Spivak's approach to conic sections (Michael Spivak - Calculus - p.81.)

Question

Conic Sections

I don't understand when Spivak says (see the picture)

we can make things a lot simpler for ourselves if we rotate everything so that this intersection line points straight out from the plane of the paper, while the first axis is in the usual position that we are familiar with. The plane $P$ is thus viewed “straight on,” ...

Can someone please clarify it? How does he exactly rotate things to get that view?

score 2 · Answer 1 · answered Aug 06 '15 at 15:37

2

The plane $P$ intersects the horizontal plane in a line. In particular, a line within the horizontal plane. Rotate “everything” (that is, space) around the vertical ($z$-) axis so that line rotates to one perpendicular to the $x$-axis. So if you look at the $xz$-plane that line is coming right at you. The intersection of $P$ with the $xz$-plane is the line $L$.

answered Aug 06 '15 at 15:37

Matthew Leingang

26,112

"The plane P intersects the horizontal plane in a line." It intersects the xy-plane? – lcaop828 Aug 06 '15 at 15:49
I mean is the horizantal plane the xy-plane? – lcaop828 Aug 06 '15 at 16:30
I believe so, yes. Spivak isn't using $x$, $y$, or $z$, just talking about three perpendicular axes. – Matthew Leingang Aug 06 '15 at 20:17

Spivak's approach to conic sections (Michael Spivak - Calculus - p.81.)

1 Answers1