What is the diameter of the ellipse called and what is the diameter of the ellipse?

Ellipse is a kind of conic curve (also called conic cut), and now there are two definitions in high school textbooks: 1: the sum of the distances from two points on the plane is the set of points with a fixed value (the fixed value is greater than the distance between two points) (these two fixed points are also called the focal points of the ellipse, and the distance between the focal points is called the focal length); 2：

The set of points on a plane where the ratio of the distance to the fixed point to the distance to the fixed line is a constant (the fixed point is not on the fixed line, the constant is a positive number less than 1) (the fixed point is the focal point of the ellipse, which is called the alignment of the ellipse). These two definitions are equivalent.

Since the figure obtained by a planar truncated conic (or cylinder) may be an ellipse, it is a type of conic cross-section. As shown in the figure, there is a cylinder, which is truncated to a cross-section, and it is proved below that it is an ellipse (with the first definition above):

As shown in the figure, if you squeeze two hemispheres with equal radius to the radius of the cylinder from the middle of the two ends of the cylinder, and stop when they touch the cross-section, then you will get two common points, which are obviously the tangent points between the cross-section and the sphere. Let the two points be f1 and f2

For any point p on the cross-section, the bus bar q1 and q2 of the cylinder are made through p, and the large circles tangent to the ball and the cylinder are intersected at q1 and q2 respectively

Then pf1=pq1, pf2=pq2, so pf1+pf2=q1q2

From definition 1, it is known that the cross-section is an ellipse with f1 and f2 as the focus.

In the same way, it can be proved that the oblique section of the cone (which does not pass through the bottom surface) is an ellipse.

In the plane Cartesian coordinate system, high school textbooks describe ellipses with equations, and the standard equation for ellipses is: x 2 a 2+y 2 b 2=1

Among them, a>0, b> the larger is the length of the major semi-axis of the ellipse, and the shorter is the length of the short semi-axis (the ellipse has two axes of symmetry, the symmetry axis is truncated by the ellipse, and there are two line segments, which are called the major semi-axis and the minor semi-axis of the ellipse) When a>b, the focus is on the x-axis, the focal length is 2*(a 2-b 2), and the alignment equation is x=a2 c and x=-a2 c

The area of the ellipse is ab. An ellipse can be seen as a stretch of a circle in a certain direction, and its parametric equation is: x=acos, y=bsin

The standard equation for an ellipse x2 a2 + y2 b2 = 1 (a>0, b>0) major axis length = 2a

Minor axis length = 2b

This is the so-called elliptical diameter, is it as complicated as the first floor?

The diameter of the ellipse: the distance between the straight line perpendicular to the x-axis (or y-axis) through the focal point and the two intersections of the ellipse a,b, the value = 2b 2 a.

The line segment that connects any two points on the ellipse is called the ellipse string, the string that passes through the focus is called the focus chord of the ellipse (so the long axis of the ellipse is also the focus chord), and the focus string perpendicular to the major axis is called the path of the ellipse (the focus chord). The segment of the line that connects any point on the ellipse to a focal point (or the length of that segment) is called the focal radius of the ellipse at that point, and any point on the ellipse has two focal radii.

The line segment that connects any two points on the ellipse is called the ellipse string, the string that passes through the focus is called the focus chord of the ellipse (so the long axis of the ellipse is also the focus chord), and the focus string perpendicular to the major axis is called the path of the ellipse (the focus chord). The segment of the line that connects any point on the ellipse to a focal point (or the length of that segment) is called the focal radius of the ellipse at that point, and any point on the ellipse has two focal radii.

The diameter of the ellipse is the length of the line segment obtained by intersecting the ellipse by the straight line with the focal point perpendicular to the major axis, so by substituting x in the elliptic equation to c, we can get y1=b a, y2=-b a, so the length of the diameter is y1-y2=2b a, where b represents the square of b.

The ellipse is the sum of the distances in the plane to the fixed point f1, f2 equal to the constant (greater than f1f2|The trajectories of the moving point p, f1 and f2 are called the two foci of the ellipse. The mathematical expression is: pf1|+|pf2|=2a（2a>|f1f2|）。

An ellipse is a type of conic curve, that is, the section between a cone and a plane. The circumference of an ellipse is equal to the length of a particular sinusoidal curve in a cycle.

Optical properties

An elliptical mirror (a three-dimensional shape formed by rotating the ellipse 180 degrees on the long axis of the ellipse, and making all the inner surfaces of the ellipse into a reflective surface, hollow) can reflect all the light emitted from one focal point to another focal point.

Elliptical lenses (some of which are elliptical) have the effect of concentrating light (also called convex lenses), such as reading glasses, magnifying glasses, and farsighted glasses (these optical properties can be proved by the method of rebuttal).

The diameter of the ellipse is the length of the line segment obtained by intersecting the ellipse through the direct combustion line with the ellipse with the focal point perpendicular to the long axis, so the x in the elliptic equation is replaced by c, and y1 = b a, y2 = -b a, so the length of the diameter is y1-y2 = 2b a, where b represents the square of b.

The ellipse is the sum of the distances from the plane to the fixed points f1 and f2, and the constant is called (greater than |f1f2|The trajectories of the moving point p, f1 and f2 are called the two foci of the ellipse. The mathematical expression is: |pf1|+|pf2|=2a（2a>|f1f2|）。

An ellipse is a type of conic curve, that is, the section between a cone and a plane. The circumference of an ellipse is equal to a particular sinusoidal curve in one period.

The path formula is d 2ep (p = distance from focus to alignment).

Focus on the x-axis: |pf1|=a+ex |pf2|=a-ex(f1, f2 are the left and right focus, respectively).

The radius of the ellipse over the right focal point r=a-ex.

The radius of the left focal point r=a+ex.

Focus on the y-axis: |pf1|=a+ey |pf2|=a-ey(f2,f1 are the upper and lower focus, respectively).

The diameter of the ellipse: the distance between the straight line perpendicular to the x-axis (or y-axis) of the focal point and the two intersections of the ellipse a,b, i.e., |ab|=2*b^2/a。

Geometric properties of ellipses

1. Range: The focus is on the x-axis -a x a, -b y b; The focus is on the y-axis -b x b, -a y a.

2. Symmetry: symmetry on the x-axis, symmetry on the y-axis, symmetry on the center of the origin.

3. Vertices: (a,0)(-a,0)(0,b)(0,-b).

4. Eccentricity range: 05. The smaller the eccentricity, the closer it is to the circle, and the larger the ellipse, the flatter the ellipse.

6. Focus (when the center is the origin) :(c,0),(c,0) or (0,c),(0,-c).

An ellipse is the distance to two fixed points and the trajectory of a fixed point.

An ellipse is an extension of the definition of a circle, and it is a graph of all the points in the plane whose sum of the distances to two points is a fixed value, which is called the focal point, and the distance between the two points is called the focal length.

In mathematics, an ellipse is a curve in a plane around two focal points such that for each point on the curve, the sum of the distances to the two focal points is constant. Therefore, it is a 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 it "elongates") is indicated by its eccentricity, and for an ellipse it can be any number from 0 (the limit case of the circle) to anything that is arbitrarily close but less than 1.

The ellipse is the sum of the distances from the plane to the fixed points f1 and f2 equal to the constant (greater than |f1f2|The trajectories of the moving point p, f1 and f2 are called the two foci of the ellipse. The mathematical expression is: |pf1|+|pf2|=2a（2a>|f1f2|）

Oval: An oblong shape that changes from a circle.