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92-454: Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship x y = 1. {\displaystyle xy=1.} In practical applications,

184-437: A {\displaystyle \left|\left|PF_{2}\right|-\left|PF_{1}\right|\right|=2a} can be viewed in a different way (see diagram): If c 2 {\displaystyle c_{2}} is the circle with midpoint F 2 {\displaystyle F_{2}} and radius 2 a {\displaystyle 2a} , then the distance of a point P {\displaystyle P} of

276-496: A {\displaystyle a} to the center. The distance c {\displaystyle c} of the foci to the center is called the focal distance or linear eccentricity . The quotient c a {\displaystyle {\tfrac {c}{a}}} is the eccentricity e {\displaystyle e} . The equation | | P F 2 | − | P F 1 | | = 2

368-394: A ) 2 a 2 − y 2 b 2 = 1 , {\displaystyle {\frac {(x+a)^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1\,,} which is the equation of a hyperbola with center ( − a , 0 ) {\displaystyle (-a,0)} , the x -axis as major axis and the major/minor semi axis

460-457: A 2 x 2 − b 2 {\displaystyle y^{2}={\tfrac {b^{2}}{a^{2}}}x^{2}-b^{2}} satisfy the equation | P F 1 | 2 − c 2 a 2 | P l 1 | 2 = 0   . {\displaystyle |PF_{1}|^{2}-{\frac {c^{2}}{a^{2}}}|Pl_{1}|^{2}=0\ .} The second case

552-423: A 2 − y 2 b 2 = 1 {\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1} one uses the pencils at the vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} . Let P = ( x 0 , y 0 ) {\displaystyle P=(x_{0},y_{0})} be

644-391: A 2 = 1 {\displaystyle {\tfrac {x^{2}-y^{2}}{a^{2}}}=1} (whose semi-axes are equal) has the new equation 2 ξ η a 2 = 1 {\displaystyle {\tfrac {2\xi \eta }{a^{2}}}=1} . Solving for η {\displaystyle \eta } yields η =

736-481: A 2 / 2 ξ   . {\displaystyle \eta ={\tfrac {a^{2}/2}{\xi }}\ .} Thus, in an xy -coordinate system the graph of a function f : x ↦ A x , A > 0 , {\displaystyle f:x\mapsto {\tfrac {A}{x}},\;A>0\;,} with equation y = A x , A > 0 , {\displaystyle y={\frac {A}{x}}\;,A>0\;,}

828-400: A , b {\displaystyle a,b} so that e 2 − 1 = b 2 a 2 ,  and    p = b 2 a {\displaystyle e^{2}-1={\tfrac {b^{2}}{a^{2}}},{\text{ and }}\ p={\tfrac {b^{2}}{a}}} , and then the equation above becomes ( x +

920-468: A , b {\displaystyle a,b} . Because of c ⋅ a 2 c = a 2 {\displaystyle c\cdot {\tfrac {a^{2}}{c}}=a^{2}} point L 1 {\displaystyle L_{1}} of directrix l 1 {\displaystyle l_{1}} (see diagram) and focus F 1 {\displaystyle F_{1}} are inverse with respect to

1012-404: A , b , p , c , e {\displaystyle a,b,p,c,e} remain unchanged. The two lines at distance d = a 2 c {\textstyle d={\frac {a^{2}}{c}}} from the center and parallel to the minor axis are called directrices of the hyperbola (see diagram). For an arbitrary point P {\displaystyle P} of

SECTION 10

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1104-413: A collection or family , especially when its elements are themselves sets. Roster or enumeration notation defines a set by listing its elements between curly brackets , separated by commas: This notation was introduced by Ernst Zermelo in 1908. In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in

1196-599: A parabola and if e < 1 {\displaystyle e<1} an ellipse .) Let F = ( f , 0 ) ,   e > 0 {\displaystyle F=(f,0),\ e>0} and assume ( 0 , 0 ) {\displaystyle (0,0)} is a point on the curve. The directrix l {\displaystyle l} has equation x = − f e {\displaystyle x=-{\tfrac {f}{e}}} . With P = ( x , y ) {\displaystyle P=(x,y)} ,

