Intersection of a few surfaces

[brotherbobby]

Summary:: Describe what the intersection of the following surfaces - one on one - would look like? Cone, sphere and plane.

My answers :

(1) A cone intersects a sphere forming a circle.

(2) A sphere intersects a plane forming a circle.

(3) A plane intersects a cone forming (a pair of?) straight lines.

Are these correct?

I wish we could have a 3-D tool to visualise. Any suggestions?

 

[BvU]

I wish we could have a 3-D tool to visualise. Any suggestions?

A pencil

 

[brotherbobby]

We can't visualise 3-D figures on paper. A software would be ideal.

 

[fresh_42]

We can't visualise 3-D figures on paper. A software would be ideal.

Oh, we can! At least since the last 33,000 years:

(Wikipedia)

You may want to look up perspective painting and conic sections.

Btw.: The answer to your question is no. And here is your software.

 

[BvU]

We can't visualise 3-D figures on paper. A software would be ideal

You mean a program to https://all3dp.com/1/best-free-3d-printing-software-3d-printer-program/ ? Still difficult to meaningfully visualize the intersections

 

[brotherbobby]

Something on the computer will do. I am trying to understand how, for a point on the Earth's surface, the (unit) vector $\hat e_r$ points in a direction vertically upwards, $\hat e_{\theta}$ points along the south at that point and $\hat e_{\phi}$ points along the east. I have used a program to do something of the kind which I paste below.

I can get the directions of directions of $\hat e_r$ and $\hat e_{\theta}$. It is $\hat e_{\phi}$ pointing to the east that am strugging with.

 

[BvU]

On a serious note: I am really impressed by nowadays means to visualize things and to support the didactic process with software. But to get anywhere, students should at some point also train their own powers of abstraction and skills of visually rendering.

Perhaps you'll like Sketchup (Google) or SketchBook (AutoCad) ?

 

[fresh_42]

These are very specific planes and by no means generic: leave the equatorial plane, move the tangential plane inwards.

 

[mfb]

(1) A cone intersects a sphere forming a circle.
(3) A plane intersects a cone forming (a pair of?) straight lines.

There are more (and far more interesting) options for these.

 

[robphy]

Something on the computer will do. I am trying to understand how, for a point on the Earth's surface, the (unit) vector $\hat e_r$ points in a direction vertically upwards, $\hat e_{\theta}$ points along the south at that point and $\hat e_{\phi}$ points along the east. I have used a program to do something of the kind which I paste below.

[snip]

I can get the directions of directions of $\hat e_r$ and $\hat e_{\theta}$. It is $\hat e_{\phi}$ pointing to the east that am strugging with.

GeoGebra is great!

This visualization I made might help
https://www.geogebra.org/m/sjzxecxm

To get $\hat e_{\phi}$ , use the cross product of $\hat e_r$ and $\hat e_{\theta}$.

 

[brotherbobby]

Thank you. I can "see" how $\hat e_{\phi}$ is directed to the "east" at the point in question. I was trying to imagine myself sitting at the center of the sphere and looking at the point and struggling.

 

[fresh_42]

Thank you. I can "see" how $\hat e_{\phi}$ is directed to the "east" at the point in question. I was trying to imagine myself sitting at the center of the sphere and looking at the point and struggling.

Plane and sphere is easy, since you can always find a coordinate system such that the plane is parallel to the equatorial plane. It's the other two which weren't correct.

 

[kuruman]

Plane and cone intersections have been investigated since Euclid's days by people who didn't even have pencils as such. These intersections are known as conic sections.

 

[haruspex]

If you care for a real challenge, try a plane intersecting Platonic solids.