Design (26) Pascal Theorem Software Design

Design (26) Pascal Theorem Software Design

Pascal's Theorem

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Verification of more than 400 Special Cases

0. This is the cover of the vopt software, but it will be automatically deleted in just an instant.

0. The screen menu is displayed after the cover disappears. The function of the screen menu is the same as that of the top menu bar, but the function annotation is added, and"....The pop-up menu is displayed.

0 B is the one on the screenC: hexagonal menuA drop-down menu is displayed.

1. The above is an overview of Pascal Theorem and the functions of this software (Pascal Theorem verification software. For more information about the theorem, see section 60-70 in Chapter 4th of Lehmer, or click the menu Theorem | proof of the ry synthesis method,This is a comprehensive proof. You can also click the proof of resolution. Theorem | proof of the analytical projection ry Method, Or see the bottom of this page (Figure 14-2)

2. The above describes the various cone curves involved in Pascal's theorem, including five kinds of degraded and non-degraded cone curves. Six points have been selected for each of the five curves and will be used as the hexagonal vertices .. [Note] selecting points on a conical curve can be arbitrary without affecting the validity of the theorem. I can also enable the reader to select the vertex later, but the self-selected vertex usually degrades the image quality (asymmetry ). 3-0. the preceding figure is displayed from the drop-down menu of 6 edges. The following describes the general concept of polygon and examples of hexagonal or polygon related to Pascal's theorem, as shown in Figure 6 and Figure 8.
[Note] For convenience, see Figure 7. Figure 8 is also called a six-point or hexagonal graph. K-points are counted as K points, regard the K-heavy tangent as K edge .. 3-1. The above details the mysterious Pascal hexagonal or polygon and their naming methods (coding methods), such as 123456,123465, 135246, etc.

3-2. the above is composed of 6 numbers 1... 6 is a 720 arrangement (dictionary order) formed. Each sort can form a Pascal hexagonal structure at the beginning and end, but a large number of them are equivalent.

3-3. The above are 60 kinds of unequal Pascal hexagonal, which are composed of, 123456, 154326, but in different orders: From. It is worth noting that these 60 hexagonal forms are not formed by the first 60 in the last 720 arrangement. They must pass the 1-1 check before they can be obtained. For example, the 720 and 36th are arranged as 134652. They are only the difference between clockwise and counterclockwise. As a loop, they are equivalent. Therefore, only one of them can be selected.

3-4A. The first 30 images in the preceding 60 images are displayed in greater detail.

[Note] Any Polygon is usually named after their vertices. For example, A, B, and C are called ABC triagons (or ABC triangles ). However, the reading of vertices must be performed in sequence. Any of the headers, clockwise or counterclockwise, can be used at will. This Convention gives each n-side polygon a 2n naming convention. For example, the triagon ABC has 2x3 = 6 reading methods. You can call it ABC or cab, BCA ,... and so on; each of the six sides in the figure above has 2x6 = 12 naming conventions. For example, in the first image: 123456, it can be called 123456, but it can also be called 234561,
345612,456 123, 561234,612 345, or read 6 points counter-clockwise, that is, 543216,432 165, 321654,216 543, 165432. All the 12 sequences have the right to represent the hexagonal Figure 1. The so-called difference between polygon with the same vertex is that their vertex order is different. The difference between them is not the order of order or the starting point is different.

3-4B. The last 30 images in the preceding 60 images are displayed in a enlarged manner.

3-5. Examine the 193 different partitioning methods (equivalence classes) of the six words above, and find out the eight classes that can form the Pascal hexagonal, including the degraded hexagonal.

3-6. The above is the Graph Representation of eight non-equivalence classes of the distribution of six points of Pascal hexagonal 3-7. The above are the parameters required to distinguish Pascal hexagonal (for example)

3-8. the relationship between Pascal polygon and complete polygon is discussed above. 4. the preceding figure shows the 60 Pascal hexagonal interfaces that are enclosed in an ellipse. Click any of them to enter the unified verification program (as shown in the following figure) to verify the Pascal Theorem of the image. 5. The above is the 60 Pascal hexagonal built on the hyperbolic curve. Click any of them to enter the unified verification program and verify the Pascal Theorem.

6. The above are 60 Pascal hexagonal links connected to the parabolic curve. Click any of them to enter the uniform verification program and verify the Pascal Theorem.

7. the preceding six points are 60 Pascal hexagonal distributed on two intersecting straight lines. Click any one of them to go to the unified verification program and verify it using Pascal's theorem.

8. the preceding six points are 60 Pascal hexagonal distributed on two parallel lines. Click any one of them to go to the unified verification program and verify it using Pascal's theorem.

9-1. The above are all 12 different pentagons, excluding the coincidence point 9-2. The generation of pentagons is described above. To apply the Pascal Theorem to pentagons, We need to regard one point as the two key points, which are 6 points in total. 2. The two points in the key point are connected, that is, the tangent of the point.

