Design (25) desargues theorem Software Design

Design (25) Software Design of desargue Theorem

Desargues theorem and 3D plot

.

(Draw only 3D images related to the desargues configuration)

Introduction

Introduction.

This paper introduces the desargues theorem, which is a beautiful geometric theorem recognized by the Chinese and foreign mathematics circles and is also a mathematical theorem most closely related to 3D plot. But what is the beauty of this? Not everyone actually knows, but foreigners say this, so we also talk about it. What does it have to do with plotting? It is not that easy to understand. I have made it easier. I would like to introduce the problems I understand below, which may be helpful to you.

I started to explain how to convert the graph (the so-called desargues configuration) used to prove the desargues theorem from a two-dimensional plan to a three-dimensional space map, continue to gain an in-depth understanding of the practical meaning of the theorem and the beauty and abundant symmetry of the configuration, and add a lot of knowledge in graphic programming.

Figure 1. ClickDesignMenu. The image in the upper-left corner of the image isDesarguesConfiguration. The upper-right corner shows a construction process of the image. The lower-right corner shows that the 10 lines in the graph constructed by this method are real straight lines (3 points do not constitute an area triangle ). In the red box at the bottom leftDesarguesTheorem and Its Inverse Theorem. The figure involved is the configuration in the upper left corner. It is displayed after you click it to verify that it is listed below:

Proof: Set a straight line14, 25, and 36The common intersection0.We think of this graphic plane as a three-dimensional image in a three-dimensional space.,0Is123And456The two triangles are the vertices of the triangle cone of the cross section,
Line Segment12 and 45On the same side of the triangle cone, the point of intersection after they are extended7It is also contained in the two planes where the two triangles are located. Therefore, it is the public line (that is, the line) that is included in the two planes.789. Likewise, it can be explained that23 and 56,Or31 and 64Intersection Point after Extension8 and 9It is also in the joint intersection of two planes that contain two triangles.789.
Now you can view the chart as a floor plan again.7, 8, and 9Three points are also on the "always-Online" page. ThereforeDesarguesTheorem proof.

[Inverse Theorem] triangle123 and 456Corresponding edge12, 45, 23, 56, 31, and 64The three intersections of are on the same line, and they correspond to the vertex line.14, 25, and 36At the same point. The certification method is similar to the above, slightly.

[Note 0] The above proof is to regard Figure 1 as a spatial 3D image. A 2D plot can be viewed as a 3D space map with conditions, that is, it is actually a projection of a 3D image. If the projection is not a 3D image, it cannot be regarded as a 3D image. In the projective ry of a plane, if the space ry is not used, the desargues theorem must be recognized as the principle.
[NOTE 1] triangle123 and 456MutualView correspondenceTriangle,0For their perspective center,789The pivot axis.Pivot center and pivot axisWhat is the actual meaning?.
[NOTE 2] The desargues theorem and its inverse theorem are actually dual, that is, the premise of the point and line interchange is turned into a conclusion, and the conclusion is turned into a premise. This dual relationship is called self-dual.
 *It is proved that the perspective actually used is the perspective between two spatial images (two triangles not in the same plane). It is a promotion of the Perspective Concept Between Two triangles or other figures on the plane, this perspective is exactly the most common 3D perspective for us. It is proved that the perspective is the perspective between two flat graphs (two triangles in the same plane.
 *We will point out that figure 1 above is an extremely interesting figure. It contains 10 points and 10 lines. We previously thought of 123,456 as the perspective center with point 0 and two mutual perspective triangles with 789 as the perspective axis, but you can think of any of the 10 points as the perspective center, any of the 10 lines is used as the pivot axis. Of course, the center and axis of the perspective are changed. The two basic points of mutual perspective are no longer original, but must be changed accordingly. As an exercise, you can find out the two pivoting triangles and pivoting axes corresponding to each vertex.
[Special Case 1] In the above theorem, the zero point as the pivot center can be thought of as infinite, that is, the three straight lines, 36 can be parallel, then the theorem is also true, an image in the middle of the following edge is shown.
[Special Case 2] in the above theorem, the corresponding edges of the two pivoting triangles can be parallel and not intersecting, that is, the two planes of the triangle 123 and the triangle 456 are parallel, the three intersections are all infinite, and they are in the same infinite straight line. (Two infinity edges can form an infinite straight line. Three infinity edges constitute an infinite straight line. There are three infinity edges in total. It is a special case that three intersections are on the same infinite straight line .)
[Special Case 3] combines all the features of the above two special cases. That is to say, the two triangles are all parallel to the same surface. At this time, the vertex lines 123 and 456 are parallel to each other, and the two triangles correspond to the sides 12 and 45, 23 and 56, 31 and 64, that is, the intersection is infinite, that is, the point of intersection is on an infinite straight line, so it is also linear.

