Ellipse is the sum of the distances from the plane to the fixed points F1 and F2constantThe track of moving point P (greater than | F1F2 |), F1 and F2 are called two ellipsesfocus。The mathematical expression is: | PF1 |+| PF2 |=2a (2a>| F1F2 |).[1]
In mathematics, an ellipse is a curve in a plane around two focal points, so that for each point on the curve, the sum of the distances to the two focal points is constant.Therefore, it is the generalization of a circle, which is a special type of ellipse with two focal points at the same position.The shape of an ellipse (how to "stretch") is represented by its eccentricity. For an ellipse, it can be from 0 (the limit case of a circle) to any number close to but less than 1.
An ellipse is a closed cone section: a plane curve intersected by a cone and a plane.The ellipse has many similarities with the other two forms of cone section: parabola and hyperbola, both of which are open and unbounded.The cross section of a cylinder is elliptical unless it is perpendicular to the axis of the cylinder.
An ellipse can also be defined as a group of points, so that the ratio of the distance between each point on the curve and the distance between a given point (called the focus) and the distance between the same point on the curve is a constant.This ratio is called ellipticalEccentricity。
You can also define an ellipse in this way. An ellipse is a collection of points, and the sum of the distances between a point and its two focal points is a fixed number.
Ellipses are common in physics, astronomy and engineering.
Called ellipticalGuide lineThe equation of the fixed line is
(Focus onxOn the shaft), or
(Focus onyOn the shaft).
Other definitions
According to an important property of an ellipse: the point on the ellipse is connected to the two ends of the ellipse's major axis (in fact, as long as it is diameter)SlopeThe product is a fixed value, and the fixed value is
The premise is that the major axis is parallel to the x-axis.If the long axis is parallel to the y-axis, such as the ellipse whose focus is on the y-axis, the slope product can be obtained as - a ²/b ²=1/(e ² - 1)>, and the following can be obtained:
）The slope product of is equal to the constant m (- 1<m<0).
Note: Considering that the product is not constant when the slope does not exist
It cannot be obtained, that is, the definition is only an ellipse with four points removed.
Ellipses can also be regarded as circles that are compressed or stretched in a certain directionproportionThe resulting graph.
equation
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Standard equation
Ellipse definition
In the plane rectangular coordinate system, the ellipse is described by the equation. The "standard" in the standard equation of the ellipse refers to the center at the origin, and the axis of symmetry is the coordinate axis.
There are two standard equations of ellipse, depending on the coordinate axis of focus:
1. When the focus is on the X axis, the standard equation is:
2. When the focus is on the Y axis, the standard equation is:
The sum of distances from any point on the ellipse to F1 and F2 is 2a, and the distance between F1 and F2 is 2c.In the formula, b ²=a ² - c ².B is for writing parameters that can be set easily.
Also: if the center is at the origin, but the focus position is not clear on the X or Y axis, the equation can be set as mx ²+ny ²=1 (m>0, n>0, m ≠ n). That is the unified form of the standard equation.
The area of the ellipse is π ab.The ellipse can be regarded as the stretching of a circle in a certain direction, and its parameter equation is: x=acos θ, y=bsin θ
Point and ellipse
The standard form of the ellipse at (x0, y0) pointtangentThat is: xxzero/a²+yyzero/b²=1。Tangent to ellipseSlopeYes: - b ² xzero/a²yzero, which can be obtained through complex algebraic calculation.[3]
Parametric equation
x=acosθ ， y=bsinθ。
When solving the maximum distance from a point on an ellipse to a fixed point or to a fixed straight line, the problem can be transformed intotrigonometric functionproblem solving
X=a × cos β, y=b × sin β a isthe major axisThe long half b is half of the short axis
polar coordinates
(One focus is onPolar coordinate systemOrigin, the other is in the positive direction of θ=0)
(e is the eccentricity of the ellipse=c/a).
Geometric properties
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Basic properties
1. Scope: Focus on
On shaft
，
；Focus on
On shaft
，
。
2、Symmetry: Symmetry about X axis, Y axis, and origin center.
6. The smaller the eccentricity, the closer to the circle, and the larger the ellipse, the more flat.
7. Focus (when the center is the origin): (- c, 0), (c, 0) or (0, c), (0, - c).
8、
And
(m is a real number) is an ellipse with the same eccentricity.