9-3. The above describes 60 kinds of Pascal hexagonal shapes formed by five kinds of conical curves with 12 pentagons. Click any of them to verify the Pascal Theorem.

10-1. The preceding description shows that there are only three different quadrilateral shapes, but nine Pascal hexagonal shapes can be formed after different points are added, and then applied to five cone curves. There are 45 different special cases.

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10.2. 45 different quadrilateral types. By now, we have obtained 5*60 hexagonal, 60 pentagons, and 45 quadrilateral. A total of 405 polygon that are applicable to Pascal's theorem.

11. finally, we examine the triagon. There are only one triagon, but there are three kinds of 6-point coincidence modes, which can be applied to five curve types. Therefore, there are 15 kinds of polygon that are applicable to Pascal Theorem, so we will not elaborate on them.

In this way, we add a total of 420 polygon (but the Pentagon 15 only draw six internally connected to the elliptic and parabolic, so the actual draw is 411 ), they can have or do not have a focus.K-Focus onK points,Tangent as edgeAnd they all become hexagonal: degradation and non-degradation of the Pascal hexagonal, which can be verified by the Pascal Theorem.

The following describes how to verify the Pascal Theorem for over 400 polygon (including six sides, five sides, four sides, and three sides) listed above.

Vp1. the above figure shows the Pascal Theorem unified verification program. It can verify any polygon listed above,

The verification process can be performed step by step from left to right by menu, or automatically step by step.

[Note]In this form (form2), click Figure 4 ~ 11. A polygon automatically pops up after a polygon and cannot be called independently.

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VP2. the above is the result of Pacal theorem verification (that is, figure 4 13th hexagonal). The three groups of vertices are always online, such as l, m, and N. Where the intersection of point M is 23 and 56 is infinitely far, but it cannot be drawn because the two sides are also parallel to ln, that is, the intersection of the infinity and LN, as described in the upper left corner.
[Note] As long as the intersection m is over the ln extension line, infinity is allowed, because this is the theorem of the projection space. It should also be noted that there can be one or two or three vertices at infinity. That is to say, not three finite points, but only two, one, or zero points are allowed .. VP3. the above is the result of verifying the hexagonal number 136425 in Figure 4. The three sets of vertices l, m, and n are also online. VP4. the Pascal Theorem is used to verify the hexagonal number 134625 in Figure 4. l, m, and n are always online, but n is an infinite point in the ml direction.

. Vp5. the Pascal Theorem of the hexagonal number 123456 in Figure 4 is verified. l, m, and n are always online, but all three points are infinite.

Vp6. the Pascal Theorem is used to verify the hexagonal 123564 on the hyperbolic curve in Figure 5. l, m, and n are always online.

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Vp7. the above is the Pascal Theorem for the 123465 hexagonal hyperbola. L, m, n are always online,
M (23-56 intersection) is an infinite point in the ln direction and cannot be drawn.

.. Vp8. the above is the Pascal Theorem for the 125634 hexagonal parabolic objects. L, m, n are always online,
M (23-56 intersection) is an infinite point in the ln direction. Vp9. the above is the Pascal Theorem verification of the 125463 hexagonal on the intersection line. L, m, n are always online. Vp10. the above is the validation of the Pascal Theorem of the hexagonal composition of the 125436 six points on the parallel line. L, m, n are always online .. Vp11a. The Pascal Theorem is used for step-by-step verification on Pentagon 9.3 in Figure 12534. The results of l, m, n (n outside the screen) have been obtained but are not connected.
Because point 1 and point 6 overlap together to form a 2-key, the line between point 1 and point 6 is the tangent of the 2-key.

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Vp11b. the Pentagon 12534 of the preceding pair is automatically verified by Pascal Theorem. after finding the intersection of the edge, M, N, and then connecting the intersection of 12 and 45, L is at (-, 2941). Although it is not infinitely far away, it is far away from the screen.

Interactive verification is not used as an example. The automatic batch verification of Pascal Theorem is introduced below. 12-4.0. The above is the Pascal Theorem automatic demo program-1 (cover ). There are a total of three similar programs, respectively, to automatically verify the 60 hexagonal products connected to the elliptic, hyperbolic, and parabolic products 1-1.

12-. The above is a picture after entering the Pascal Theorem automatic demonstration program (1). the red line is the Pascal line.

12-. The preceding figure shows another image in the Pascal Theorem automatic demonstration program (1). the red line is the Pascal line.

12-4.3. The above is another image that is displayed in the Pascal Theorem automatic demonstration program (1). There are 60 images in a similar way.

12-4.4. The above is an automatic demonstration program using Pascal's theorem (1). It overlays all 60 hexagonal Pascal lines connected to an elliptic to display them together.

12-5.1. The above is a picture seen in the Pascal Theorem automatic demonstration program (II). Similarly, it can automatically verify 60 hexagonal s connected to the hyperbolic curve.

12-5.2. The above is an automatic demonstration program using Pascal's theorem (2) overlays 60 Pascal lines on the hexagonal shape on the hyperbolic state for display.