Figure 2. Special Cases and promotion

This strong three-dimensional figure is also a special case of the desargues architecture: Two pivoting triangles and three sides have a pair of sides parallel, that is, their intersection points are infinite, therefore, there are only two dots on the pivot axis.

Same as above: it is also a special case. Take the yellow points in the lower left corner as the center of the perspective. The corresponding two perspective triangles have green, purple, purple, yellow, green, and green vertices. The three sides of the two spatial triangles have one edge: purple-purple, green-green,The third pair is also parallel,The intersection of them is infinite, and only two yellow finite points are displayed on the pivot axis.

[NOTE 3] We combine the above special cases with the limitations of the original situation, indicating that the theorem is established not only in the Euclidean space, but also in the projection space,DesarguesTheorem is a theorem of the inner space. In the projection space, the two lines always intersect, regardless of whether the points are finite or infinite. On the contrary, the two points can always be connected into a line, regardless of whether there are infinite points or several infinite points. If both are infinitely far, the connected straight line is an infinitely far line.
[NOTE 4] desargues theorem applies not only to triangles, but also to quadrilateral, Pentagon, or any polygon. The bottom graph on the right shows the four edges. In this case, the pivot center is connected with four lines, the pivot axis has four points, and the entire desargues configuration has 13 points. The proof method is similar to that of the Double Triangle. You may think about it yourself.

Based on the above discussion, we can easily find out what kind of drawing is correct and what kind of drawing is incorrect. As shown in the following examples:

Perspective of real photos: The same wide road converges to an infinite distance, and the horizon through an infinite distance is an infinite line

Correct drawing: parallel lines of equal height converge to the same infinity point, with two Infinity points x1 and X2

An example of an incorrect stereoscopy.

The reason for this figure error is simply that there is no perspective. Without perspective, it is against the desargues theorem. Here we will illustrate this point.

This figure includes the largest mauma area. There are a total of eight cubes, each of which has 12 edges. They are divided into three groups. The three edges drawn here are parallel to each other. But this is impossible. Because they are all cubes of limited size. If we observe them from a finite point of view (perspective Center), there may be at most two groups of edges parallel, and the third group must converge to one point, for example, if the viewpoint is on the top, the four vertical edges converge. If the viewpoint is on the front, a group of edges in the depth direction converge. The three groups of edges do not converge, indicating that the viewpoint (perspective Center) must be infinitely far away. However, if the viewpoint is infinitely far away, the limited things should be infinitely small and cannot be drawn as large.

In actual drawing, it is easier to find whether a large object violates the perspective principle of proximity, size, and size. In this example, the muma district is much clearer than the shunma district.

[Note 5] in addition to a desargues inverse theorem, the desargues theorem also has a space dual (see later ). In order to distinguish, we call the theorem currently investigated as the desargues theorem of the plane.

[NOTE 6] There are more than one desargues theorem. For example, there is also a desargues theorem for the conical curve: any cut-off line of the conical curve is connected to the cut-off points of the four-point sides of the curve and the curve. see Chapter 17 "cremona17. For difference, we call the theorem we have investigated as the desargues theorem about two triangles.

Figure 2. ClickCenterThe menu item can be used to obtain the screen. The picture in the middle is the previous one. The top view has ten points on one line, indicating that the picture is flat.

Figure 3 click3DThe menu item can be used. This is the same as the preceding figure, but it has been 3-dimensional. Different from the vertical projection of 3-dimension to 2-dimension, it is infinite to change from 2-dimension to 3-dimension. This process is illustrated in the lower left corner. The top view is in the middle. The ten vertices in the figure are not in a straight line, indicating that they have different zcoordinates, and the front view is indeed three-dimensional.

Figure 4 clickShapeMenu items can be in the shape, where points and lines are replaced by bulb and rods, 3D is obvious.