1288-429: A sequence , a tuple , or a permutation of a set, the ordering of the terms matters). For example, {2, 4, 6} and {4, 6, 4, 2} represent the same set. For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ' ... '. For instance, the set of the first thousand positive integers may be specified in roster notation as An infinite set

1380-502: A circle, line, etc.). For example, the locus of the inequality 2 x + 3 y – 6 < 0 is the portion of the plane that is below the line of equation 2 x + 3 y – 6 = 0 . Set (mathematics) In mathematics , a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. A set may have

1472-488: A curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that

1564-466: A definition is called a semantic description . Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set F can be defined as follows: F = { n ∣ n  is an integer, and  0 ≤ n ≤ 19 } . {\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.} In this notation,

1656-399: A finite number of elements or be an infinite set . There is a unique set with no elements, called the empty set ; a set with a single element is a singleton . Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set). This property is called extensionality . In particular, this implies that there

1748-432: A fixed side [ AB ] with length c . Determine the locus of the third vertex C such that the medians from A and C are orthogonal . Choose an orthonormal coordinate system such that A (− c /2, 0), B ( c /2, 0). C ( x ,  y ) is the variable third vertex. The center of [ BC ] is M ((2 x  +  c )/4,  y /2). The median from C has a slope y / x . The median AM has slope 2 y /(2 x  + 3 c ). The locus of

1840-433: A hyperbola are easily proven using the representation of a hyperbola introduced in this section. Locus (mathematics) In geometry , a locus (plural: loci ) (Latin word for "place", "location") is a set of all points (commonly, a line , a line segment , a curve or a surface ), whose location satisfies or is determined by one or more specified conditions. The set of the points that satisfy some property

1932-402: A hyperbola can arise as the path followed by the shadow of the tip of a sundial 's gnomon , the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle , among others. Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from

SECTION 20

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2024-509: A hyperbola in this section gives a parametric representation of an arbitrary hyperbola, even in space, if one allows f → 0 , f → 1 , f → 2 {\displaystyle {\vec {f}}\!_{0},{\vec {f}}\!_{1},{\vec {f}}\!_{2}} to be vectors in space. Because the unit hyperbola x 2 − y 2 = 1 {\displaystyle x^{2}-y^{2}=1}

2116-461: A hyperbola uses affine transformations : An affine transformation of the Euclidean plane has the form x → → f → 0 + A x → {\displaystyle {\vec {x}}\to {\vec {f}}_{0}+A{\vec {x}}} , where A {\displaystyle A} is a regular matrix (its determinant

2208-472: A point of the hyperbola and A = ( a , y 0 ) , B = ( x 0 , 0 ) {\displaystyle A=(a,y_{0}),B=(x_{0},0)} . The line segment B P ¯ {\displaystyle {\overline {BP}}} is divided into n equally-spaced segments and this division is projected parallel with the diagonal A B {\displaystyle AB} as direction onto

2300-441: A set S , denoted | S | , is the number of members of S . For example, if B = {blue, white, red} , then | B | = 3 . Repeated members in roster notation are not counted, so | {blue, white, red, blue, white} | = 3 , too. More formally, two sets share the same cardinality if there exists a bijection between them. The cardinality of the empty set is zero. The list of elements of some sets

2392-465: A set A to a set B is a rule that assigns to each "input" element of A an "output" that is an element of B ; more formally, a function is a special kind of relation , one that relates each element of A to exactly one element of B . A function is called An injective function is called an injection , a surjective function is called a surjection , and a bijective function is called a bijection or one-to-one correspondence . The cardinality of

