ( y ) for all x , y . {\displaystyle f(x+y)=f(x)+f(y)\qquad {\text{ for all }}x,y.} Functional derivatives are used in Lagrangian mechanics . They are derivatives of functionals; that is, they carry information on how
620-420: Is zero is a vector subspace of X , {\displaystyle X,} called the null space or kernel of the functional, or the orthogonal complement of x → , {\displaystyle {\vec {x}},} denoted { x → } ⊥ . {\displaystyle \{{\vec {x}}\}^{\perp }.} For example, taking
651-502: The intersecting chords theorem (also known as power of a point or secant tangent theorem) it is possible to calculate the radius r of a circle given the height H and the width W of an arc: Consider the chord with the same endpoints as the arc. Its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is W , and it is divided by the bisector into two equal halves, each with length W / 2 . The total length of
SECTION 20
#1732775873768682-403: The boundary condition, the real solution of C {\displaystyle C} is zero. Thus, ϕ ′ = 0 {\displaystyle \phi '=0} and θ {\displaystyle \theta } can be any value between 0 and θ 0 {\displaystyle \theta _{0}} , indicating that the curve must lie on
713-407: The circle center — i.e., the central angle — is This is because Substituting in the circumference and, with α being the same angle measured in degrees, since θ = α / 180 π , the arc length equals A practical way to determine the length of an arc in a circle is to plot two lines from the arc's endpoints to the center of the circle, measure the angle where
744-527: The diameter is 2 r , and it is divided into two parts by the first chord. The length of one part is the sagitta of the arc, H , and the other part is the remainder of the diameter, with length 2 r − H . Applying the intersecting chords theorem to these two chords produces whence so The arc, chord, and sagitta derive their names respectively from the Latin words for bow, bowstring, and arrow . Functional (mathematics) In mathematics ,
775-491: The final result: Using the conversion described above, we find that the area of the sector for a central angle measured in degrees is The area of the shape bounded by the arc and the straight line between its two end points is To get the area of the arc segment , we need to subtract the area of the triangle, determined by the circle's center and the two end points of the arc, from the area A {\displaystyle A} . See Circular segment for details. Using
806-935: The inner product with a fixed function g ∈ L 2 ( [ − π , π ] ) {\displaystyle g\in L^{2}([-\pi ,\pi ])} defines a (linear) functional on the Hilbert space L 2 ( [ − π , π ] ) {\displaystyle L^{2}([-\pi ,\pi ])} of square integrable functions on [ − π , π ] : {\displaystyle [-\pi ,\pi ]:} f ↦ ⟨ f , g ⟩ = ∫ [ − π , π ] f ¯ g {\displaystyle f\mapsto \langle f,g\rangle =\int _{[-\pi ,\pi ]}{\bar {f}}g} If
837-536: The map defined by y → ↦ x → ⋅ y → {\displaystyle {\vec {y}}\mapsto {\vec {x}}\cdot {\vec {y}}} is a linear functional on X . {\displaystyle X.} The set of vectors y → {\displaystyle {\vec {y}}} such that x → ⋅ y → {\displaystyle {\vec {x}}\cdot {\vec {y}}}
868-416: The minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all regular paths from a point p {\displaystyle p} to another point q {\displaystyle q} . Introduce spherical coordinates so that p {\displaystyle p} coincides with
899-479: The north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by provided ϕ {\displaystyle \phi } is allowed to take on arbitrary real values. The infinitesimal arc length in these coordinates is So the length of a curve γ {\displaystyle \gamma } from p {\displaystyle p} to q {\displaystyle q}
930-491: The two great-circle arcs between two distinct points on the sphere is called the minor arc , and is the shortest surface-path between them. Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with
961-417: The two lines meet the center, then solve for L by cross-multiplying the statement: For example, if the measure of the angle is 60 degrees and the circumference is 24 inches, then This is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional. The upper half of a circle can be parameterized as Then the arc length from x =