The circle and projective homogeneous coordinates | Universal Hyperbolic Geometry 7a | NJ Wildberger

Video Description

Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine space of one higher dimension. Thus the projective line is viewed as the space of lines through the origin in two dimensional space, while the projective plane deals with one dimensional and two dimensional subspaces of a 3 dimensional affine xyz space (called respectively projective points and projective lines). This relates to Renassiance artists attempts to render perspectives correctly; we illustrate by looking at a parabola in a somewhat novel way. The usual two dimensional view of the projective plane emerges by intersecting with the plane z=1 in the ambient x,y,z space. This way the circle is the two dimensional representation of a cone: a view relating back to the ancient Greeks. Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary. CONTENT SUMMARY: Projective Geometry: Affine and projective geometry @00:23 Perspective and points at infinity @08:00 example of affine vs projective view @13:11 One dimensional geometry as starting point @17:48 affine space and vector space @22:30 page change @22:28 One dimensional projective geometry @26:27 page change @31:04 (THANKS to EmptySpaceEnterprise) ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger My blog is at http://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 https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated. Here are the Insights into Mathematics Playlists: https://www.youtube.com/playlist?list=PL55C7C83781CF4316 https://www.youtube.com/playlist?list=PL3C58498718451C47 https://www.youtube.com/playlist?list=PL5A714C94D40392AB https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf https://www.youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw https://www.youtube.com/playlist?list=PL8403C2F0C89B1333 https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua https://www.youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa https://www.youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8 https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF https://www.youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW https://www.youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7 Here are the Wild Egg Maths Playlists (some available only to Members!) https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB https://www.youtube.com/playlist?list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG- https://www.youtube.com/playlist?list=PLzdiPTrEWyz6zpIZ4Y_RK9zyJ9OufqNJv https://www.youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6 https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_ https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HWgaVkhIwpGVKi6fciRxW https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM https://www.youtube.com/playlist?list=PLzdiPTrEWyz7LKbuJAHaXAhRj2ylD0OoX https://www.youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM https://www.youtube.com/playlist?list=PLzdiPTrEWyz6MwUTOHRgC0oIxVtaHGQBd https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS ***********************