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Almost all painters can skillfully apply the principle of perspective. Because the perspective principle can help the painter to make a correct and scientific observation of the shape of the object.

From the perspective of painting, the so-called perspective is to see the object through a layer of flat glass standing between the human eye and the object. At this time, we can regard the flat glass as a picture, and the microcosm of the object seen on the flat glass is the image we need for the picture.

When du Fu, a poet of the Tang Dynasty, described the scenery around the thatched cottage in Chengdu, he left a unique sentence through the ages:

Two golden orioles sing amid the willows green; A flock white egrets flies into the blue sky.

My window frames the snow -crowned western mountains scene; My door oft says to eastward-going ships "Goodbye! ".

This poem is actually an exquisite description of du Fu's perspective of the external environment through the doors and windows sitting in the thatched cottage study.

In painting, the point on the picture that is facing the painter's eyes is called the heart point. All the straight lines perpendicular to the picture disappear in the heart. Figure (a) is a plan from top to bottom, with three parallel train tracks and a row of trees on the right side of the track. Figure (b) is a perspective view of the person standing at the "×" point in figure (a). The trees, tracks and telephone poles perpendicular to the picture are all intersected at the center.

During the European Renaissance, the achievements of perspective reflected the brilliance of the history of painting. Many famous painters, including the versatile Leonardo da Vinci, have made outstanding contributions to the study of perspective with their extraordinary skills and talents. Their achievements soon affected geometry and gave birth to a new branch of geometry-projective geometry.

In Leonardo da Vinci's Last Supper, the lines of window frames on both sides of the walls are parallel to each other and perpendicular to the picture. As the viewpoint gets farther and farther away, it becomes shorter and shorter. As shown in the following figure, the so-called projection refers to the light projection cone emitted from the center O, so that the figure Ω on the plane Q obtains a screenshot Ω'on the plane P. Then Ω'is called the projection of Ω with respect to the center O on the plane P.

Projective geometry is the geometry that studies the invariant properties under the above projective transformation.

It was two French mathematicians who laid the foundation for the birth of projective geometry: Gillard de Sager (Girard Desargues,1591-1661) and Brespascal (BrycePascal,1623-1662).

In 1636, de Sager published a book, the General way of representing objects in Perspective. In this book, de Sager first gave the concept of "measuring ruler" of height, width and depth, thus linking the theory of painting with strict science. Incredibly, for this scientific progress, it was attacked from many aspects at that time, which made de Sager resentful! He publicly announced that anyone who could find a mistake in his method would be awarded a hundred Spanish pesetas, and that he would pay 1000 francs who could come up with a better way. This is really a mockery of history!

In 1639, de Sager made a new breakthrough in the study of the intersecting of plane and cone. He discussed that all three kinds of conic can be obtained from a plane truncated cone, so that all three kinds of curves can be regarded as perspective figures of circles, as shown in the figure.

This gives a particularly concise form to the study of conic curves. However, Desag's above-mentioned works were unfortunately lost until 200 years later, one day in 1845, French mathematician Charles accidentally found a transcript of Desag's manuscript at a used bookstall in Paris. so that Desag's buried achievement can be reissued to glory!

The reason why Desag can make a name in history is also due to the following theorem: if the three connections of the corresponding vertices of two spatial triangles are common, then the intersection of the edge lines is collinear, as shown in the figure. This theorem was later named after Desag.

Interestingly, the proposition obtained by changing "point" in de Sager's theorem to "straight line" and "straight line" to "point" is still valid. That is, if the three intersections of the corresponding edge lines of two spatial triangles are collinear, then the connections of their corresponding vertices are at the same point.

In projective geometry, the above phenomena are universal. Generally speaking, the words in a known proposition or composition are translated according to the following "dictionary":

You will get a "dual" proposition.

Two propositions that are dual to each other are either true at the same time or not at the same time. This is the "dual principle" unique to projective geometry.

Another founder of Brespas projective geometry is a recognized "prodigy" in the history of mathematics-Blaise Pascal, a French mathematician. His achievements are full of legends. Pascal's father was also a mathematician. For some reason, he strongly opposed Pascal's study of mathematics and even hid all his math books.

Unexpectedly, all this made the brain-thinking Pascal more yearning for the "mysterious forbidden area" of mathematics, and at a young age, he independently proved an important theorem in plane geometry: the sum of interior angles of triangles is equal to 180 °.

Pascal's mathematical talent made his father burst into tears and changed his old attitude. He not only stopped opposing Pascal's study of mathematics, but also fully supported him and personally led Pascal to attend a seminar chaired by Mason, founder of the French Academy of Sciences. Pascal was 14 years old.

In 1639, Pascal discovered the theorem that made him famous: if A, B, C, D, E, F are any six points on the conic curve, then the three intersecting points formed by AB and DE,BC and EF,CD and FA are collinear! As shown in figure 18.5.

Pascal's theorem is exquisite! It shows that a conic curve can be determined by only five points, and the sixth point can be derived from the collinear condition in the theorem. There are more than 400 corollaries of this theorem, which is worth a masterpiece.

Unexpectedly, Pascal's brilliant achievement aroused the suspicion of some people, including the famous Descartes, who did not believe that it would be the thinking of a 16-year-old child, but that it was the ghostwriting of Pascal's father!

Since then, however, Pascal has made a lot of achievements: he invented the desktop addition and subtraction computer at the age of 19, discovered the famous law of fluid pressure in physics at 23, co-founded the theory of probability with Fermat at 31, and made great achievements in the study of cycloids at 35.

Pascal's series of achievements finally convinced all the skeptics! So far, people all praise the brilliance of the wisdom of this French genius!

Unfortunately, Desag and Pascal, the forerunners of projective geometry, died in 1661 and 1662 respectively. Since then, the study of projective geometry has not been given due attention, and therefore remained silent for a whole century and a half, until the arrival of another French mathematician, Poncelet.

Source: "Mathematics Story Book for Children" by Zhang Yuannan, Zhang Chang, Editor: Zhang Runxin, this article comes from the official account of Wechat: ID:tupydread, author: Zhang Yuannan, Zhang Chang, Editor: Zhang Runxin

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