Affine Combinations and Barycentric Coords | Algebraic Calculus One | Wild Egg

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In this video we show how affine combinations and barycentric coordinates express mathematically what Archimedes' Law of the Lever captures in terms of the centre of mass of a triangle. We examine both the one dimensional case of a segment, as well as the more general two dimensional case where we use three points as a framework for a coordinate system in the plane. Ceva's theorem plays an important role.
This lecture is part of the Algebraic Calculus One course, which you can find at openlearning.com, and which is setting out to restructure modern mathematics education by creating, essentially for the first time, a solid logical foundation for Calculus.
Video Contents:
0:00 Intro
0:13 Affine cobinations and Centres of mass
3:55 The 1-dim version
6:49 More generally
9:19 The 2-dim version
11:46 Example
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My research papers can be found at my Research Gate page, at www.researchgate.net/profile/Norman_Wildberger
My blog is at njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
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Опубликовано:

13 янв 2018

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Комментарии : 11

@federicofresneda7021 6 лет назад

The power of linear algebra!!

@rolfdoets 4 года назад

Very powerful lecture again!

@steffenkarl7967 6 месяцев назад

As if one is freely wielding the aspect of a circle centered at infinity in the form of a straight line and the aspect of the circle located at infinity in the form of a point. A compass reaching from zero to Infinity of displacement one. Visible magic how fun thank you😊❤

@Jaras8884 5 лет назад

great lecture!thank you

@hassanraza-gh5to 4 года назад

Why the sum of combination of coefficients equal to 1 in affine combinations?

@dnaviap 6 лет назад

Dear professor Wildberger, if the barycentric coordinates are points why are in rounded brackets? and if they are vectors then are necessary two points K and other, Which is the other?

@WildEggmathematicscourses 6 лет назад

Barycentric coordinates are defined with respect to an ordered triple of (non-collinear) points. I think of them as something like vectors, so round brackets seemed most natural to me.

@dnaviap 6 лет назад

That's true, round brackets seems most natural to me too, but I think that when are used in this way is a more general concept

@jehovajah 6 лет назад

Let us suppose we have a point vector .bold a.Let 0a be the beginning of the vector and 1a be the end of the vector . Call the beginning of the vector O and the end of the vector A. We can now define O and A as points the beginning point and the endpoint of the vector bold a.Any pair of points linked in this way can be so defined as beginning and end points of a vector. Thus a point B is either the endpoint of a vector bold b or the beginning point of a vector bold b. Whichever it is we have named the other point associated with the vector called bold b naturally arises.If we name the beginning of all vectors O then we will be able to associate all the endpoints to a common O which we can call the originWe can now add and subtract points through their associated vectors. We can also set up any given vector as an an affine combination of subsectors.

@WildEggmathematicscourses 6 лет назад

It is true that once we specify an origin O as a distinguished point, then we get a canonical identification of points and vectors. But in affine geometry we do not really want to do this, since we want translations to be an important symmetry. The notational significance of O=[0,0] is rather arbitrary. Vectors are relations between points, and so support an arithmetic that the points do not. But still the points have a more limited arithmetic, that of affine combinations.

@jehovajah 6 лет назад

Wild Egg mathematics courses any point is arbitrary . Without going into too much depth a point is a primitive defined by lines either as where lines meet or intersect. Alternatively we can say they are a focus of apprehension . It makes sense to set the line segment as the fundamental primitive and then to define beginning and end of such a segment as a point. The letter O can be used arbitrarily as the beginning or end point of a line segment . The affine geometry arises from composing the line with a scalar as proposed . Translation of a line segment is defined by the parallel notion. Thus if a pair of line segments are not parallel they are translated by a rotation into a parallel orientation . These fundamental primitives are at the base of affine geometry . Along with the proportions of line segments we can establish a full algebra of point vectors . A position vector is a specific arrangement of non parallel point vectors around a common arbitrary beginning or end point of a specified line segment . We call the line segment itself a vector but few understand the meaning of the term . A definite segment that is carried by a radial construction line which construction line is the Trãger or vector or carrier, thus giving the segment orientation , is mistakenly called the vector. On a construction line one can carry a line segment of a specific direction( beginning to end) and of a specific magnitude of extension . A geometry is affine when the segments are related by proportion not standard measures