9. P is a point on the ellipse, a-c ≤ PF1 (or PF2) ≤ a+c.[1]
10. The perimeter of an ellipse is equal to the length of a specific sine curve in a period.
Tangent Normal
Theorem 1:Let F1 and F2 be two ellipses CfocusP is any point on C.If straight line AB cuts ellipse C at point P, and A and B lie on both sides of P on the straight line, ∠ APF1=∠ BPF2.(That is to say, the tangent line of the ellipse at point P is the straight line where the external corner bisector of ∠ F1PF2 is located).
Theorem 2:Let F1 and F2 be the two focal points of ellipse C, and P be any point on C.If straight line AB is the normal of C at point P, AB bisects ∠ F1PF2.
The proofs of the above two theorems can be found in reference materials.
The tangent theorem of ellipse is proved by analytic geometry:
Solution: Let C: ((x ^ 2)/(a ^ 2))+((y ^ 2)/(b ^ 2))=1 ----- Formula 1;
（a^2）-（b^2）=（c^2）；
F1（-c，0）；F2（c，0）；P（xp，yp）
AB：（y-yp）=k（x-xp）=>y=kx+（yp-kxp）；Let m=yp kxp=>AB: y=kx+m ---- Formula 2;
When y is deleted from simultaneous formula 1 and formula 2, ((k ^ 2)+((b ^ 2)/(a ^ 2))) (x ^ 2)+2kmx+((m ^ 2) - (b ^ 2))=0;
Since straight line AB cuts ellipse C at point P, the above formula has only one solution, then:
=>The equation ((xp ^ 2)/(a ^ 2))+((yp ^ 2)/(b ^ 2))=1 is established, ∠ APF1=∠ BPF2 is proved.
optical properties
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Elliptical surface mirror (taking the long axis of the ellipse as the axis, turning the ellipse 180 degrees to form a three-dimensional figure, with all its internal surfaces made into reflective surfaces, hollow) can reflect all the light emitted from one focal point to another focal point;Ellipticallens(Some sections are elliptical) have the function of converging light (also called convex lens), and presbyopia glasses, magnifying glasses and hyperopia glasses are all such lenses (these optical properties can be proved by the method of proof to the contrary).
Related formula
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Area formula
(of which
Is the length of the long half axis and the short half axis of the ellipse, respectively), or
(of which
They are the major axis of the ellipse and the length of the minor axis.
Certificate:
The area of the first quadrant is multiplied by 4 because of the symmetry of the figure.
T is an ellipsecoefficient, you can find the coefficient T value by looking up the table from the value of r/R;R is the short radius of the ellipse;R is the long radius of the ellipse.
Ellipse perimeter theorem: the perimeter of an ellipse is equal to the product of the sum of the short radius and long radius of the ellipse and the ellipse coefficient (including the positive circle).
A simple table of ellipticity coefficient is attached:
Elliptical coefficient table
r / R
coefficient
r / R
coefficient
r / R
coefficient
r / R
coefficient
zero point zero one
three point nine six one four eight three four nine five
zero point two six
three point four one eight nine two zero four three nine
zero point five one
three point two two four eight five six two two five
zero point seven six
three point one five six two one four two one seven
zero point zero two
three point nine two five three three two five zero nine
zero point two seven
three point four zero six nine five six eight five
zero point five two
three point two two zero four one five seven three five
zero point seven seven
three point one five four eight six eight four zero three
zero point zero three
three point eight nine one one seven four two two three
zero point two eight
three point three nine five four five seven six nine eight
zero point five three
three point two one six one five four nine zero three
zero point seven eight
three point one five three six zero one seven seven six
zero point zero four
three point eight five eight seven nine one six four seven
zero point two nine
three point three eight four four zero three eight zero three
zero point five four
three point two one two zero six seven six one six
zero point seven nine
three point one five two four one one nine zero three
zero point zero five
three point eight two eight zero two four three nine nine
zero point three
three point three seven three seven seven six nine seven six
zero point five five
three point two zero eight one four eight
zero point eight
three point one five one two nine six four three two
zero point zero six
three point seven nine eight seven four three six one six
zero point three one
three point three six three five five nine nine five four
zero point five six
three point two zero four three nine zero four one one
zero point eight one
three point one five zero two five three zero eight nine
zero point zero seven
three point seven seven zero eight four one zero five nine
zero point three two
three point three five three seven three six three three five
zero point five seven
three point two zero zero seven eight nine four two two
zero point eight two
three point one four nine two seven nine six seven seven
zero point zero eight
three point seven four four two two three two six five
zero point three three
three point three four four two nine zero five three two
zero point five eight
three point one nine seven three three nine eight one five
zero point eight three
three point one four eight three seven four zero six seven
zero point zero nine
three point seven one eight eight zero eight zero one three
zero point three four
three point three three five two zero seven seven one two
zero point five nine
three point one nine four zero three six five seven one