2484-2597: A vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter t 0 {\displaystyle t_{0}} of a vertex from the equation p → ′ ( t ) ⋅ ( p → ( t ) − f → 0 ) = ( f → 1 sinh ⁡ t + f → 2 cosh ⁡ t ) ⋅ ( f → 1 cosh ⁡ t + f → 2 sinh ⁡ t ) = 0 {\displaystyle {\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}_{0}\right)=\left({\vec {f}}_{1}\sinh t+{\vec {f}}_{2}\cosh t\right)\cdot \left({\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\right)=0} and hence from coth ⁡ ( 2 t 0 ) = − f → 1 2 + f → 2 2 2 f → 1 ⋅ f → 2   , {\displaystyle \coth(2t_{0})=-{\tfrac {{\vec {f}}_{1}^{\,2}+{\vec {f}}_{2}^{\,2}}{2{\vec {f}}_{1}\cdot {\vec {f}}_{2}}}\ ,} which yields t 0 = 1 4 ln ⁡ ( f → 1 − f → 2 ) 2 ( f → 1 + f → 2 ) 2 . {\displaystyle t_{0}={\tfrac {1}{4}}\ln {\tfrac {\left({\vec {f}}_{1}-{\vec {f}}_{2}\right)^{2}}{\left({\vec {f}}_{1}+{\vec {f}}_{2}\right)^{2}}}.} The formulae cosh 2 ⁡ x + sinh 2 ⁡ x = cosh ⁡ 2 x {\displaystyle \cosh ^{2}x+\sinh ^{2}x=\cosh 2x} , 2 sinh ⁡ x cosh ⁡ x = sinh ⁡ 2 x {\displaystyle 2\sinh x\cosh x=\sinh 2x} , and arcoth ⁡ x = 1 2 ln ⁡ x + 1 x − 1 {\displaystyle \operatorname {arcoth} x={\tfrac {1}{2}}\ln {\tfrac {x+1}{x-1}}} were used. The two vertices of

2576-407: Is perpendicular to k . The angle α {\displaystyle \alpha } between k and m is the parameter. k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines. A locus of points need not be one-dimensional (as

2668-426: Is a rectangular hyperbola entirely in the first and third quadrants with A rotation of the original hyperbola by − 45 ∘ {\displaystyle -45^{\circ }} results in a rectangular hyperbola entirely in the second and fourth quadrants, with the same asymptotes, center, semi-latus rectum, radius of curvature at the vertices, linear eccentricity, and eccentricity as for

2760-403: Is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If A is a subset of B , then the region representing A is completely inside the region representing B . If two sets have no elements in common, the regions do not overlap. A Venn diagram , in contrast, is a graphical representation of n sets in which

2852-402: Is a point of the hyperbola. The tangent vector is p → ′ ( t ) = f → 1 − f → 2 1 t 2 . {\displaystyle {\vec {p}}'(t)={\vec {f}}_{1}-{\vec {f}}_{2}{\tfrac {1}{t^{2}}}.} At a vertex the tangent is perpendicular to

Hyperbola - Misplaced Pages Continue

2944-422: Is a set with an infinite number of elements. If the pattern of its elements is obvious, an infinite set can be given in roster notation, with an ellipsis placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is and the set of all integers is Another way to define a set is to use a rule to determine what the elements are: Such

3036-506: Is a set with exactly one element; such a set may also be called a unit set . Any such set can be written as { x }, where x is the element. The set { x } and the element x mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat. If every element of set A is also in B , then A is described as being a subset of B , or contained in B , written A ⊆ B , or B ⊇ A . The latter notation may be read B contains A , B includes A , or B

3128-435: Is a superset of A . The relationship between sets established by ⊆ is called inclusion or containment . Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B . If A is a subset of B , but A is not equal to B , then A is called a proper subset of B . This can be written A ⊊ B . Likewise, B ⊋ A means B is a proper superset of A , i.e. B contains A , and

3220-830: Is affinely equivalent to the hyperbola y = 1 / x {\displaystyle y=1/x} , an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola y = 1 / x {\displaystyle y=1/x\,} : x → = p → ( t ) = f → 0 + f → 1 t + f → 2 1 t , t ≠ 0 . {\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}+{\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}},\quad t\neq 0\,.} M : f → 0 {\displaystyle M:{\vec {f}}_{0}}

