WT35: Affine geometry and barycentric coordinates

Affine geometry is the geometry of parallel lines. Using parallelism, we show how to construct a ruled line, how to find the midpoint of a segment, and divide a segment into a given ratio. We connect this to Archimedes law of the lever, and then extend to barycentric coordinates with respect to a triangle.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.

Category:

Science & Technology

Tags:

• Rational
• trigonometry
• geometry
• affine
• parallel
• ruler
• Archimedes
• barycentric
• Wildberger

[euchyhuc]

Hello, you define the first two points as units and then build from them a ruler then you draw a third line with four points. are the four points are equally spaced as the units points? another question is why affine space is not including perpendicularity?

[njwildberger]

@euchyhuc We are not supposing we have any means of measuring separation of points. Affine geometry avoids a notion of perpendicularity, and also of metrical notions---at least at first when we set up the theory.

[steve9340]

Wait a second. How (at 3:02 or so) do you draw "equally spaced peices" (From 3:06 to 3:25) without a ruler ? (Or compass, which you are not allowed). You say, "just as we've done earlier." But I don't see the conncection between this and what you did between 2:00 and 3:00.

[njwildberger]

Hi@steve9340 Looking at the diagram, we already see equally spaced points on the two parallel lines going up the page. The previous minute was devoted to showing how to construct these. So we now do exactly the same on the third line visible at 3:15.

[Julimbob]

Thanks for the help! :)

[CyberneticOrganism01]

@CyberneticOrganism01

Oh I see in affine geometry the coordinate frame can be skewed, that's the result of giving up the metric but retaining parallelism, I guess.

[CyberneticOrganism01]

Thanks a lot =)

How come affine geometry still appears visually the same as Euclidean geometry? I thought giving up the metric would make it look different...

[19916718514]

Norman, you rock !

[gruner819]

holy crap!!!!!! i love you. i havent been able to find a simple xplanation to barycentric coordinates almost ANYWHERE, please keep making making videos even if it is for fun, they are very useful and dispell any myths of math sucking, which if it werent for this video i would have to start thinking

[786HKHM]

really many thanks.