zero point eight four
three point one four seven five three four two zero four
zero point one
three point six nine four five two one nine eight two
zero point three five
three point three two six four seven three seven five eight
zero point six
three point one nine zero eight seven four eight five eight
zero point eight five
three point one four six seven five eight zero nine seven
zero point one one
three point six seven one two nine nine one two one
zero point three six
three point three one eight zero seven five two one nine
zero point six one
three point one eight seven eight five zero zero two nine
zero point eight six
three point one four six zero four three eight two two
zero point one two
three point six four nine zero seven nine four five five
zero point three seven
three point three zero nine nine nine nine two seven six
zero point six two
three point one eight four nine five seven six zero eight
zero point eight seven
three point one four five three eight nine five one four
zero point one three
three point six two seven eight zero eight one seven seven
zero point three eight
three point three zero two two three three seven zero two
zero point six three
three point one eight two one nine three two eight six
zero point eight eight
three point one four four seven nine three three seven one
zero point one four
three point six zero seven four three four nine four one
zero point three nine
three point two nine four seven six six eight two eight
zero point six four
three point one seven nine five five two nine one one
zero point eight nine
three point one four four two five three six four six
zero point one five
three point five eight seven nine one three two nine nine
zero point four
three point two eight seven five eight seven five one four
zero point six five
three point one seven seven zero three two four eight four
zero point nine
three point one four three seven six eight six four nine
zero point one six
three point five six nine two zero zero two three eight
zero point four one
three point two eight zero six eight five one one five
zero point six six
three point one seven four six two eight one five one
zero point nine one
three point one four three three three six seven four two
zero point one seven
three point five five one two five five seven nine nine
zero point four two
three point two seven four zero four nine four five nine
zero point six seven
three point one seven two three three six one nine five
zero point nine two
three point one four two nine five six three four
zero point one eight
three point five three four zero four two seven six two
zero point four three
three point two six seven six seven zero eight one nine
zero point six eight
three point one seven zero one five three zero three four
zero point nine three
three point one four two six two five nine zero seven
zero point one nine
three point five one seven five two six three six eight
zero point four four
three point two six one five three nine eight eight six
zero point six nine
three point one six eight zero seven five two one four
zero point nine four
three point one four two three four three nine five six
zero point two
three point five zero one six seven four zero nine
zero point four five
three point two five five six four seven seven five four
zero point seven
three point one six six zero nine nine four zero one
zero point nine five
three point one four two one zero nine zero four four
zero point two one
three point four eight six four five five four two nine
zero point four six
three point two four nine nine eight five eight nine three
zero point seven one
three point one six four two two two three seven nine
zero point nine six
three point one four one nine one nine seven seven five
zero point two two
three point four seven one eight four one seven four one
zero point four seven
three point two four four five four six one three two
zero point seven two
three point one six two four four one zero four six
zero point nine seven
three point one four one seven seven four seven nine four
zero point two three
three point four five seven eight zero six zero seven seven
zero point four eight
three point two three nine three two zero six three nine
zero point seven three
three point one six zero seven five two four zero seven
zero point nine eight
three point one four one six seven two seven eight eight
zero point two four
three point four four four three two three zero four nine
zero point four nine
three point two three four three zero one nine zero nine
zero point seven four
three point one five nine one five three five six eight
zero point nine nine
three point one four one six one two four eight six
zero point two five
three point four three one three six eight seven one
zero point five
three point two two nine four eight two seven four
zero point seven five
three point one five seven six four one seven three seven
one
π
Brief Table of Elliptical Coefficient for Engineering Application
r / R
coefficient
r / R
coefficient
r / R
coefficient
r / R
coefficient
one
π
zero point four seven eight seven
three point two four
zero point two zero one one
three point five
zero point zero seven three nine
three point seven six
zero point nine five five five
three point one four two
zero point four five nine nine
three point two five
zero point one nine four six
three point five one
zero point zero seven zero three
three point seven seven
zero point nine one eight eight
three point one four three
zero point four four two two
three point two six
zero point one eight eight four
three point five two