3312-414: Is called the circular directrix (related to focus F 2 {\displaystyle F_{2}} ) of the hyperbola. In order to get the left branch of the hyperbola, one has to use the circular directrix related to F 1 {\displaystyle F_{1}} . This property should not be confused with the definition of a hyperbola with help of a directrix (line) below. If

3404-399: Is endless, or infinite . For example, the set N {\displaystyle \mathbb {N} } of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality . Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that

3496-423: Is equivalent to t 0 = ± 1 {\displaystyle t_{0}=\pm 1} and f → 0 ± ( f → 1 + f → 2 ) {\displaystyle {\vec {f}}_{0}\pm ({\vec {f}}_{1}+{\vec {f}}_{2})} are the vertices of the hyperbola. The following properties of

3588-482: Is in B ". The statement " y is not an element of B " is written as y ∉ B , which can also be read as " y is not in B ". For example, with respect to the sets A = {1, 2, 3, 4} , B = {blue, white, red} , and F = { n | n is an integer, and 0 ≤ n ≤ 19} , The empty set (or null set ) is the unique set that has no members. It is denoted ∅ , ∅ {\displaystyle \emptyset } , { }, ϕ , or ϕ . A singleton set

3680-549: Is mapped onto the hyperbola x → = p → ( t ) = f → 0 ± f → 1 cosh ⁡ t + f → 2 sinh ⁡ t   . {\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}\pm {\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\ .} f → 0 {\displaystyle {\vec {f}}_{0}}

3772-578: Is not Euclidean ). The word "hyperbola" derives from the Greek ὑπερβολή , meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube , but were then called sections of obtuse cones. The term hyperbola is believed to have been coined by Apollonius of Perga ( c.  262  – c.  190 BC ) in his definitive work on

Hyperbola - Misplaced Pages Continue

3864-611: Is not 0) and f → 0 {\displaystyle {\vec {f}}_{0}} is an arbitrary vector. If f → 1 , f → 2 {\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}} are the column vectors of the matrix A {\displaystyle A} , the unit hyperbola ( ± cosh ⁡ ( t ) , sinh ⁡ ( t ) ) , t ∈ R , {\displaystyle (\pm \cosh(t),\sinh(t)),t\in \mathbb {R} ,}

3956-428: Is not equal to A . A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset), while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B . Examples: The empty set is a subset of every set, and every set is a subset of itself: An Euler diagram

4048-413: Is often called the locus of a point satisfying this property. The use of the singular in this formulation is a witness that, until the end of the 19th century, mathematicians did not consider infinite sets . Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be located or may move. Until the beginning of the 20th century, a geometrical shape (for example

4140-400: Is only one empty set. Sets are ubiquitous in modern mathematics. Indeed, set theory , more specifically Zermelo–Fraenkel set theory , has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Mathematical texts commonly denote sets by capital letters in italic , such as A , B , C . A set may also be called

4232-479: Is proven analogously. The inverse statement is also true and can be used to define a hyperbola (in a manner similar to the definition of a parabola): For any point F {\displaystyle F} (focus), any line l {\displaystyle l} (directrix) not through F {\displaystyle F} and any real number e {\displaystyle e} with e > 1 {\displaystyle e>1}

4324-397: Is the center of the hyperbola, the vectors f → 1 , f → 2 {\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}} have the directions of the asymptotes and f → 1 + f → 2 {\displaystyle {\vec {f}}_{1}+{\vec {f}}_{2}}

4416-744: Is the center, f → 0 + f → 1 {\displaystyle {\vec {f}}_{0}+{\vec {f}}_{1}} a point of the hyperbola and f → 2 {\displaystyle {\vec {f}}_{2}} a tangent vector at this point. In general the vectors f → 1 , f → 2 {\displaystyle {\vec {f}}_{1},{\vec {f}}_{2}} are not perpendicular. That means, in general f → 0 ± f → 1 {\displaystyle {\vec {f}}_{0}\pm {\vec {f}}_{1}} are not

