8.7.2 By considering the homogenous polynomial equation x3 - yz = 0, show that the cubic curves y = x5 and y2 = x3 have the same projective completion. More surprisingly, a great simplification of cubic curves also occurs when they are viewed projectively. As mentioned in Section 7.4, Newton (1695) classified cubic curves into 72 types (and missed 6). However, in his Section 29, "On the Genesis of Curves by Shadows," Newton claimed that each cubic curve can be projected onto one of just five types. As mentioned in Section 7.4, this includes the result that y y x. The proof of this is an easy calculation when coordinates are introduced (see Exercise 8.7.2), but one already gets an inkling of it from the perspective view of y cusp is the view of y projecting the view behind one's head through P to the picture plane in front x can be projected onto x. See Figure 8.16. The lower half of the x3 below the horizon; the upper half comes from Copyrighted material
8.7.2 By considering the homogenous polynomial equation x3 - yz = 0, show that the cubic curves y = x5 and y2 = x3 have the same projective completion. More surprisingly, a great simplification of cubic curves also occurs when they are viewed projectively. As mentioned in Section 7.4, Newton (1695) classified cubic curves into 72 types (and missed 6). However, in his Section 29, "On the Genesis of Curves by Shadows," Newton claimed that each cubic curve can be projected onto one of just five types. As mentioned in Section 7.4, this includes the result that y y x. The proof of this is an easy calculation when coordinates are introduced (see Exercise 8.7.2), but one already gets an inkling of it from the perspective view of y cusp is the view of y projecting the view behind one's head through P to the picture plane in front x can be projected onto x. See Figure 8.16. The lower half of the x3 below the horizon; the upper half comes from Copyrighted material
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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need help with 8.7.2 please
Transcribed Image Text:8.7.2 By considering the homogenous polynomial equation x3 - yz = 0, show that the
cubic curves y = x5 and y2 = x3 have the same projective completion.
Transcribed Image Text:More surprisingly, a great simplification of cubic curves also occurs
when they are viewed projectively. As mentioned in Section 7.4, Newton
(1695) classified cubic curves into 72 types (and missed 6). However, in his
Section 29, "On the Genesis of Curves by Shadows," Newton claimed that
each cubic curve can be projected onto one of just five types. As mentioned
in Section 7.4, this includes the result that y
y x. The proof of this is an easy calculation when coordinates are
introduced (see Exercise 8.7.2), but one already gets an inkling of it from
the perspective view of y
cusp is the view of y
projecting the view behind one's head through P to the picture plane in
front
x can be projected onto
x. See Figure 8.16. The lower half of the
x3 below the horizon; the upper half comes from
Copyrighted material
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