The Projective Plane 8.5 Figures 8.13 and 8. 14 were made by taking Alberti's veil to be the (x,z)-plane in (x, y, z)-space, with the "eye" at (0, -4,4) viewing the (x, y)-plane. 8.4.2 Find the parametric equations of the line from (0,-4,4) to (x,y,0), and 141 hence show that this line meets the veil where 4x' 4y Z = y' +4 y' 4 843 Renaming the coordinates x, z in the veil as X, Y respectively, show that 4X 4Y V' 4 Y 4-Y 8.4.4 Deduce from Exercise 8.4.3 that the points (x', y') on the parabola y = have image on the veil (Y-2) = 1, X2 + 4 and check that this is the ellipse shown in Figure 8.13. 8.5 The Projective Plane The way in which projective geometry allows infinity to be put on the same footing of the horizon in a picture, which is a line like any other. But what, math ematically speaking, is this line we see? We can model the situation math as the finite points of the plane is intuitively clear when one thinks ha the plane z = -1 in th - eve 8.4.2 Find the parametric equations of the line from (0, -4,4) to (x', y', 0) and hence show that this line meets the veil where: 4x 4y z=_ y4 y'4 х — х' у-у' Z 0 0 x'0 y'4 4 х — х' у —у' Z .. (1) = х' y' 4 -4 х — х' у-у' х' y'4 x'(y-' x(y' 4) x'(4 — х(у' + 4) — х'(у' + 4+у-у) %3D0 — х(у' + 4) %3 х"(4 + у) x'(4y) (2) y' 4) у-у' z y'4 -4 z(y'+ 4)4(y-y') 4y'y - (3) Z= y'4 The line (1) meets the veil at y = 0 hence (2) (3) 4x 4y' ;2= y' 4 y'4 0 у-у' y'4 Suppose = k -4 yy'ky'4) y k(y' +4) y z=-4k Also suppose y+4 4(1 a) x ax', z ay',4 = ay' + 4a »y' ; z = 4(1 -a) a (k 1)x'; y (k+1)y' 4k;z = -4k Hence parametric equation is given by: x =
The Projective Plane 8.5 Figures 8.13 and 8. 14 were made by taking Alberti's veil to be the (x,z)-plane in (x, y, z)-space, with the "eye" at (0, -4,4) viewing the (x, y)-plane. 8.4.2 Find the parametric equations of the line from (0,-4,4) to (x,y,0), and 141 hence show that this line meets the veil where 4x' 4y Z = y' +4 y' 4 843 Renaming the coordinates x, z in the veil as X, Y respectively, show that 4X 4Y V' 4 Y 4-Y 8.4.4 Deduce from Exercise 8.4.3 that the points (x', y') on the parabola y = have image on the veil (Y-2) = 1, X2 + 4 and check that this is the ellipse shown in Figure 8.13. 8.5 The Projective Plane The way in which projective geometry allows infinity to be put on the same footing of the horizon in a picture, which is a line like any other. But what, math ematically speaking, is this line we see? We can model the situation math as the finite points of the plane is intuitively clear when one thinks ha the plane z = -1 in th - eve 8.4.2 Find the parametric equations of the line from (0, -4,4) to (x', y', 0) and hence show that this line meets the veil where: 4x 4y z=_ y4 y'4 х — х' у-у' Z 0 0 x'0 y'4 4 х — х' у —у' Z .. (1) = х' y' 4 -4 х — х' у-у' х' y'4 x'(y-' x(y' 4) x'(4 — х(у' + 4) — х'(у' + 4+у-у) %3D0 — х(у' + 4) %3 х"(4 + у) x'(4y) (2) y' 4) у-у' z y'4 -4 z(y'+ 4)4(y-y') 4y'y - (3) Z= y'4 The line (1) meets the veil at y = 0 hence (2) (3) 4x 4y' ;2= y' 4 y'4 0 у-у' y'4 Suppose = k -4 yy'ky'4) y k(y' +4) y z=-4k Also suppose y+4 4(1 a) x ax', z ay',4 = ay' + 4a »y' ; z = 4(1 -a) a (k 1)x'; y (k+1)y' 4k;z = -4k Hence parametric equation is given by: x =
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter12: Conic Sections
Section12.CR: Chapter Review
Problem 6CC
Related questions
Question
100%
I need ass istance with this for History of math 8.4.3
I have attached how I did 8.4.2 for reference
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,