12-6.1. The above is an automatic demonstration program using Pascal Theorem (3) overlays 60 hexagonal Pascal lines on a parabolic curve for display.

Points, lines, and theorems related to 60 Pascal lines are further discussed below.

13-7. Another independent program. Click 60 Pascal ln on the right. It also draws 60 Pascal lines connected to the basic hexagonal lines in the garden. This figure is called pascalhexagrammum.
Mysticum (Pascal hexagonal Ecstasy) can be generated from any 6 points connected to any taper line. There are many important concepts related to this graph, including: Steiner point, pluker line, Kirkman point, Cayley line, and salmon point. For the relationship between these points and lines, see

(If it is not clear because of the restriction on the outer frame, click it with the mouse to enlarge the original size to for clear display ). the right half of this image, starting from top to bottom and starting from, leads to 15 edges (red and black numbers marked next to and below ). 6, 5, 2Representative: 6Points, each point and the rest 5Point connection, 6* 5= 30 sides, but yes 2Points form one edge, and the number of edges is calculated repeatedly. Therefore, after 6*5, Division is required. 2, 15 is the number of edges. Others have the same meaning ),
45 vertices (edge meet), 60 Pascal lines, then 20 steiner points and 15 pluker lines, or. 60 Kirkman points, 20 Cayley lines, and 15 salmon points. The line of the left half side of the graph and the point of the right half side or the dual of the line. For example, the 60 brianchon points on the left side are the dual Of the 60 Pascal lines on the right side. To find out ConceptThe relationship with each other is not easy. There are still many theorems to understand. See website http://www.paideiaschool.org/teacherpages/steve_sigur/resources/pascal2/index.html

Next we will examineSteinerTheorem and15SteinerPoint, From 60 Pascal lines15SteinerPointIn the middle, 15 edges and 45 edgemeet are required.

13.8a. The above is the software developed by the translator to examine the above relationship between points and lines. Here we will introduce the Steiner Theorem and Its verification. The Steiner theorem says: 60 Pascal lines are divided into 20 groups, and each three points are handed over to the Steiner point. The 20 buttons in the bottom row represent the Steiner point to be found. Click any of the buttons in the left table to find three two-digit codes,
Based on the three codes, we can find three hexagonal patterns from the top table, and then draw the hexagonal and the corresponding Pascal lines in the three pictures on the right, finally, in the big picture in the middle, the three hexagonal and three lines are combined, and it is proved that the three lines are handed over to a point, that is, the Steiner point. 60 Pascal line how to divide 20 groups please refer to the description of the next page, theorem proof please see http://www.cut-the-knot.org/Curriculum/Geometry/PascalLines.shtml#Explanation

13.8b. the above is the partitioning method of 20 three arrays used by the Steiner theorem (detailed steps). Each group corresponds to three Pascal lines and is intersecting with one Steiner point.

13C. A simple software dedicated to Steiner theorem verification. The usage and graphical meaning are shown in the middle column of the upper part.

We will not discuss the remaining points. Next we will discuss the problem of quadratic curve plotting, including using or not using Pascal's theorem.13-0. The above is the classification of the Cone Curve plot method and some examples of specific plotting problems. The following describes two examples.

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13-9. The above is a program that uses two interactive points in the upper left and lower right to create a standard cone curve. Two straight lines (parallel or cross) represent the degraded cone curve.

Graph is an important application of the 13-10.pascal theorem. The above two kinds of cone curves are directly produced by applying Pascal's theorem at any five points (see Section 74 in Lehmer ). These curves are not smooth enough and need to be improved. An improvement method is indirect plotting, that is, using the Gauss elimination method to obtain the cone curve equation based on the five-point coordinate, and then you can draw a standard cone curve like above, draw a very fine curve point by point. In addition to the 5-5 points on the curve, there are two points marked with the letter L, which are under the elliptic curve and the hyperbolic curve respectively, which are the auxiliary points required for drawing, further explanation is not provided.

Next we will discuss the transfer to Pascal's theorem proof.14-2. There are many ways to prove Pascal's theorem. Apart from the projection geometric proof method in the book, it can also be proved by the elementary geometric method or the analytical method. The elementary geometric proof method can be found on the Internet, but the proof process is complicated. We will not introduce it here. The above figure shows the proof of resolution. It can be seen that the process is very simple, but it is not easy to understand. In addition, it is only proved by the circle. Generally, the proof of quadratic curve should also use the fact that the circular and quadratic curve can be used to projectize the corresponding facts.
[NOTE 1] refer to renewal.
[NOTE 2] In Lehmer's book, we will not repeat it here, but we should also add two points: proof of any quadratic curve. Can be mapped by projectionAnd proof of Pascal's theorem for any Pascal hexagonal on any quadratic curve Can be mapped by projectionIn this way, readers will feel steadfast after reading. For the previous proof, refer to Lehmer
For the Chinese translation of the book, refer to Cremona's book for the next proof.

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