WT32: Projective geometry and perspective

[auspicious99]

The parts on drawing lines and conics in perspective were very clear. The last part, on perspectivity, was not so clear. What is perspectivity fundamentally about? Is it about distance between 3 points, co-linearity of 3 points, or about always being able to find A', B' and C' that are a kind of mirror image of A, B and C, or?

[njwildberger]

@auspicious99 It is about finding a point for the eye so that A,A' are collinear with the eye, also B,B' and also C,C'. In other words if you are looking from this point with one eye only, then the three points A,B,C are superimposed on A',B',C'.

[auspicious99]

@njwildberger thanks!

[ic3sk83r]

Wow, this was very well done and very helpful. I have to do a 20 minute presentation in my math class about Projective geometry, and I've been trying to do some research on it but have been very unsuccessful hahah. I feel like I understand things way more now! Thanks for the video! :)

[Anonymystik]

Tell me, if I see a finite part of parabola (non-cyclic curve), then any image of it must be also a non-cyclic curve. But an elipse is a cyclic-curve. So the image of part of parabola should be only a part of elipse (also non-cyclic), is that right?

[njwildberger]

Hi Anonymystik, Your question is unclear to me, since I do not know what you mean by a cyclic-curve, or by a non-cyclic curve. These are not standard notions I believe.

[Anonymystik]

@njwildberger

I am sorry, but my English is quite weak, especially in this precise terminology. I was thinking about picture you drew and I think I finally understood it. Therefore, my question was a mistake.

Keep on that great work!

[Algebrodadio]

@Anonymystik This is true. One way to see this is to imagine drawing the graph of our favorite parabola ... say y = x^2 ... on the "ground" using the rules of perspective in this video. The slope of the parabola increases without bound to the left and right. So as the x variable (left and right position) increases to infinity, the "sides" parabola eventually become parallel. Therefore the two sides meet at point on the horizon, and the parabola "closes up" and looks like an ellipse.

[mellowmelmel]

@Anonymystik Yes, but if you extend the two arms of the parabola to infinity, then it is possible for their projection to meet at a vanishing point, which would complete the ellipse.

[EmptyInfiniti]

Actually, never mind. Starting to understand.

[EmptyInfiniti]

Excuse me, I've watched some of your videos pertaining to "projective geometry" and such, but I'm still unclear how one goes on directly channeling the mathematics of perspective into a drawing. That technique you explained, the one in which spacing between horizontal lines down the sight can easily be symmetrical, was good. It helped me confirm some of what I know. But, quite frankly, I don't understand the 'drawing' as 'math.' I cannot see it. Please help. You seem most approachable.

[toadenk]

wow I learned more from you in this vid than from my geometry in art teacher all semester, and it is not cuz you use a drum stick as a pointer

[jaredds66]

The idea of having the eye placed where point O was in order to understand why l1 and l2 were perspective has helped to understand the idea quite a bit. Before I had the math, and now I have the concept.

Thanks

[RationArtiste]

I had some basic understanding of geometry that I picked up as a kid, but I had never thought of this idea of taking a single quadrilateral and extending it to a set of tiles. I had never imagined that a parabola could look like an elipse either.

Thankyou.