Figure 5Color DisplayColor charts are available for menu items. The text in this figure shows that the painter algorithm (based on the depth of the object to determine the order of the object) is unreliable and uses the segmentation method to solve this problem.

Figure 6FillThe menu items enable the two corresponding triangles to be colored. The right side is the ten-color palette used for drawing the rod.

Figure 7 below is a variety of tools for rotating, translating, or enlarging a graph. For example, clickZoom in
Click the button to obtain the enlarged color image.

Figure 8 clickPerspectiveThe menu shows the above eight figures. For more information about the perspective, see the explanations at the bottom. In the above 8 figures, the above 4 figures are not pivoting, And the next 4 figures are pivoting. 3D image perspective, as long as the rotation around X and Y axis plus an external box can be seen clearly.

From these figures, especially from figure 6, figure 7, or figure 8, you can see that the desargues configuration is a non-flat graph in 3D space. It is somewhat in contact with the six sides of the outer frame.

By now, we have drawn a perspective of a 3D desargues configuration. The following describes the abundant symmetry of the desargues configuration.

Figure 8a clickViewpointMenu is available. This figure shows that each point in the ten points can be treated as a viewpoint (pivot center), with a pair of perspective triangles and a corresponding pivot axis. Here we use the dark blue points and dark blue lines to represent the pivot center and pivot axis, respectively, and draw the two triangles corresponding to the perspective with orange and sky blue. (Note: Be careful !)

Figure 1 is the name of ten vertices unchanged, while Figure 2 replaces the vertex names of two of them. In either case, the symmetry of the configuration is illustrated.

Figure 8a is in the same pivot center0Under, other vertices can have 12 replacements, so that the theorem is still true. The same is true when other points are in the pivot center. This figure further illustrates the abundant symmetry (symmetric replacement) of the desargues configuration ).

Figure 9 clickDualMenu items are available. This figure shows that the desargues theorem has another dual: three-dimensional space.Points and surfacesDual. Please read the following note (1) carefully ).

As shown in Figure 9, the cross section of any vertex that is dual by space is a plane desargues configuration, which attempts to describe this relationship using a plane image of the entire form. But it does not work, so we can change it to a full window for painting.

The space dual of the desargues configuration is a very complex image, because it is a graph composed of ten faces. These ten faces overlap with each other, and sometimes even cover five times, it is very difficult for you to clearly see ten sides, even if you are using any transparent technology.

Note: This is not a simple polygon. Its relationship between the number of point and line surfaces does not meet the Euler's formula: P + F = e + 2. Here P = 5, E = 10, F = 10, P + F = e + 5.

We have discussed the space dual of the plane desargues configuration above, regardlessDesarguesThe configuration is in a plane or space, and their points and lines are expressed as balls and rods. The following describes how the balls and rods are drawn?

Figure 10 ball and Rod painting can be used in many ways, the simplest of which is also the painter algorithm and the Z buffer method. ClickProgrammingThe menu item will pop up. ClickPainter algorithm ball painting. The following ball always overwrites the first ball.

Figure 11 clickProgrammingIn the drop-down menu of the menu itemProgrammingButtonDeep buffer algorithm ball painting. This algorithm draws the ball to cover the ball far from the near ball by the depth, and the ball will be embedded into each other in the same depth or almost depth.

Figure 12 click againProgrammingButtonDepth buffering methodSpatial desargues configuration,The aboveAndDual-plane desargues.However, it is not shown that there are 10 planes corresponding to the 10 points of the plane configuration. Because these 10 planes overlap with each other, it is not easy to draw them out using a semi-transparent graph, which is expressed only by points and lines. This figure correctly draws the spatial relationship between the five points and the line of the spatial configuration. For example, P3 (244,199, 500) is the deepest, that is, Z = 500 is the largest, and P1 (475,366,100) is the shortest, that is, the minimum z = 100, and the depth of the remaining three points = 300. As a result, the shape of the rod (line) connected to the five balls (points) varies with each other; the depth of the line is clear at a glance. The two pictures on the left are two types of images that are hard to represent without the Z buffer method.

Figure 13 drawn with a translucent ScreenSpatial desargues Configuration(To be painted)

Now, you have finished talking about the things you want to talk about. You should have some knowledge about the desargues theorem, the rich nature of the desargues architecture, and the 3D drawing technology.

Figure 0. Finally, as an exercise, let's take a look at the cover and find out what it represents? Note the red letter above and the comment below.