4508-442: Is the equation of an ellipse ( e < 1 {\displaystyle e<1} ) or a parabola ( e = 1 {\displaystyle e=1} ) or a hyperbola ( e > 1 {\displaystyle e>1} ). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram). If e > 1 {\displaystyle e>1} , introduce new parameters

4600-545: Is the intersection of the asymptote with its perpendicular through F 1 {\displaystyle F_{1}} (see diagram). The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see diagram: red curve). In order to prove the defining property of a hyperbola (see above) one uses two Dandelin spheres d 1 , d 2 {\displaystyle d_{1},d_{2}} , which are spheres that touch

4692-412: Is the perpendicular to line F 1 F 2 ¯ {\displaystyle {\overline {F_{1}F_{2}}}} through point E 1 {\displaystyle E_{1}} . Alternative construction of E 1 {\displaystyle E_{1}} : Calculation shows, that point E 1 {\displaystyle E_{1}}

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4784-886: Is the solution of the equation ( y − y 1 ) ( x − x 1 ) ( x − x 2 ) ( y − y 2 ) = ( y 3 − y 1 ) ( x 3 − x 1 ) ( x 3 − x 2 ) ( y 3 − y 2 ) {\displaystyle {\frac {({\color {red}y}-y_{1})}{({\color {green}x}-x_{1})}}{\frac {({\color {green}x}-x_{2})}{({\color {red}y}-y_{2})}}={\frac {(y_{3}-y_{1})}{(x_{3}-x_{1})}}{\frac {(x_{3}-x_{2})}{(y_{3}-y_{2})}}} for y {\displaystyle {\color {red}y}} . Another definition of

4876-440: Is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of S with the elements of P ( S ) will leave some elements of P ( S ) unpaired. (There is never a bijection from S onto P ( S ) .) A partition of a set S is a set of nonempty subsets of S , such that every element x in S is in exactly one of these subsets. That is,

4968-474: The Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. To prove a geometric shape is the correct locus for a given set of conditions, one generally divides the proof into two stages: the proof that all the points that satisfy the conditions are on the given shape, and the proof that all the points on the given shape satisfy

5060-394: The circle inversion at circle x 2 + y 2 = a 2 {\displaystyle x^{2}+y^{2}=a^{2}} (in diagram green). Hence point E 1 {\displaystyle E_{1}} can be constructed using the theorem of Thales (not shown in the diagram). The directrix l 1 {\displaystyle l_{1}}

5152-506: The conic sections , the Conics . The names of the other two general conic sections, the ellipse and the parabola , derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than

5244-456: The foci of the hyperbola. The definition of a hyperbola by its foci and its circular directrices (see above) can be used for drawing an arc of it with help of pins, a string and a ruler: The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section : For the generation of points of the hyperbola x 2

5336-842: The n loops divide the plane into 2 zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are A , B , and C , there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist). There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them. Many of these important sets are represented in mathematical texts using bold (e.g. Z {\displaystyle \mathbf {Z} } ) or blackboard bold (e.g. Z {\displaystyle \mathbb {Z} } ) typeface. These include Each of

5428-462: The vertical bar "|" means "such that", and the description can be interpreted as " F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar. Philosophy uses specific terms to classify types of definitions: If B is a set and x is an element of B , this is written in shorthand as x ∈ B , which can also be read as " x belongs to B ", or " x

5520-628: The xy -coordinate system is rotated about the origin by the angle + 45 ∘ {\displaystyle +45^{\circ }} and new coordinates ξ , η {\displaystyle \xi ,\eta } are assigned, then x = ξ + η 2 , y = − ξ + η 2 {\displaystyle x={\tfrac {\xi +\eta }{\sqrt {2}}},\;y={\tfrac {-\xi +\eta }{\sqrt {2}}}} . The rectangular hyperbola x 2 − y 2

5612-399: The above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it. Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, Q + {\displaystyle \mathbf {Q} ^{+}} represents the set of positive rational numbers. A function (or mapping ) from

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5704-638: The asymptotes are the two coordinate axes . Hyperbolas share many of the ellipses' analytical properties such as eccentricity , focus , and directrix . Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry ( Lobachevsky 's celebrated non-Euclidean geometry ), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which

