In geometry, focus (focus or foci) (UK:/fo ʊ ka ɪ/, USA:/fo ʊ sa ɪ/), focus refers to the special point where the curve is constructed.For example, one or two focal points can be used to define a cone section. The four types arecircular，ellipse，parabolaandhyperbola。 In addition, two focus points are used to defineCassiniEllipse andDescartes Ellipses, and n-ellipses are defined using more than two focal points.
In geometry, focus (focus or foci) (UK:/fo ʊ ka ɪ/, USA:/fo ʊ sa ɪ/), focus refers to the special point where the curve is constructed.For example, one or two focal points can be used to define a conical section, and its four types are circle, ellipse,parabolaAnd hyperbola.In addition, two focus points are used to defineCassiniEllipse andDescartes Ellipses, and n-ellipses are defined using more than two focal points.
Conical section
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Locate ellipse
Define the cone according to two focal points
An ellipse can be defined as the trajectory of a point whose sum of the distances to two given focal points is a constant.
A circle is a special case of an ellipse in which two focal points coincide with each other.Therefore, a circle can be more simply defined as the trajectory of each point at a fixed distance from a single given focus.You can also define a circle asApolloniusCircle, as far as two different focal points are concerned, serves as a set of points with a fixed proportion of the distance from the two focal points.
Parabola is the limit case of ellipse, in which one focus is the point at infinity.
The hyperbola can be defined as the locus of a point whose absolute value of the difference between the distances to two given focal points is constant.
Define cone according to focus and direction
All cone sections can also be described according to focus and line, which is a given line without focus.A cone is defined as a fixed positive number, calledEccentricitye。 If e is between 0 and 1, the cone is an ellipse;If e=1, the cone is a parabola;If e>1, the conic curve is hyperbolic.If the distance to the focus is fixed, and the line is an infinite line, then the eccentricity is zero, and the cone is a circle.
Define cone according to focus and straight circle
All cone sections can also be described as tracks of points equidistant from a single focus and a single circular matrix.For an ellipse, the focus and center of the center of the circle have finite coordinates, and the radius of the center of the circle is greater than the distance between the center of the circle and the focus;Therefore, the focus is within the inner coil.The second focus of the generated ellipse is at the center of the circle center, and the ellipse is completely inside the circle.
For parabolas, the center of the array moves to infinity (see Projection Geometry).A straight line "circle" becomes a curve with zero curvature, which is indistinguishable from a straight line.The two arms of the parabola become more and more parallel with their extension, and the "infinity" becomes parallel;Using the principle of projection geometry, two parallel lines intersect at infinity, and the parabola becomes a closed curve (elliptic projection).
In order to generate hyperbola, the radius of the selected straight circle is less than the distance between the center of the circle and the focus;Therefore, the focus is outside the straight circle.The double arms of the hyperbola close to the asymptote and the "right hand" arm of one branch of the hyperbola meet the "left hand" arm of the other branch of the hyperbola at the infinity;This is based on the principle that in projective geometry, a single line meets itself at infinity.Thus, the two branches of a hyperbola are two (twisted) halves of an infinitely distant curve.
In projection geometry, all conic curves are the same, because each definition can be another definition.
Astronomical significance
In the gravitational two body problem, the orbits of two bodies are described by two overlapping cone sections, where the focus of one object coincides with one of the focuses of the other object at the center of gravity of the two objects.
Therefore, for example, Pluto's smallest moon has an elliptical orbit, and there is a point in the center of gravity of Pluto's system, which is a space point between two points.And Pluto also moves to the same center of gravity between the bodies with a focus in the ellipse.Pluto's ellipse is completely within Charon's ellipse.
In contrast, the moon and one of the focuses of the earth are located in the ellipse of the moon and the earth's center of gravity, which is located in the earth itself, while the earth (more accurately, its center) moves to the same center of gravity in the earth in an ellipse with one focus.The center of gravity is three quarters from the center of the earth to the ground.
In addition, Pluto system moves an ellipse around its center of gravity with the sun, as does the Earth Moon system (and every other planetary moon system or moonless planet in the solar system).In both cases, the center of gravity is in the solar body.[1]
Cartesian and Cassini ellipses
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Descartes An ellipse is a set of points, and the weighted sum of the distances from two given focal points is a constant.If the weights are equal, a special case of ellipses occurs.
Cassini ellipse is a set of points, where the product of the distances between two given focuses is a constant.
extension
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An n-ellipse is a set of points with the same distance sum as the n focal points.(The case of n=2 is the traditional ellipse)
The concept of focus can be extended to any algebraic curve.Let C be a curve like m, and let I and J represent infinitely far dots.Draw the m tangent to C through each of I and J.There are two sets of m rows that will have mtwoPoint intersection points are different due to singular points in some cases.These intersections are defined as the focus, in other words, if PI and PJ are tangent to C, then point P is the focus.When C is a real curve, only the intersection of the conjugate pair is true, so when the actual focus and mtwo-M Hypothetical focus.When C is a conic, the real focus defined in this way is exactly the focus that can be used for the geometric construction of C.[2]
Confocal curve
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Let P1, P2,..., Pm be the focus of curve C of class m.Let P be the product of tangent equations of these points, and Q be the product of tangent equations of infinite circular points.Then all lines of the common tangent of P=0 and Q=0 are tangent to C.Therefore, through AF+BG theorem, the tangent equation of C has the form HP+KQ=0.Since C has grade m, H must be a constant K but less than or equal to m-2.The case of H=0 can be eliminated as degradation, so the tangent equation of C can be written as P+fQ=0, where f is an arbitrary polynomial m-2
For example, let P1=(1,0), P2=(- 1,0).The tangent equation is X+1=0, X-1=0, so P=Xtwo-1 = 0。The tangent equation of the infinite loop point is X+iY=0, X-iY=0, so Q=Xtwo+ Ytwo。Therefore, the tangent equation of the conic with a given focus is Xtwo-1 + c（Xtwo+ Ytwo）=0 or (1+c) Xtwo+ cYtwo=1, where c is an arbitrary constant.At point coordinates this becomes
