WT34: Lines and planes in projective geometry

How to think about both projective points and projective lines via lines and planes in 3D geometry. Also we discuss some basic facts about 3D geometry, relating perpendicularity and quadrances.

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
• mathematics
• geometry
• projective
• points
• lines
• planes
• Wildberger

[momoney26]

Maybe this doesnt relate specifically to this video but Im a bit unclear about what the basic space is, so is the projective plane just represented by a single point O (the origin) and some object in space?? To which we then insert the plane z=1 and use it essentially as our canvas, and transpose the image of the object onto the plane z=1 via the projective points?? (i.e lines through the origin.

[njwildberger]

@momoney26 The fundamental object which we call a projective point is a line in three dimensional space through the origin. Most of the time, it is represented as an honest point in the z=1 plane, namely the point at which it meets that plane.

[MsDirectionable]

Dr. Wildberger, can you please explain why we don't need to square the quadrants of (O,P) (O,X) and (X,P) to verify that OX and OP are perpendicular to one another? Thank you for your videos.

[njwildberger]

Hi MsDirectionable,

Please have a look at the first few videos in this series, where quadrance between points is the square of the usual distance and where Pythagoras' theorem takes the form Q_1+Q-2=Q_3.

[auspicious99]

@MrFunatabi Yes, as I understand it, in projective geometry, at least one of a and b MUST be nonzero. So, as you say "a = 0 and b =/= 0, or a =/=0 and b = 0" and ax+by+cz intersects the plane z=1 in a line.

[MrFunatabi]

For the projective line, a can be 0 if b =/= 0 ? And vice versa ?

If a = b = 0, either z = 0 and ax + by + cz = 0 is identical to the xy plane, or c = 0 and we have the whole xyz space. So in either way it doesn't intersect the XY plane in a line. When a = 0 and b =/= 0, or a =/=0 and b = 0, it seems to intersect the XY plane in a line.

[gaguilardelgado]

Dr. Wildberger thank you a lot. I'm student from Spain that begins with Computer Vision this year and this set of tutorials saved my a lot of pain. Thank you for your easy and progressive introduction to proy. geometry!

[786HKHM]

Awesome!!!