5796-400: The cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane , and indeed of any finite-dimensional Euclidean space . The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and

5888-470: The cardinality of a straight line. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice . (ZFC is the most widely-studied version of axiomatic set theory.) The power set of a set S is the set of all subsets of S . The empty set and S itself are elements of the power set of S , because these are both subsets of S . For example,

5980-479: The case of + 45 ∘ {\displaystyle +45^{\circ }} rotation, with equation y = − A x ,     A > 0 , {\displaystyle y=-{\frac {A}{x}}\;,~~A>0\;,} Shifting the hyperbola with equation y = A x ,   A ≠ 0   , {\displaystyle y={\frac {A}{x}},\ A\neq 0\ ,} so that

6072-463: The center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve y ( x ) = 1 / x {\displaystyle y(x)=1/x}

6164-436: The circle is the set of points that are at a given distance from the center. In contrast to the set-theoretic view, the old formulation avoids considering infinite collections, as avoiding the actual infinite was an important philosophical position of earlier mathematicians. Once set theory became the universal basis over which the whole mathematics is built, the term of locus became rather old-fashioned. Nevertheless,

6256-489: The conditions. Find the locus of a point P that has a given ratio of distances k = d 1 / d 2 to two given points. In this example k = 3, A (−1, 0) and B (0, 2) are chosen as the fixed points. This equation represents a circle with center (1/8, 9/4) and radius 3 8 5 {\displaystyle {\tfrac {3}{8}}{\sqrt {5}}} . It is the circle of Apollonius defined by these values of k , A , and B . A triangle ABC has

6348-441: The cone along circles c 1 {\displaystyle c_{1}} , c 2 {\displaystyle c_{2}} and the intersecting (hyperbola) plane at points F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} . It turns out: F 1 , F 2 {\displaystyle F_{1},F_{2}} are

6440-422: The elements outside the union of A and B are the elements that are outside A and outside B ). The cardinality of A × B is the product of the cardinalities of A and B . This is an elementary fact when A and B are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true. The power set of any set becomes a Boolean ring with symmetric difference as

6532-1226: The following statement is true: The four points are on a hyperbola with equation y = a x − b + c {\displaystyle y={\tfrac {a}{x-b}}+c} if and only if the angles at P 3 {\displaystyle P_{3}} and P 4 {\displaystyle P_{4}} are equal in the sense of the measurement above. That means if ( y 4 − y 1 ) ( x 4 − x 1 ) ( x 4 − x 2 ) ( y 4 − y 2 ) = ( y 3 − y 1 ) ( x 3 − x 1 ) ( x 3 − x 2 ) ( y 3 − y 2 ) {\displaystyle {\frac {(y_{4}-y_{1})}{(x_{4}-x_{1})}}{\frac {(x_{4}-x_{2})}{(y_{4}-y_{2})}}={\frac {(y_{3}-y_{1})}{(x_{3}-x_{1})}}{\frac {(x_{3}-x_{2})}{(y_{3}-y_{2})}}} The proof can be derived by straightforward calculation. If

6624-399: The hyperbola are f → 0 ± ( f → 1 cosh ⁡ t 0 + f → 2 sinh ⁡ t 0 ) . {\displaystyle {\vec {f}}_{0}\pm \left({\vec {f}}_{1}\cosh t_{0}+{\vec {f}}_{2}\sinh t_{0}\right).} Solving

6716-505: The hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity: | P F 1 | | P l 1 | = | P F 2 | | P l 2 | = e = c a . {\displaystyle {\frac {|PF_{1}|}{|Pl_{1}|}}={\frac {|PF_{2}|}{|Pl_{2}|}}=e={\frac {c}{a}}\,.} The proof for

6808-580: The line segment A P ¯ {\displaystyle {\overline {AP}}} (see diagram). The parallel projection is part of the projective mapping between the pencils at V 1 {\displaystyle V_{1}} and V 2 {\displaystyle V_{2}} needed. The intersection points of any two related lines S 1 A i {\displaystyle S_{1}A_{i}} and S 2 B i {\displaystyle S_{2}B_{i}} are points of

