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Dimensions | Chapter 9

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needs above all proving what 1 claim we have seen that this very graphic projection since circles on sphere I'm not going to the polls 2 circles in the plane and now we're going to prove it even tho this has been known for many centuries desire be on how the man who will present this proves to you or I'm frequently honored since 1 speaks today of the renounce here proving is much more than Shelley is not enough to see in a movie that some curve looks like a cynical To be sure that it is indeed a circle show a mathematical proof must use reasoning to be convincing and has to explain why it is indeed a circle the grade you played during the 3rd century before Christ formulated the rules of the mathematical games in his book The Elements the proof has to rely on facts that themselves have to be approved but 1 has to start with something so some statements have to be accepted without proof these are the axioms therefore mathematics appears as a gigantic construction whose foundations consist of the axioms such that each brick rests on the previous 1 In order to prove the theory about the mystery graphic projection of circles we should in principle start with the axioms Of course we have no time for that now we assume that we already know that their arms of geometry which is studied saying in secondary schools and we will prove this theory but In the past you start with something simple the intersection of the sphere and a plane we see that if a plane Cox's fear and if it is not tangent to the street and then the intersection is a circle we can see it but why is it true How do you prove it In a well let's consider an arbitrary playing college in moved we control the perpendicular from the center see this here to the it's called Peter footage is perpendicular consider 2 points a and B on the intersection of the sphere and the plane and let's look at the 2
triangles CP and and CPB they share a common side CPE had both have a right angle to
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right had since the plane is that particular to CPU but not the hypotenuse is AC and BC had the same length because a and B here on this year and our hands in the same distance from the center but remember Python dresses theorem since to right-angled triangles had 2 sides of the same length the 3 sides must have the same length and we approved the PA and PCB have the same length but is that a and B are on the same circle with centerpiece In the Blue Plains therefore we have proved that all points which are both on the streets and the plane belonged to some circles does that imply that all points on his serve quarter on the sphere and on the plane take
priority to note we still have to prove it this may be a point which is almost here and the plane consider the Circle in the Blue playing with centipede and the goes through that we will prove that this circle is contained in this sphere the next be be some points on the circuit look at the 2 triangles CPI and CPB they share side CPU both a
right-angle triangles since the angle at P has a right angle but the length of P S & P B are equal since they and be on the same circle with centipede again using by progress is there and we conclude the hypotenuse is have the same lengths CIA equals CBD and this means that the point B also relies on the sphere since it is at the same distance from city as but 6 we have proved that when a
plane Cox's fear the cross section is a circle or what now let's look at a diameter APB of our circle and let's place it in the plane of the screen the Blue Plains appears a straight line on the screen and the sphere abuses circle this tensions to the circle of faith and be they intersect in point yes From the cost the lines CDs is against him a check for a few why well because the triangle CAS and CBS soaring you know why because they're both right-angled triangle having a common hypotenuse underside CIA NCB have the same length why well because these are 2 radio costs we see if we had to go right to the end of all the arguments this movie would be the longest in the history of the cinema look we've just proved that any circle drawn on his sleeve can always be thought of as the contact Lucas between a coming of revolution and a tendency if you like series like ice cream in occur well we mustn't forget when ironies the show that historiographic projection carries circles onto circles this 1st prove what mathematicians call member and Bloomberg here is the time to play into this city at some point seen from side the problem and I'm not here attention playing at some other point B also seem to these 2 planes intersect maligned but at present we only see 1 point since this line is perpendicular to the screen the figure that you're looking at is symmetric with respect to the bisecting line of the 2 lines that we see this three-dimensional pictures symmetric with respect to the bisecting the plane of the 2 attention playing a but the true some plane containing segment AB it intersects the line D in .period M solicitors parallel to of course the symmetry of the figure with respect to the bisecting plane shows that a M and B M at the same length borough the Triangle ABC and his our subsidies here it is right that was Alabama will now we can prove our fear of using what we have just learned a considers circle on the stand which does not go to the North Pole we want to show that its projection is a cynical man I look
if instead of projecting onto the tangent playing to the South Pole we projected onto some other parallel playing the famous theorem fatalities would imply that all the projections a seminar and in order to prove all Fareham we may choose the projection plane as we wish Of course as long as it is parallel to the tension playing to the South
Pole well let's play yellow circle in a can remember the Board of of ice cream on account with the takes tests well we're going to project onto the horizontal plane through S the point B projects on tour .period D the American
but look at the figure the
triangles a M B and
D S a similar what will
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ABM is subsidies hence the same is true the triangle PDS so that the has the same length as the and In the end of the 2 when the moon among the analysts the 2nd B S keeps attention to this head its length is therefore constant seemed to be S & S have the same length the moving segment of the US also retains control let's see saying that has a constant length it means precisely that the describes a circle with standard tests so the projection of our yellow circle on the horizontal plane through is contained in a circle we have seen that face there and this implies that the projection onto the tension playing to the South Pole is also contained in this car the number of I want to hear evidence you the U.S. do
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Ebene
Punkt
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Mathematik
sinc-Funktion
Schlussregel
Element <Mathematik>
Physikalische Theorie
Computeranimation
Gradient
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Dimensionsanalyse
Vorlesung/Konferenz
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Ordnung <Mathematik>
Drei
Axiom
Geometrie
Euklidische Ebene
Unterhaltungsmathematik
Rechter Winkel
Dreieck
Computeranimation
Ebene
Länge
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Verband <Mathematik>
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Ebene
Länge
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Annulator
Symmetrische Matrix
Computeranimation
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Dimension 3
Projektive Ebene
Theorem
Projektive Ebene
Ordnung <Mathematik>
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Ebene
Polstelle
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Kreisfläche
Exakter Test
Projektive Ebene
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Dreieck
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Standardabweichung
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Metadaten

Formale Metadaten

Titel Dimensions | Chapter 9
Serientitel Dimensions
Teil 9
Anzahl der Teile 9
Autor Leys, Joe (Images and Animations)
Ghys, Étienne (Scenario and Mathematics)
Alvarez, Aurélien (Image Rendering and Post-production)
Mitwirkende Bullet, Shaun (Speaker)
Delong, Matt (Speaker)
Guaschi, John (Speaker)
Grant, John Lewis (Music)
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/14675
Herausgeber Joe Leys, Étienne Ghys, Aurélien Alvarez
Erscheinungsjahr 2008
Sprache Englisch
Produzent École Normale Supérieure de Lyon (ENS-Lyon)

Technische Metadaten

Dauer 13:57

Inhaltliche Metadaten

Fachgebiet Mathematik

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