Undergrad Spivak's Calculus: clarification on Conic Sections appendix

[Adgorn]

Hello everyone.
This was originally a homework problem but I realized my misunderstanding stems from the explanation given before the problem so here we are. The thread deals with these 3 pages from Spivak's Calculus:
https://ibb.co/kAKyVU
https://ibb.co/jXVSPp
https://ibb.co/kwRdVU
I'm pretty sure this is copyright infringement but quoting all this without being able to show the illustrations would just serve to confuse anyone who reads it, so i'll take down the links after I figured things out.

So here's my problem: In page 81 the author claims that after combining equations (1) and (2) on can conclude that a point $(x,y,z)$ is in the intersection between the plane and the cone if and only if $Mx+B=±C\sqrt{x^2+y^2}$. My first problem comes from this equation; although every point in the intersection adheres to this equation, the opposite is not true. You could find infinite points with $x$ and $y$ coordinates that adhere to this equation but with some random $z$ coordinate that are not in the intersection. The proper equation (or system of 2 equations) to my understanding should be $Mx+B=±C\sqrt{x^2+y^2}=z$.

After developing this equation, Spivak changes the coordinate system to one in the intersecting plane and somehow we end up with the final equation of the 2d intersection. So it seems to me that from equations (1) and (2) to equation (*) Spivak somehow went from the 3d space to the 2d plane and didn't bother telling about it. This just makes me more confused because the equation (*) is still expressed in terms of the original $x$ and $y$ coordinates, but the intersection is not in the $[x,y]$ plane, so I don't get how you would describe it using a 2d coordinate system when it is not in that system.

I'm sorry if this all sounds messy and unclear, since I don't really know where my misunderstanding comes from myself it's hard to express it in writing. I just hope I articulated my problem quickly enough to receive some clarification. Thanks for reading all this.

 

[Mark44]

Hello everyone.
This was originally a homework problem but I realized my misunderstanding stems from the explanation given before the problem so here we are. The thread deals with these 3 pages from Spivak's Calculus:
https://ibb.co/kAKyVU
https://ibb.co/jXVSPp
https://ibb.co/kwRdVU
I'm pretty sure this is copyright infringement but quoting all this without being able to show the illustrations would just serve to confuse anyone who reads it, so i'll take down the links after I figured things out.

So here's my problem: In page 81 the author claims that after combining equations (1) and (2) on can conclude that a point $(x,y,z)$ is in the intersection between the plane and the cone if and only if $Mx+B=±C\sqrt{x^2+y^2}$. My first problem comes from this equation; although every point in the intersection adheres to this equation, the opposite is not true.

I don't understand what you mean by "the opposite is not true."

You could find infinite points with $x$ and $y$ coordinates that adhere to this equation but with some random $z$ coordinate that are not in the intersection. The proper equation (or system of 2 equations) to my understanding should be $Mx+B=±C\sqrt{x^2+y^2}=z$.

The two equations (1) and (2) are given in terms of z. For any point (x, y, z) on the intersection of the cone and plane, the z coordinate must satisfy both equations. If the z coordinate doesn't satisfy both equations, the point isn't anywhere on the intersecting curve.

After developing this equation, Spivak changes the coordinate system to one in the intersecting plane and somehow we end up with the final equation of the 2d intersection. So it seems to me that from equations (1) and (2) to equation (*) Spivak somehow went from the 3d space to the 2d plane and didn't bother telling about it. This just makes me more confused because the equation (*) is still expressed in terms of the original $x$ and $y$ coordinates, but the intersection is not in the $[x,y]$ plane, so I don't get how you would describe it using a 2d coordinate system when it is not in that system.

I'm sorry if this all sounds messy and unclear, since I don't really know where my misunderstanding comes from myself it's hard to express it in writing. I just hope I articulated my problem quickly enough to receive some clarification. Thanks for reading all this.

 

[tnich]

In page 81 the author claims that after combining equations (1) and (2) on can conclude that a point $(x,y,z)$ is in the intersection between the plane and the cone if and only if $Mx+B=±C\sqrt{x^2+y^2}$.

You could view this as either an elliptical cylinder perpendicular to the $x,y$ plane, or as a projection of the intersection on the $x,y$ plane. I think either would work for the purposes of the derivation.

My first problem comes from this equation; although every point in the intersection adheres to this equation, the opposite is not true.

But it is true for points in the intersection. Since the intersection is what you are concerned with, why worry about points that are not in the intersection? You now have a nice 2-D representation of points in the intersection, obtained by eliminating $z$.

After developing this equation, Spivak changes the coordinate system to one in the intersecting plane and somehow we end up with the final equation of the 2d intersection. So it seems to me that from equations (1) and (2) to equation (*) Spivak somehow went from the 3d space to the 2d plane and didn't bother telling about it.

The final step is to express the intersection in coordinates embedded in the plane. It might help to imagine this as projecting the ellipse in the $x,y$ plane onto the plane $Mx+B$.

 

[mathwonk]

i think you are understanding it and i agree it is a little unclearly described. there should be some z's that are omitted, since a single equation in three space does not in general define a curve, but rather a surface. also he is compounding the description of a curve in space with the equation of its projection into a plane.

By the way that appendix does not even seem to appear in the old edition I had and I never missed it.

 

[Adgorn]

I think I got it. What the equation $Mx+B=±C\sqrt{x^2+y^2}$ essentially says is "The points in the intersection have $x$ and $y$ values described by this equation". Since we don't actually care about the $z$ value, but rather the shape of the 2 dimensional intersection, that's all we need. From there we just change the coordinate system to see how that shape looks on the intersecting plane and we got the final equation.

Thanks for the help everyone.