How to prove that a cylindrical oblique section is an ellipse

Because of the principle of perspective, [perspective] is a theoretical term for painting. The word "perspective" is derived from the Latin "perspclre" (to see through). The initial study of perspective is to take the method of looking at the scenery through a transparent plane, and accurately depict the seen scenery on this plane, that is, the perspective of the scene.

Later, the science of using lines to show the spatial position, contour, and projection of objects on a flat canvas according to certain principles is called perspective.

There are three types of perspective: color perspective, vanishing perspective, and line perspective. This is what Leonardo da Vinci summarized, and one of the most commonly used is line perspective.

Perspective plays a large part in painting, and its basic principle is to imagine a glass between the painter and the object being painted, fix the position of the eye (see with one eye), connect the key points of the object with the eye to form a line of sight, and then intersect with the imaginary glass. The position of each point on the glass is the position of the point on the two-dimensional plane of the three-dimensional object you want to draw. This is the application method of Western classical painting perspective.

Such as "The Last Supper".

The application of perspective in Chinese painting:

1) Multiple viewpoints.

Chinese painting is good at expressing rich plots, while Western painting focuses on a single point of view (similar to photography).The rich plot of Chinese painting cannot be completed with a single point of view. Therefore, Chinese painting uses multiple viewpoints (similar to splitting and reassembling multiple shots of a camera).

Such as "Qingming Riverside Map".

2) High vision.

It is performed from a slightly tilted perspective"The distant mountains are high"Mountains are often painted in the distance, and they are connected by clouds and mist. It expresses a mood in which people are taller than mountains. Chinese painting does not use a close-up view to represent the mountains.

2) Far sight.

Chinese painting is demanding"Fighting mountains and trees, inch horses and beans"The object in the painting is required to conform to the normal proportions of the object, so the painter must use a distance of vision to represent it.

Picasso's works break the basic laws of perspective, and express the front and back of an object, both visible and invisible, in a two-dimensional space. To understand Picasso's paintings, one must first abandon perspective.

Today's painters have begun to ignore all the rules and try to break them. But these practices are within a basic philosophical rule--- breaking an old rule and creating a new one.

This question generally does not need to be proven. It's natural that the transverse surface is a circle, and the oblique section is an ellipse. The definition of an ellipse is used to prove an ellipse. It's more troublesome. Or you can think about using symmetry to prove it.

In short, it is not easy to prove.

The plane obliquely crosses the cylinder, and the cross-section is elliptical, as demonstrated by the Dandelin double-sphere method.

Cut diagonally are all ellipses.

It's very ......It's very complicated to give birth to a brother.

There is a method in math textbooks.

Let the elliptic equation be

x^2/a^2+y^2/b^2=1

Both sides have derivatives for x.

2x/a^2+2yy'/b^2=0

y'=-xb^2/(a^2y)

Because the derivative represents the tangent slope.

Theorem 1: If there are five points in the plane, any three of which are not collinear, then there is only one conic curve passing through these five points.

Theorem 1: If there are five straight lines in a plane, and any three of them are not at the same point, then there is only one conic curve tangent to all five straight lines.

Theorem 2: (Paqingze's theorem): three sets of opposite-sided intersection points of a hexagon bordered by a non-degenerate conic curve (ellipse, hyperbola, parabola, circle) are collinear.

1. The first and second definitions of the ellipse.

This is often used in problem solving, especially when combining numbers and shapes, and the efficiency of solving problems will be greatly improved after use.

2. The relationship between the parameters of the ellipse (a, b, c) This is used in almost every question and needs to be kept in mind.

3. The length of the ellipse cut by a straight line is usually the equation between a simultaneous circle and a straight line. We get a quadratic equation that collapses like x or y. Then use the formula l=sqrt(1+k2)|x1-x2|or l=sqrt(1+(1 k) 2) |y1-y2|(k is the slope of a straight line).

4. The tangent equation for elliptic passing (m, n) is mx a + ny b 2=1

First, let's look at the relationship between the two types of ellipses.

Regardless of how the ellipse rotates, the OA length does not change, and understanding this, it is easier to understand this, such as the x-coordinate of point A before it is rotatedx= b sin a, after rotation x= b sin (a+b), we can derive the formula x=z*sin(b)+x*cos(b), the same is true for z-coordinates, z=z*cos(b)-x*sin, to clarify: in order to distinguish between before and after spinning, add something to Zheng coarsex、zis not rotated before.

Usually in CNC turning, we usually take z as an independent variable.

Therefore, according to the above formula, we only need to find the z-coordinates of the start and end points of the ellipse of the processed part, and the start and end coordinates must be in the unrotated elliptic coordinate system.

So let's establish the coordinate system according to the rotation angle, as shown in the figure below.

It can be seen from the figure that the arc of AB is to be processed, wherein the Z coordinate of point A is the starting point, the coordinate of point B is the end point, in the coordinate system XOZ, the z coordinate is the starting point of 9, it is easier to see that the end point of the Z coordinate needs to be calculated, or directly found in the software, as shown in the figure below, the end point of the Z coordinate is.

After understanding the above knowledge, it is easy to program it. First, in the unrotated ellipse, z[9, is used as the independent variable 1 to compile the dependent variable.

x is 3=15*sqrt[1- 1 81], and then x and z are brought into the parametric equations of the rotated ellipse:

x=#*sin(25)+#*cos(25)；

z=#*cos(25)-#*sin(25), and finally use g01 to interpolate.

In particular, it is necessary to consider the offset of the elliptic center, and the ellipse center of the part diagram in this article is (,, I don't know if your peers have understood it?

A represents the ellipse formed by the beveled cut of the cylinder.

Well, let's solve it according to the concept of "cylindrical side"!

There are three types of data that need to be set:

The radius of the cylinder is r;

The angle between the oblique section and the positive section (the section perpendicular to the central axis) is;

The distance from the center of the inclined plane to the normal section is h.

Then according to the symmetry of the crack of the violent, cut and paved along the FG when setting, if A is the coordinate origin and AE is the Y axis, then the circumference is the X axis of the graph. Laugh.

The moving point m on the circumference of the cross-section, the abscissa on the diagram is the ap "branch closed arc" (plus or minus), and the ordinate is the high pm.

Take the central angle (-) of the ap "arc" (directional) as a parameter, then.

x=rθ，y=pm=cd=on=oo'-no'=h-dn*tanα=h-oc*tanα=h-rcosθtanα。

Remove the argument and get.

y=h-(rtanα)cos(x/r)，-r≤x≤πr。

This is available in the textbook of the People's Education Edition.

It is roughly like this: take two balls, one placed on the top of the section m is marked as ball A, and the other is placed below the section as ball B, and both of them are tangent to the section and the cone at the same time. Ball A and B are cut to C and D respectively to the cross-section, and to the cone are cut to the circle E and F. respectively

For any point i on the ellipse, take a conical bus bar l of i, and note that l and circles a and b intersect at points g and h respectively, and with the cross-section at point i. Since the straight line ig and ic are tangent to the sphere a, ig=ic, and in the same way, ih=idTherefore, IC+ID=IG+IH=GH is the fixed value.

You can draw a picture of it yourself and think about it. This is a very classic proof.

Mathematical dandelin cleverly places two spheres inside the cone, thus proving the shape of the cross-section.