6900-982: The major axis. Hence p → ′ ( t ) ⋅ ( p → ( t ) − f → 0 ) = ( f → 1 − f → 2 1 t 2 ) ⋅ ( f → 1 t + f → 2 1 t ) = f → 1 2 t − f → 2 2 1 t 3 = 0 {\displaystyle {\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}_{0}\right)=\left({\vec {f}}_{1}-{\vec {f}}_{2}{\tfrac {1}{t^{2}}}\right)\cdot \left({\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}}\right)={\vec {f}}_{1}^{2}t-{\vec {f}}_{2}^{2}{\tfrac {1}{t^{3}}}=0} and

6992-513: The new center is ( c 0 , d 0 ) {\displaystyle (c_{0},d_{0})} , yields the new equation y = A x − c 0 + d 0 , {\displaystyle y={\frac {A}{x-c_{0}}}+d_{0}\;,} and the new asymptotes are x = c 0 {\displaystyle x=c_{0}} and y = d 0 {\displaystyle y=d_{0}} . The shape parameters

7084-590: The pair F 1 , l 1 {\displaystyle F_{1},l_{1}} follows from the fact that | P F 1 | 2 = ( x − c ) 2 + y 2 ,   | P l 1 | 2 = ( x − a 2 c ) 2 {\displaystyle |PF_{1}|^{2}=(x-c)^{2}+y^{2},\ |Pl_{1}|^{2}=\left(x-{\tfrac {a^{2}}{c}}\right)^{2}} and y 2 = b 2

7176-531: The parameter of a vertex is t 0 = ± f → 2 2 f → 1 2 4 . {\displaystyle t_{0}=\pm {\sqrt[{4}]{\frac {{\vec {f}}_{2}^{2}}{{\vec {f}}_{1}^{2}}}}.} | f → 1 | = | f → 2 | {\displaystyle \left|{\vec {f}}\!_{1}\right|=\left|{\vec {f}}\!_{2}\right|}

7268-1195: The parametric representation for cosh ⁡ t , sinh ⁡ t {\displaystyle \cosh t,\sinh t} by Cramer's rule and using cosh 2 ⁡ t − sinh 2 ⁡ t − 1 = 0 {\displaystyle \;\cosh ^{2}t-\sinh ^{2}t-1=0\;} , one gets the implicit representation det ( x → − f → 0 , f → 2 ) 2 − det ( f → 1 , x → − f → 0 ) 2 − det ( f → 1 , f → 2 ) 2 = 0. {\displaystyle \det \left({\vec {x}}\!-\!{\vec {f}}\!_{0},{\vec {f}}\!_{2}\right)^{2}-\det \left({\vec {f}}\!_{1},{\vec {x}}\!-\!{\vec {f}}\!_{0}\right)^{2}-\det \left({\vec {f}}\!_{1},{\vec {f}}\!_{2}\right)^{2}=0.} The definition of

7360-709: The points are on a hyperbola, one can assume the hyperbola's equation is y = a / x {\displaystyle y=a/x} . A consequence of the inscribed angle theorem for hyperbolas is the 3-point-form of a hyperbola's equation  —  The equation of the hyperbola determined by 3 points P i = ( x i , y i ) ,   i = 1 , 2 , 3 ,   x i ≠ x k , y i ≠ y k , i ≠ k {\displaystyle P_{i}=(x_{i},y_{i}),\ i=1,2,3,\ x_{i}\neq x_{k},y_{i}\neq y_{k},i\neq k}

7452-410: The power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} . The power set of a set S is commonly written as P ( S ) or 2 . If S has n elements, then P ( S ) has 2 elements. For example, {1, 2, 3} has three elements, and its power set has 2 = 8 elements, as shown above. If S is infinite (whether countable or uncountable ), then P ( S )