zero point zero six six six
three point seven eight
zero point eight nine five one
three point one four four
zero point four two six three
three point two seven
zero point one eight two four
three point five three
zero point zero six three one
three point seven nine
zero point eight seven six four
three point one four five
zero point four one one one
three point two eight
zero point one seven six four
three point five four
zero point zero five nine five
three point eight
zero point eight six zero seven
three point one four six
zero point three nine six six
three point two nine
zero point one seven zero seven
three point five five
zero point zero five six one
three point eight one
zero point eight four six eight
three point one four seven
zero point three eight two nine
three point three
zero point one six five one
three point five six
zero point zero five two six
three point eight two
zero point eight four three three
three point one four eight
zero point three six nine nine
three point three one
zero point one five nine five
three point five seven
zero point zero four nine three
three point eight three
zero point eight two three one
three point one four nine
zero point three five seven seven
three point three two
zero point one five four one
three point five eight
zero point zero four six one
three point eight four
zero point eight one two six
three point one five
zero point three four five nine
three point three three
zero point one four eight nine
three point five nine
zero point zero four two eight
three point eight five
zero point seven six eight nine
three point one five five
zero point three four one four
three point three four
zero point one four three seven
three point six
zero point zero three nine six
three point eight six
zero point seven three four seven
three point one six
zero point three two three nine
three point three five
zero point one three eight seven
three point six one
zero point zero three six four
three point eight seven
zero point seven zero five eight
three point one six five
zero point three one three six
three point three six
zero point one three three seven
three point six two
zero point zero three three three
three point eight eight
zero point six eight zero six
three point one seven
zero point three zero three six
three point three seven
zero point one two eight nine
three point six three
zero point zero three zero three
three point eight nine
zero point six five eight four
three point one seven five
zero point two nine four one
three point three eight
zero point one two four two
three point six four
zero point zero two seven three
three point nine
zero point six three eight three
three point one eight
zero point two eight four eight
three point three nine
zero point one one nine five
three point six five
zero point zero two four four
three point nine one
zero point six one nine nine
three point one eight five
zero point two seven five nine
three point four
zero point one one four nine
three point six six
zero point zero two one five
three point nine two
zero point six zero two eight
three point one nine
zero point two six seven four
three point four one
zero point one one zero five
three point six seven
zero point zero one eight six
three point nine three
zero point five eight seven one
three point one nine five
zero point two five nine one
three point four two
zero point one zero six two
three point six eight
zero point zero one five eight
three point nine four
zero point five seven two two
three point two
zero point two five one one
three point four three
zero point one zero one nine
three point six nine
zero point zero one three one
three point nine five
zero point five five eight three
three point two zero five
zero point two four three two
three point four four
zero point zero nine seven seven
three point seven
zero point zero one zero three
three point nine six
zero point five four five two
three point two one
zero point two three five seven
three point four five
zero point zero nine three five
three point seven one
zero point zero zero seven seven
three point nine seven
zero point five three two eight
three point two one five
zero point two two eight four
three point four six
zero point zero eight nine five
three point seven two
zero point zero zero five one
three point nine eight
zero point five zero nine seven
three point two two five
zero point two two one two
three point four seven
zero point zero eight five five
three point seven three
zero point zero zero two five
three point nine nine
zero point four nine eight nine
three point two three
zero point two one four three
three point four eight
zero point zero eight one six
three point seven four
zero point zero zero one two
three point nine nine five
zero point four eight eight six
three point two three five
zero point two zero seven six
three point four nine
zero point zero seven seven seven
three point seven five
zero point zero zero zero two
three point nine nine nine
Relationship between ellipse and trigonometric function
The proof that the perimeter of an ellipse is equal to the length of a specific sine curve in a period:
The cylinder with radius r intersects with an inclined plane to obtain an ellipse. The angle between the inclined plane and the horizontal plane is α, and a circle passing through the short diameter of the ellipse is intercepted.Turn a θ angle starting from a certain intersection point of the circle and ellipse.Then the height of the point on the ellipse and the point perpendicular to the circle can get f (c)=r tan α sin (c/r).
