WT32: Projective geometry and perspective

Projective geometry has close links to artist's attempts to render accurately three dimensional objects in two dimensions. We introduce three basic rules for such perspectivities, concerning images of lines, images of parallel lines, and images of conic sections.

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:

• Projective
• geometry
• rational
• trigonometry
• perspective
• conic
• sections
• wildberger
• mathematics

[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'.

[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.

[auspicious99]

@njwildberger thanks!

[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.

[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.

[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]

@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!

[EmptyInfiniti]

Actually, never mind. Starting to understand.