7544-513: The relation | P F | 2 = e 2 | P l | 2 {\displaystyle |PF|^{2}=e^{2}|Pl|^{2}} produces the equations The substitution p = f ( 1 + e ) {\displaystyle p=f(1+e)} yields x 2 ( e 2 − 1 ) + 2 p x − y 2 = 0. {\displaystyle x^{2}(e^{2}-1)+2px-y^{2}=0.} This

7636-393: The right branch to the circle c 2 {\displaystyle c_{2}} equals the distance to the focus F 1 {\displaystyle F_{1}} : | P F 1 | = | P c 2 | . {\displaystyle |PF_{1}|=|Pc_{2}|.} c 2 {\displaystyle c_{2}}

7728-475: The segment or exceed the segment. A hyperbola can be defined geometrically as a set of points ( locus of points ) in the Euclidean plane: The midpoint M {\displaystyle M} of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis . It contains the vertices V 1 , V 2 {\displaystyle V_{1},V_{2}} , which have distance

7820-620: The set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of N {\displaystyle \mathbb {N} } are called countable sets ; these are either finite sets or countably infinite sets (sets of the same cardinality as N {\displaystyle \mathbb {N} } ); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of N {\displaystyle \mathbb {N} } are called uncountable sets . However, it can be shown that

7912-474: The set of points (locus of points), for which the quotient of the distances to the point and to the line is e {\displaystyle e} H = { P | | P F | | P l | = e } {\displaystyle H=\left\{P\,{\Biggr |}\,{\frac {|PF|}{|Pl|}}=e\right\}} is a hyperbola. (The choice e = 1 {\displaystyle e=1} yields

8004-752: The shape parameters a , b , c {\displaystyle a,b,c} uses the inscribed angle theorem for hyperbolas: Analogous to the inscribed angle theorem for circles one gets the Inscribed angle theorem for hyperbolas  —  For four points P i = ( x i , y i ) ,   i = 1 , 2 , 3 , 4 ,   x i ≠ x k , y i ≠ y k , i ≠ k {\displaystyle P_{i}=(x_{i},y_{i}),\ i=1,2,3,4,\ x_{i}\neq x_{k},y_{i}\neq y_{k},i\neq k} (see diagram)

8096-489: The subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S . Suppose that a universal set U (a set containing all elements being discussed) has been fixed, and that A is a subset of U . Given any two sets A and B , Examples: The operations above satisfy many identities. For example, one of De Morgan's laws states that ( A ∪ B )′ = A ′ ∩ B ′ (that is,

8188-603: The uniquely defined hyperbola. Remarks: A hyperbola with equation y = a x − b + c ,   a ≠ 0 {\displaystyle y={\tfrac {a}{x-b}}+c,\ a\neq 0} is uniquely determined by three points ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) {\displaystyle (x_{1},y_{1}),\;(x_{2},y_{2}),\;(x_{3},y_{3})} with different x - and y -coordinates. A simple way to determine

8280-403: The vertex C is a circle with center (−3 c /4, 0) and radius 3 c /4. A locus can also be defined by two associated curves depending on one common parameter . If the parameter varies, the intersection points of the associated curves describe the locus. In the figure, the points K and L are fixed points on a given line m . The line k is a variable line through K . The line l through L

8372-714: The vertices of the hyperbola. But f → 1 ± f → 2 {\displaystyle {\vec {f}}_{1}\pm {\vec {f}}_{2}} point into the directions of the asymptotes. The tangent vector at point p → ( t ) {\displaystyle {\vec {p}}(t)} is p → ′ ( t ) = f → 1 sinh ⁡ t + f → 2 cosh ⁡ t   . {\displaystyle {\vec {p}}'(t)={\vec {f}}_{1}\sinh t+{\vec {f}}_{2}\cosh t\ .} Because at

8464-499: The word is still widely used, mainly for a concise formulation, for example: More recently, techniques such as the theory of schemes , and the use of category theory instead of set theory to give a foundation to mathematics, have returned to notions more like the original definition of a locus as an object in itself rather than as a set of points. Examples from plane geometry include: Other examples of loci appear in various areas of mathematics. For example, in complex dynamics ,

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