r: Cylinder radius;
α: The angle between the ellipse and the horizontal plane;
c: Corresponding arc length (moving from a certain intersection point to a certain direction);
The above is to prove the simple process, then the circumference of the ellipse (x * cos α) ^ 2+y ^ 2=r ^ 2 is equal to the length of the sine curve of f (c)=r tan α sin (c/r) in a period, and a period T=2 π r, which is exactly the circumference of a circle.
Eccentricity
Eccentricity of ellipseIs defined as the ratio of the focal length on the ellipse to the major axis, (range: 0<X<1).
E=c/a (0<e<1), because 2a>2c.The greater the eccentricity, the flatter the ellipse;The smaller the eccentricity, the closer the ellipse is to the circle.
EllipticalFocal length: ellipticalfocusThe distance from its corresponding guide line (such as focus (c, 0) and guide line x=± a ^ 2/c) is a ^ 2/c-c=b ^ 2/c
Eccentricity vs
Relationship of:
。
Focal radius
The focus is on the x-axis: | PF1 |=a+ex | PF2 |=a-ex (F1 and F2 are the left and right focus respectively).
The focus is on the y-axis: | PF1 |=a+ey | PF2 |=a-ey (F2 and F1 are the upper and lower focus respectively).
EllipticalDrift diameter: the distance between the line perpendicular to the x-axis (or y-axis) passing through the focus and the intersection points A and B of the ellipse, that is, | AB |=2 * b ^ 2/a.[1]
Geometric relationship
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Point and ellipse
Point M (x0, y0) ellipse x ^ 2/a ^ 2+y ^ 2/b ^ 2=1;
Point in circle: xzerotwo/atwo+yzerotwo/btwo<1；
Point on circle: xzerotwo/atwo+yzerotwo/btwo=1；
Point outside the circle: xzerotwo/atwo+yzerotwo/btwo>1；
For example, there is acylinder, interceptedsection, the following proves that it is an ellipse (using the first definition above):
Squeeze two hemispheres with the same radius as the cylinder from both ends of the cylinder to the middle. When they touch the section, they stop. Then two common points will be obtained. Obviously, they are the tangent points of the section and the ball.
Set two points as F1 and F2
For any point P on the section, make a cylinder through Pa bus or bus barQ1、Q2，The big circle tangent to the ball and cylinder intersects Q1 and Q2 respectively
Then PF1=PQ1, PF2=PQ2, so PF1+PF2=Q1Q2
From definition 1, it is known that the section is an ellipse with F1 and F2 as the focus
Given the size of major axis and minor axis, the drawing method of two focal length ruler and gauge
In the same way, it can also be proved that the oblique section of the cone (not passing through the bottom) is an ellipse
For example, the eccentricity of known ellipse C: x ^ 2/a ^ 2+y ^ 2/b ^ 2=1 (a>b>0) is √ 6/3, and the distance from one endpoint of the minor axis to the right focus is √ 3
1. Find the equation of ellipse C
2. The straight line l: y=x+1 intersects the ellipse at two points A and B, and P is a point on the ellipse. Find the maximum value of △ PAB area
3. Calculate the area of △ AOB on the basis of ⑵
I Analysis of minor axisEndpointTo the left and rightfocusThe sum of distances is 2a, and the distance from the end point to the left and right focus is equal (the definition of ellipse). It can be seen that a=√ 3, and c/a=√ 6/3. Substituting it, we can get c=√ 2, b=√ (a ^ 2-c ^ 2)=1, and the equation is x ^ 2/3+y ^ 2/1=1,
II Requirementsthe measure of areaObviously, with ab as the bottom edge of the triangle, the simultaneous solution of x ^ 2/3+y ^ 2/1=1, y=x+1 yields x1=0, y1=1, x2=- 1.5, y2=- 0.5. Using the chord length formula, there is √ (1+k ^ 2)) [x2-x1] (the square brackets represent the absolute value), chord length=3 √ 2/2. For the largest area of point p, its distance to the chord should be the largest, assuming that the maximum distance from p to the chord has been found, and the chord is made through pParallel lineIt can be found that the tangent line of the parallel line is the tangent line of the ellipse, so the tangent line and chord are parallel, so the slope and chord slope=, let y=x+m, and use the discriminant equal to 0 to find m=2, - 2. Combined with the graph, m=- 2. x=1.5, y=- 0.5, p (1.5, - 0.5),
Three straight line equation x-y+1=0, use the distance formula from point to straight line to get √ 2/2, area 1/2 * √ 2/2 * 3 √ 2/2=3/4,[2]
Manual drawing
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Hand Drawing Method I
(1) : Draw the long axis AB, the short axis CD, AB and CD are mutually perpendicular and evenly divided at point O.
(2) : Connect AC.
(3) : With O as the center and OA as the radius, make an arc to intersect OC extension line at point E.
(5) : Make the vertical bisector of AF intersect the CD extension line at point G, and intersect AB at point H.
(6) : Intercept H, G for the symmetric points H ', G' of point O (7): H, H 'is the center of the long axis circle, and HA, H' B are the radii for making circles respectively;G. G 'is the center of the short axis circle, and GC and G' D are used as the radii to make circles.
The method of drawing ellipses with a line or thin copper wire, pencil, two thumbnails or pins: first draw the cross lines of the long and short axes, first find two points on the long axis with the dot as the center that are larger than the radius of the short axis, first fix one point with a line tied by a thumbnail or pin, and then do not fix the line of the other point. Use a pen to hold the line to find the four vertices of the long and short axes. This step requires multiple positioning,Fix the two points until they are just able to coincide with the vertex. Use a pen to lead the line and draw an ellipse directly:) It is best to use thin copper wire because the lineelasticLarger drawings may not be accurate.
Hand Drawing Method II
Ellipticalfocal length│ FF '│ (Z) defines that the major axis X (ab) and the minor axis Y (cd) formed by a known ellipse draw an arc with one end of the major axis A as the center of the circle and the minor axis Y as the radius, and the line segment tangent to the arc drawn from the other segment of point B of the major axis is the focal length of the ellipse. The proof formula is 2 √ {(Z/2) ^ 2+(Y/2) ^ 2}+Z=X+Z (in the plane and two fixed points FThe locus of the moving point P whose sum of the distances of F 'is equal to the constant 2a (2a>| FF' |) is called an ellipse), which can evolve into z=√ x ^ 2-y ^ 2 (x>y>0).Points F and F 'at both ends of Z are fixed points.Take the line with toughness and the smaller the expansion coefficient, the better. The length of any group of surrounding line segments AF 'or FB is taken as the fixed trianglePerimeterThe ellipse is formed by taking F, F 'as fixed points and taking the third point on the triangle as the moving point to draw an arc.[4]
Ellipse diagram
Hand Drawing Method III
Loop length
。According to the graphic characteristics of the ellipse, a circular line is used to represent the distance relationship between the moving point and the focus, forming a unified circular line drawing method.Introduction to specific methods:
(1) The drawing tools are pen, pin, ruler and circular line.(Manufacturing of ring wire: take a length (30-50cm) of flexible wire with moderate thickness and small elasticity, a length of 8mm thin wire hollow plastic pipe, the flexible wire runs through the plastic pipe, and the plastic pipe clamps the flexible wire, but can be twitched by force to form a ring wire that can shrink and lengthen).
(2) Make fixed points and moving points of various circles on the drawing plane.
(3) Erect and fix the pins on the fixed points respectively;
(4) Put the matching circular line outside the pin, straighten the circular line from inside to outside, and adjust the length of the circular line so that the pen tip just falls on the moving point;
(5) Move the brush one week to make various round shapes.
The biggest feature of the loop drawing method is that the distance relationship between the moving point of a circle and the focus is linked in a loop way, and is not affected by the number of focus points. Any number of focus points can be accommodated in the loop. To explore the number of three or more focus pointsMultifocal circleProvide effective methods.The circle drawing method belongs to the continuous moving drawing method, which is suitable for circles, ellipses andOvalEtc.
If this method is used to draw an ellipse specifying semi major axis a and semi minor axis b, then
